Hierarchical matrix approximation of large covariance matrices
0 likes128 views
The document discusses the hierarchical matrix (h-matrix) approximation of large non-structured covariance matrices, emphasizing computational efficiency with log-linear cost and storage. It details the methods for calculating matrix operations like inversion and Cholesky decomposition, particularly using Matern covariance functions prevalent in spatial statistics. Additionally, the document presents theoretical insights and accuracy comparisons related to h-matrix approximations and their applications in fields such as kriging and optimal design.
1 of 1
Download to read offline
![Center for Uncertainty
Quantification
Hierarchical matrix approximation of
large covariance matrices
A. Litvinenko1
, M. Genton2
, Ying Sun2
, R. Tempone
1
SRI-UQ Center and 2
Spatio-Temporal Statistics & Data Analysis Group
at KAUST
alexander.litvinenko@kaust.edu.sa
Center for Uncertainty
Quantification
Center for Uncertainty
Quantification
Abstract
We approximate large non-structured covariance ma-
trices in the H-matrix format with a log-linear com-
putational cost and storage O(n log n). We compute
inverse, Cholesky decomposition and determinant in
H-format. As an example we consider the class of
Matern covariance functions, which are very popu-
lar in spatial statistics, geostatistics, machine learning
and image analysis. Applications are: kriging and op-
timal design
1. Matern covariance
C(x, y) = C(|x−y|) = σ2 1
Γ(ν)2ν−1
√
2ν
r
L
ν
Kν
√
2ν
r
L
,
where Γ is the gamma function, Kν is the modified
Bessel function of the second kind, r = |x − y| and
ν, L are non-negative parameters of the covariance.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.05
0.1
0.15
0.2
0.25
Matern covariance (nu=1)
σ=0.5, l=0.5
σ=0.5, l=0.3
σ=0.5, l=0.2
σ=0.5, l=0.1
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.05
0.1
0.15
0.2
0.25
nu=0.15
nu=0.3
nu=0.5
nu=1
nu=2
nu=30
As ν → ∞ [4],
C(r) = σ2
exp(−r2
/2L2
).
When ν = 0.5, the Matern covariance is identical to
the exponential covariance function.
Cν=3/2(r) = 1 +
√
3r
L
exp −
√
3r
L
Cν=5/2(r) = 1 +
√
5r
L
+
5r2
3L2
exp −
√
5r
L
.
Note: no need to assume neither C(x, y) = C(|x − y|) nor tensor
grid.
2. H-matrix approximation
25 20
20 20
20 16
20 16
20 20
16 16
20 16
16 16
19 20
20 19 32
19 19
16 16 32
19 20
20 19
19 16
19 16
32 32
20 20
20 20 32
32 32
20 19
19 19 32
20 19
16 16 32
32 20
32 32
20 32
32 32
32 20
32 32
20 19
19 19
20 16
19 16
32 32
20 32
32 32
32 32
20 32
32 32
20 32
20 20
20 20 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
20 20
20 19
20 20 32
32 32 20
20 20
32 20
32 32 20
20 20
32 32
20 32 20
20 20
32 32
32 32 20
20
20 20
19 20 32
32 32
20 20
20
32 20
32 32
20 20
20
32 32
20 32
20 20
20
32 32
32 32
20 20
20 20
20 20 32
32 32
20 19
20 19 32
32 32
20 20
19 19 32
32 32
20 20
20 20 32
32 32
32 20
32 32
32 20
32 32
32 20
32 32
32 20
32 32
32 32
20 32
32 32
20 32
32 32
20 32
32 32
20 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
20 20
20 20 2019 20
20 20 32
32 32
32 20
32 32
20 32
32 32
32 20
32 32
20 20
20 20
20 20
20 20 20
20 20
32 20
32 32 20
20 20
20 20
20 20
20 20 20
20
32 20
32 32
20 20
20 20
20 20
20 20
20 20 20
32 20
32 32
32 20
32 32
32 20
32 32
32 20
32 32
20 20
20 20
20 20
20 20
19 20
20 20 32
32 32
20 32
32 32
32 32
20 32
32 32
20 32
20
20 20
20 20
20 20
20 20
20 20
32 32
20 32 20
20
20 20
20 20
20 20
20 20
20
32 32
20 32
20 20
20
20 20
20 20
20 20
20 20
32 32
20 32
32 32
20 32
32 32
20 32
32 32
20 32
20
20 20
20 20
20 20
20 20 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
19 19
20 20 32
32 32
32 20
32 32
20 32
32 32
32 20
32 32
19 20
19 20 32
32 32
20 32
32 32
32 32
20 32
32 32
20 32
20 20
20 20 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
20 20
32 32
32 32 20
20 20
32 20
32 32 20
20 20
32 32
20 32 20
20 20
32 32
32 32 20
20
32 32
32 32
20 20
20
32 20
32 32
20 20
20
32 32
20 32
20 20
20
32 32
32 32
20 20
32 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
32 20
32 32
32 20
32 20
32 20
32 32
32 20
32 20
32 32
20 32
32 32
20 32
32 32
20 20
32 32
20 20
32 32
32 32
32 32
32 32
32 32
32 32
32 32
32 32
25 9
9 20 9
9
20 7
7 16
9
9
20 9
9 20 9
9 32
9
9
20 9
9 20 9
9 32 9
9
32 9
9 32
9
9
20 9
9 20 9
9 32 9
9
20 9
9 20 9
9 32
9
9
32 9
9 32 9
9
32 9
9 32
9
9
20 9
9 20 9
9 32 9
9
32 9
9 32
9
9
20 9
9 20 9
9 32 9
9
32 9
9 32
9
9
32 9
9 32 9
9
32 9
9 32
9
9
32 9
9 32 9
9
32 9
9 32
Figure 2: Two approximation strategies [1]: fixed rank (left) and
flexible rank (right) approximations, C ∈ Rn×n, n = 652.
I
I
I I
I
I
I I I I1
1
2
2
11 12 21 22
I11
I12
I21
I22
Q
Qt
S
dist
H=
t
s
1. Build cluster tree TI, I = {1, 2, ..., n}
2. Build block cluster tree TI×I
3. For each (t × s) ∈ TI×I, t, s ∈ TI, check admissibility
condition min{diam(Qt), diam(Qs)} ≤ η·dist(Qt, Qs).
if(adm=true) then M|t×s is a rank-k matrix block
if(adm=false) then divide M|t×s further or define as a
dense matrix block, if small enough.
Grid → cluster tree (TI) + admissibility condition →
block cluster tree (TI×I) → H-matrix → H-matrix arith-
metics.
Operation Sequential Complexity Parallel Complexity
(Hackbusch et al. ’99-’06) (Kriemann ’05)
storage(M) N = O(kn log n) N
P
Mx N = O(kn log n) N
P
M1 ⊕ M2 N = O(k2n log n) N
P
M1 M2, M−1 N = O(k2n log2 n) N
P + O(n)
H-LU N = O(k2n log2 n) N
P + O(k2
n log2
n
n1/d )
Table 1: Computational cost of H-matrix arithmetics, sequential
and parallel.
Let ε =
(C−CH
)z 2
C 2 z 2
, where z is a random vector.
n rank k size, MB t, sec. ε max
i=1..10
|λi − ˜λi|, i ε2
for ˜C C ˜C C ˜C
4.0 · 103
10 48 3 0.8 0.08 7 · 10−3
7.0 · 10−2
, 9 2.0 · 10−4
1.05 · 104
18 439 19 7.0 0.4 7 · 10−4
5.5 · 10−2
, 2 1.0 · 10−4
2.1 · 104
25 2054 64 45.0 1.4 1 · 10−5
5.0 · 10−2
, 9 4.4 · 10−6
Table 2: Accuracy of the H-matrix approx. exp. covariance function, l1 = l3 =
0.1, l2 = 0.5.
l1 l2 ε
0.01 0.02 3 · 10−2
0.1 0.2 8 · 10−3
0.5 1 2.8 · 10−5
Table 3: Dependence of the H-matrix accuracy on the covari-
ance lengths l1 and l2, n = 1292. The smaller cov. length the less
accurate is H-approximation.
0
100
200
300
0
50
100
150
200
250
300
−1
0
1
2
−1
−0.5
0
0.5
1
1.5
2
2.5
3
0
100
200
300
0
50
100
150
200
250
300
−1
0
1
2
−3
−2
−1
0
1
2
Figure 4: Two realizations of random field generated via
Cholesky decomposition of Matern covariance matrix, ν = 0.4.
3. Kullback-Leibler divergence
Measure of the information lost when distribution Q is used to
approximate P.
DKL(P Q) =
i
P(i) ln
P(i)
Q(i)
, DKL(P Q) =
∞
−∞
p(x) ln
p(x)
q(x)
dx,
where p, q densities of P and Q. For miltivariate normal distribu-
tions (µ0, C) and (µ1, CH):
2DKL(N0 N1) = tr((CH
)−1
C) + (µ1 − µ0)T
(CH
)−1
(µ1 − µ0) − n − ln
det C
det CH
.
0 10 20 30 40 50 60 70 80 90 100
−16
−14
−12
−10
−8
−6
−4
−2
0
rank k
log(rel.error)
Spectral norm, L=0.1, nu=0.5
Frob. norm, L=0.1
Spectral norm, L=0.2
Frob. norm, L=0.2
Spectral norm, L=0.5
Frob. norm, L=0.5
0 10 20 30 40 50 60 70 80 90 100
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
rank k
log(rel.error)
Spectral norm, L=0.1, ν=1.5
Frob. norm, L=0.1
Spectral norm, L=0.2
Frob. norm, L=0.2
Spectral norm, L=0.5
Frob. norm, L=0.5
Figure 5: Relative H-matrix approx. error C−CH
2 for different
cov. lengths L = {0.1, 0.2, 0.5} and ν = {0.5, 1.5}
k KLD(C, CH) C − CH
2 C(CH)−1 − I 2
L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75
5 0.51 2.3 4.0e-2 0.1 4.8 63
6 0.34 1.6 9.4e-3 0.02 3.4 22
8 5.3e-2 0.4 1.9e-3 0.003 1.2 8
10 2.6e-3 0.2 7.7e-4 7.0e-4 6.0e-2 3.1
12 5.0e-4 2e-2 9.7e-5 5.6e-5 1.6e-2 0.5
15 1.0e-5 9e-4 2.0e-5 1.1e-5 8.0e-4 0.02
20 4.5e-7 4.8e-5 6.5e-7 2.8e-7 2.1e-5 1.2e-3
50 3.4e-13 5e-12 2.0e-13 2.4e-13 4e-11 2.7e-9
Table 4: Dependence of KLD on H-matrix rank k, Matern co-
variance with L = {0.25, 0.75} and ν = 0.5, domain G = [0, 1]2,
C(L=0.25,0.75) 2 = {212, 568}.
For ν = 1.5 the KLD and the inverse (CH)−1 is hard to compute
numerically. Results in Table 4 are better since covariance ma-
trix with ν = 0.5 has smallest eigenvalues far enough from zero.
The case ν = 1.5 is more smooth, the eigenvalues decay faster,
but the smallest eigenvalues come much closer to zero than in
ν = 0.5 case.
4. Other applications
4.1 Low-rank approximation of Kriging and geo-
statistical optimal design
Let ˆs ∈ Rn to be estimated, Css covariance matrix, y ∈ Rm is
vector of measurements. The corresponding cross- and auto-
covariance matrices are denoted by Csy and Cyy, respectively,
sized n × m and m × m.
Kriging estimate ˆs = CsyC−1
yy y .
The estimation variance ˆσ is the diagonal of the cond. cov. ma-
trix Css|y: ˆσs = diag(Css|y) = diag Css − CsyC−1
yy Cys
Geostatistical optimal design:
φA = n−1 trace Css|y
φC = cT Css − CsyC−1
yy Cys c, c − a vector.
4.2 Weather forecast in Europa
180 240
30
60
Figure 6: Europa weather stations (≈ 2500). Collected data set
M ∈ R2500×365
.
0 50 100 150 200 250 300 350 400
−20
−15
−10
−5
0
5
10
15
20
Figure 7: Truth temperature forecast and its low-rank approxi-
mation (rank 50 approximation of matrix M) in one station, rel.
error=25%.
5. Open question
1. Compute the whole spectrum of large covariance matrix
2. Compute KLD for large matrices (det Σ ?)
3. How sensible is KLD to H-matrix accuracy ?
4. Derive/estimate KLD for non-Gaussian distributions.
Acknowledgements
A. Litvinenko is a member of the KAUST SRI UQ Center.
References
1. B. N. Khoromskij, A. Litvinenko, H. G. Matthies, Application of
hierarchical matrices for computing the Karhunen?Lo´eve expan-
sion, Computing, Vol. 84, Issue 1-2, pp 49-67, 2008
2. R. Furrer, M. Genton, D. Nychka, Covariance tapering for in-
terpolation of large spatial datasets, J. Comp. & Graph. Stat.,
Vol.15, N3, pp502-523.
3. M. Stein, Limitations on low rank approximations for covari-
ance matrices of spatial data, Spat. Statistics, 2013
4. J. Castrillion-Candis, M. Genton, R. Yokota, Multi-Level Re-
stricted Maximum Likelihood Cov. Estim. and Kriging for Large
Non-Gridded Datasets, 2014.](https://image.slidesharecdn.com/10litvinenko-170311115453/85/Hierarchical-matrix-approximation-of-large-covariance-matrices-1-320.jpg)
Ad
Recommended
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionAlexander Litvinenko This document discusses the data sparse approximation of the Karhunen-Loève expansion, focusing on hierarchical matrices for stochastic partial differential equations (PDEs). It outlines the methodology for approximating covariance functions, discrete eigenvalue problems, and provides numerical examples illustrating the efficiency of H-matrices in terms of computational time and memory requirements. The study emphasizes low Kronecker rank approximations and tensor decompositions to optimize storage and computation for random fields.
Talk litvinenko prior_cov
Talk litvinenko prior_covAlexander Litvinenko The document discusses the initial covariance matrix for the Kalman filter, presenting two variants: one where the matrix of snapshots is predefined and another based on specific covariance functions, particularly the Matérn class. It includes detailed equations and parameters related to the Matérn covariance functions, computational performance metrics, and the convergence of Kullback-Leibler divergence in relation to varying matrix ranks. Additionally, applications for large covariance matrices in statistical analysis and kriging are outlined, emphasizing the significance of approximating mean and variance efficiently.
Test yourself for JEE(Main)TP-2
Test yourself for JEE(Main)TP-2Vijay Joglekar This document contains 30 multiple choice questions related to mathematics for a JEE exam preparation test. It provides instructions that each question is worth 4 marks and 1 mark will be deducted for incorrect answers. The maximum total marks for the test are 120. The questions cover topics in trigonometry, algebra, geometry and calculus.
Test yourself for JEE(Main)TP-3
Test yourself for JEE(Main)TP-3Vijay Joglekar This document provides 30 multiple choice questions for a JEE mathematics exam. It includes instructions that there are 4 marks for each correct answer, a deduction of 1 mark for incorrect answers, and no deduction for unanswered questions. The maximum total marks are 120. The questions cover topics in trigonometry, coordinate geometry, calculus, matrices and other areas of mathematics.
Álgebra básica 2
Álgebra básica 2KalculosOnline The document contains a 15 question multiple choice algebra test with the following topics covered: solving systems of equations, simplifying algebraic expressions, properties of exponents, rational and irrational numbers, and evaluating expressions. The questions range in difficulty from basic operations to more complex problems involving multiple steps. An answer key is provided at the end listing the correct choice for each question.
Test yourself for JEE(Main)
Test yourself for JEE(Main)Vijay Joglekar This document contains a 30 question mathematics practice test with multiple choice answers for JEE Main exam preparation. Each question is allotted 4 marks for a correct response and there is a 1 mark deduction for an incorrect response. The maximum total marks are 120. The questions cover topics in trigonometry, coordinate geometry, calculus, and vectors. Instructions provide details on the marking scheme and conditions for the exam.
H 2004 2007
H 2004 2007sjamaths 1. The document provides instructions for a mathematics exam. It states that calculators may be used, full marks require showing working, and scale drawings will not be credited. It then lists various formulae that may be needed for the exam.
2. The exam consists of 11 multi-part questions testing a range of mathematics skills, including algebra, geometry, trigonometry, calculus and graph sketching. Candidates are advised to attempt all questions.
3. The document concludes by providing blank pages for working, followed by a notice that the exam has ended.
C3 bronze 1
C3 bronze 1Mohammed Ahmed This document provides information about an exam for the Edexcel GCE Core Mathematics C3 Bronze Level B1 qualification. It lists the paper reference, time allowed, materials required and permitted calculators. It provides instructions for candidates on writing details on the front page and information about the structure of the paper. It also lists the 9 questions that make up the exam, covering topics like functions, graphs, derivatives, iterations and logarithms. The final section suggests grade boundaries for the exam.
ITA 2010 - fechada
ITA 2010 - fechadaKalculosOnline This document contains 20 multiple choice questions regarding mathematics concepts such as sets, arithmetic progressions, matrices, trigonometry, polynomials, and complex numbers. The questions test a range of skills including evaluating statements, solving equations, finding values that maximize or minimize expressions, and determining properties of functions and their solutions.
Guess paper x
Guess paper xHumaira Ahmed The document provides 20 multiple choice questions testing mathematics objectives related to topics like sets, algebra, trigonometry, and geometry. It then lists 18 additional practice problems involving algebraic expressions, logarithms, geometry constructions, data analysis, and solving equations both algebraically and graphically. The objectives and practice problems cover a wide range of foundational mathematics skills.
Finding coordinates parabol,cubic
Finding coordinates parabol,cubicshaminakhan The document provides instructions to find the coordinates of points on curves defined by algebraic equations. It gives examples of finding the coordinates A, B, C of curves defined by equations like (x-3)(x+2) and (x-5)(x+4). It also asks to find the coordinates A, B, C, D of the curve defined by the equation y=(x^2)(1)(6)-(x) and coordinates A, B of the curve defined by y=(x^3)(4)-(x).
Spm additional-mathematical-formulae-pdf
Spm additional-mathematical-formulae-pdfMahadzir Ismail 1. The document provides formulas for algebra, calculus, geometry, trigonometry, and statistics from SPM Additional Mathematics.
2. Formulas include those for solving equations, logarithms, differentiation, integration, areas of shapes, probabilities, and trigonometric functions.
3. Fourteen trigonometric formulas are listed relating trig functions, angles, and sides of triangles.
Appendex h
Appendex hswavicky The document discusses using the discriminant of a quadratic equation to determine the type of conic section represented by the graph of the equation. It defines the discriminant as B^2 - 4AC and explains that:
(a) If the discriminant is negative, the graph is an ellipse, circle, point, or has no graph.
(b) If the discriminant is positive, the graph is a hyperbola or intersecting lines.
(c) If the discriminant is 0, the graph is a parabola, line, parallel lines, or has no graph.
Exams in college algebra
Exams in college algebraRaymond Garcia This document contains a 25 question practice exam for college algebra. It covers topics such as solving linear, quadratic, and rational equations; simplifying expressions; factoring polynomials; graphing lines and parabolas; and solving systems of equations. The exam is broken down by standard to indicate which questions relate to different skills like evaluating expressions, adding and multiplying polynomials, solving various types of equations, and modeling real world problems.
ITA 2016 - fechada
ITA 2016 - fechadaKalculosOnline This document contains 20 multiple choice questions related to mathematics:
1. The number of digits in the integer part of 7x is 288, where x is a natural number with 2015 digits.
2. Statements I and II are true - I being that a certain function is strictly increasing, and II that a quadratic equation has one real solution.
3. Statements I, II, and III are true - I and II regarding a decreasing sequence being bounded above and below by 0 and 1, respectively, and III regarding being less than 1.
4. The value of T is a particular fraction involving matrices M and N.
Summative Assessment Paper-3
Summative Assessment Paper-3APEX INSTITUTE This document contains a mathematics exam with multiple choice and multi-part questions testing concepts like factors, rational numbers, geometry proofs, and polynomials. The exam has four sections with questions of varying point values testing skills such as simplifying expressions, factoring polynomials, solving equations, proving geometric relationships, and finding zeros of polynomials.
Lista de derivadas e integrais
Lista de derivadas e integraisUniversidade de São Paulo USP This document provides formulas and properties related to calculus and trigonometry. It includes:
1) Derivative formulas for common functions like polynomials, logarithms, trigonometric functions, and hyperbolic functions.
2) Integral formulas for polynomials, logarithms, trigonometric functions, and the exponential function.
3) Trigonometric identity formulas like sum and difference formulas.
4) Limit formulas for important limits like sin(x)/x and (1+1/x)^x.
The document is a reference sheet of calculus and trigonometry formulas for students to use. It covers derivatives, integrals, identities, and limits of common functions.
Summative Assessment Paper-2
Summative Assessment Paper-2APEX INSTITUTE The document is a mathematics test paper divided into multiple sections with questions ranging from basic concepts to proofs and problem-solving. It includes a variety of question types, such as multiple choice, verification, simplification, and geometric proofs, designed to assess knowledge in different mathematical areas. Each section specifies the number of marks allocated and the total time for the test.
Lista de integrais Calculo IV
Lista de integrais Calculo IVUniversidade de São Paulo USP This document contains lists of basic power series, integrals, and trigonometric formulas. The power series section lists common power series expansions such as 1/(1-x), ln(1+x), arctan(x), e^x, sin(x), and cos(x). The integrals section provides formulas for integrals of functions such as x^n, ln(x), sin(ax), e^ax, sec(x), and elliptic integrals. The trigonometric formulas section lists trigonometric identities for sums and differences of angles.
12th mcq
12th mcqnitishguptamaps The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability, matrices, vectors, and linear transformations.
Unit 1 review packet
Unit 1 review packetmlabuski This document is a unit test review covering exponents, order of operations, variables and expressions, translating between words and math, equations and solutions. It contains 25 questions testing these concepts. The questions include writing expressions, evaluating expressions, defining math terms, solving equations, and performing calculations.
Lista de integrais2
Lista de integrais2Universidade de São Paulo USP 1) The document contains tables summarizing important limits, derivatives, integrals, and trigonometric formulas.
2) The limits table includes limits involving polynomials, logarithms, and trigonometric functions as x approaches values.
3) The derivatives table lists the derivatives of polynomials, logarithmic, trigonometric, hyperbolic, and inverse trigonometric functions.
Tabela cal1
Tabela cal1Universidade de São Paulo USP The document contains 4 sections that provide formulas and properties for calculus and trigonometry:
1. Lists 12 derivative formulas including the derivatives of x^n, a^x, ln(x), sin(x), cos(x), tan(x), etc.
2. Lists 12 integral formulas including the integrals of x^n, ln(x), sin(ax), cos(ax), tan(x), sec(x), etc.
3. Provides 6 trigonometric formulas including formulas for sin(a+b), cos(a+b), sin^2(a), sin(2a), and cos(2a).
4. Lists 3 fundamental limits including the
Math practice for final exam
Math practice for final examDulce Garza The document contains a math practice worksheet for a 7th grade final exam. It includes questions on topics like fractions, exponents, equations, word problems, and geometry. Students are asked to solve expressions, equations, and word problems involving these concepts. They must also choose the correct math terms to complete sentences and select answers to multiple choice questions testing their understanding of formulas and patterns.
Maths (2)
Maths (2)Arjun kumar Malhi This document contains 25 multiple choice questions related to mathematics topics like quadratic equations, arithmetic progressions, geometric progressions, and averages. It tests concepts such as identifying the nature of roots of quadratic equations, finding terms in sequences, determining common differences and ratios, and calculating averages. The questions are from various areas of algebra and progressions for a school exam.
Minimum mean square error estimation and approximation of the Bayesian update
Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko This document discusses methods for approximating the Bayesian update used in parameter identification problems with partial differential equations containing uncertain coefficients. It presents:
1) Deriving the Bayesian update from conditional expectation and proposing polynomial chaos expansions to approximate the full Bayesian update.
2) Describing minimum mean square error estimation to find estimators that minimize the error between the true parameter and its estimate given measurements.
3) Providing an example of applying these methods to identify an uncertain coefficient in a 1D elliptic PDE using measurements at two points.
Response Surface in Tensor Train format for Uncertainty Quantification
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko The document discusses the application of tensor train (TT) approximation for uncertainty quantification, aiming to significantly reduce computational time and storage costs. It explores various strategies for polynomial chaos expansion (PCE) and presents methods for computing stochastic Galerkin operators in TT format, achieving efficient data processing. Key outcomes include a proposed workflow for solving stochastic partial differential equations (SPDEs) and generating statistical measures in TT format.
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionAlexander Litvinenko The document discusses a sparse approximation method for the Karhunen-Loève expansion used in stochastic partial differential equations (PDEs). It details numerical techniques, including response surfaces, Monte Carlo methods, and the use of hierarchical and tensor approximations to compute eigenpairs efficiently, particularly leveraging Fast Fourier Transform (FFT) for computation. Applications are highlighted, such as analyzing covariance and higher-order moments for stochastic solutions, with a focus on eigenvalue problems and tensor decompositions.
Likelihood approximation with parallel hierarchical matrices for large spatia...
Likelihood approximation with parallel hierarchical matrices for large spatia...Alexander Litvinenko The document discusses an efficient method for approximating likelihoods in Gaussian processes using parallel hierarchical matrices for large spatial datasets. It details the mathematical framework, including the parameters of the Matérn covariance function and the computational advantages of using hierarchical matrix approximations. Various numerical examples and comparisons of setup times, memory usage, and performance metrics are also provided, demonstrating significant reductions in resource usage.
A small introduction into H-matrices which I gave for my colleagues
A small introduction into H-matrices which I gave for my colleaguesAlexander Litvinenko The document introduces hierarchical matrix techniques by Alexander Litvinenko, focusing on motivations, the concept of low-rank matrices, and applications of hierarchical matrices in computational methods. It discusses iterative and direct solvers, the efficiency of handling structured matrices, and highlights storage advantages using rank-k representations. Additionally, it touches on a specific example involving a stiffness matrix for the Poisson problem and visual representations of h-matrices.
More Related Content
What's hot (17)
ITA 2010 - fechada
ITA 2010 - fechadaKalculosOnline This document contains 20 multiple choice questions regarding mathematics concepts such as sets, arithmetic progressions, matrices, trigonometry, polynomials, and complex numbers. The questions test a range of skills including evaluating statements, solving equations, finding values that maximize or minimize expressions, and determining properties of functions and their solutions.
Guess paper x
Guess paper xHumaira Ahmed The document provides 20 multiple choice questions testing mathematics objectives related to topics like sets, algebra, trigonometry, and geometry. It then lists 18 additional practice problems involving algebraic expressions, logarithms, geometry constructions, data analysis, and solving equations both algebraically and graphically. The objectives and practice problems cover a wide range of foundational mathematics skills.
Finding coordinates parabol,cubic
Finding coordinates parabol,cubicshaminakhan The document provides instructions to find the coordinates of points on curves defined by algebraic equations. It gives examples of finding the coordinates A, B, C of curves defined by equations like (x-3)(x+2) and (x-5)(x+4). It also asks to find the coordinates A, B, C, D of the curve defined by the equation y=(x^2)(1)(6)-(x) and coordinates A, B of the curve defined by y=(x^3)(4)-(x).
Spm additional-mathematical-formulae-pdf
Spm additional-mathematical-formulae-pdfMahadzir Ismail 1. The document provides formulas for algebra, calculus, geometry, trigonometry, and statistics from SPM Additional Mathematics.
2. Formulas include those for solving equations, logarithms, differentiation, integration, areas of shapes, probabilities, and trigonometric functions.
3. Fourteen trigonometric formulas are listed relating trig functions, angles, and sides of triangles.
Appendex h
Appendex hswavicky The document discusses using the discriminant of a quadratic equation to determine the type of conic section represented by the graph of the equation. It defines the discriminant as B^2 - 4AC and explains that:
(a) If the discriminant is negative, the graph is an ellipse, circle, point, or has no graph.
(b) If the discriminant is positive, the graph is a hyperbola or intersecting lines.
(c) If the discriminant is 0, the graph is a parabola, line, parallel lines, or has no graph.
Exams in college algebra
Exams in college algebraRaymond Garcia This document contains a 25 question practice exam for college algebra. It covers topics such as solving linear, quadratic, and rational equations; simplifying expressions; factoring polynomials; graphing lines and parabolas; and solving systems of equations. The exam is broken down by standard to indicate which questions relate to different skills like evaluating expressions, adding and multiplying polynomials, solving various types of equations, and modeling real world problems.
ITA 2016 - fechada
ITA 2016 - fechadaKalculosOnline This document contains 20 multiple choice questions related to mathematics:
1. The number of digits in the integer part of 7x is 288, where x is a natural number with 2015 digits.
2. Statements I and II are true - I being that a certain function is strictly increasing, and II that a quadratic equation has one real solution.
3. Statements I, II, and III are true - I and II regarding a decreasing sequence being bounded above and below by 0 and 1, respectively, and III regarding being less than 1.
4. The value of T is a particular fraction involving matrices M and N.
Summative Assessment Paper-3
Summative Assessment Paper-3APEX INSTITUTE This document contains a mathematics exam with multiple choice and multi-part questions testing concepts like factors, rational numbers, geometry proofs, and polynomials. The exam has four sections with questions of varying point values testing skills such as simplifying expressions, factoring polynomials, solving equations, proving geometric relationships, and finding zeros of polynomials.
Lista de derivadas e integrais
Lista de derivadas e integraisUniversidade de São Paulo USP This document provides formulas and properties related to calculus and trigonometry. It includes:
1) Derivative formulas for common functions like polynomials, logarithms, trigonometric functions, and hyperbolic functions.
2) Integral formulas for polynomials, logarithms, trigonometric functions, and the exponential function.
3) Trigonometric identity formulas like sum and difference formulas.
4) Limit formulas for important limits like sin(x)/x and (1+1/x)^x.
The document is a reference sheet of calculus and trigonometry formulas for students to use. It covers derivatives, integrals, identities, and limits of common functions.
Summative Assessment Paper-2
Summative Assessment Paper-2APEX INSTITUTE The document is a mathematics test paper divided into multiple sections with questions ranging from basic concepts to proofs and problem-solving. It includes a variety of question types, such as multiple choice, verification, simplification, and geometric proofs, designed to assess knowledge in different mathematical areas. Each section specifies the number of marks allocated and the total time for the test.
Lista de integrais Calculo IV
Lista de integrais Calculo IVUniversidade de São Paulo USP This document contains lists of basic power series, integrals, and trigonometric formulas. The power series section lists common power series expansions such as 1/(1-x), ln(1+x), arctan(x), e^x, sin(x), and cos(x). The integrals section provides formulas for integrals of functions such as x^n, ln(x), sin(ax), e^ax, sec(x), and elliptic integrals. The trigonometric formulas section lists trigonometric identities for sums and differences of angles.
12th mcq
12th mcqnitishguptamaps The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability, matrices, vectors, and linear transformations.
Unit 1 review packet
Unit 1 review packetmlabuski This document is a unit test review covering exponents, order of operations, variables and expressions, translating between words and math, equations and solutions. It contains 25 questions testing these concepts. The questions include writing expressions, evaluating expressions, defining math terms, solving equations, and performing calculations.
Lista de integrais2
Lista de integrais2Universidade de São Paulo USP 1) The document contains tables summarizing important limits, derivatives, integrals, and trigonometric formulas.
2) The limits table includes limits involving polynomials, logarithms, and trigonometric functions as x approaches values.
3) The derivatives table lists the derivatives of polynomials, logarithmic, trigonometric, hyperbolic, and inverse trigonometric functions.
Tabela cal1
Tabela cal1Universidade de São Paulo USP The document contains 4 sections that provide formulas and properties for calculus and trigonometry:
1. Lists 12 derivative formulas including the derivatives of x^n, a^x, ln(x), sin(x), cos(x), tan(x), etc.
2. Lists 12 integral formulas including the integrals of x^n, ln(x), sin(ax), cos(ax), tan(x), sec(x), etc.
3. Provides 6 trigonometric formulas including formulas for sin(a+b), cos(a+b), sin^2(a), sin(2a), and cos(2a).
4. Lists 3 fundamental limits including the
Math practice for final exam
Math practice for final examDulce Garza The document contains a math practice worksheet for a 7th grade final exam. It includes questions on topics like fractions, exponents, equations, word problems, and geometry. Students are asked to solve expressions, equations, and word problems involving these concepts. They must also choose the correct math terms to complete sentences and select answers to multiple choice questions testing their understanding of formulas and patterns.
Maths (2)
Maths (2)Arjun kumar Malhi This document contains 25 multiple choice questions related to mathematics topics like quadratic equations, arithmetic progressions, geometric progressions, and averages. It tests concepts such as identifying the nature of roots of quadratic equations, finding terms in sequences, determining common differences and ratios, and calculating averages. The questions are from various areas of algebra and progressions for a school exam.
Viewers also liked (20)
Minimum mean square error estimation and approximation of the Bayesian update
Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko This document discusses methods for approximating the Bayesian update used in parameter identification problems with partial differential equations containing uncertain coefficients. It presents:
1) Deriving the Bayesian update from conditional expectation and proposing polynomial chaos expansions to approximate the full Bayesian update.
2) Describing minimum mean square error estimation to find estimators that minimize the error between the true parameter and its estimate given measurements.
3) Providing an example of applying these methods to identify an uncertain coefficient in a 1D elliptic PDE using measurements at two points.
Response Surface in Tensor Train format for Uncertainty Quantification
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko The document discusses the application of tensor train (TT) approximation for uncertainty quantification, aiming to significantly reduce computational time and storage costs. It explores various strategies for polynomial chaos expansion (PCE) and presents methods for computing stochastic Galerkin operators in TT format, achieving efficient data processing. Key outcomes include a proposed workflow for solving stochastic partial differential equations (SPDEs) and generating statistical measures in TT format.
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionAlexander Litvinenko The document discusses a sparse approximation method for the Karhunen-Loève expansion used in stochastic partial differential equations (PDEs). It details numerical techniques, including response surfaces, Monte Carlo methods, and the use of hierarchical and tensor approximations to compute eigenpairs efficiently, particularly leveraging Fast Fourier Transform (FFT) for computation. Applications are highlighted, such as analyzing covariance and higher-order moments for stochastic solutions, with a focus on eigenvalue problems and tensor decompositions.
Likelihood approximation with parallel hierarchical matrices for large spatia...
Likelihood approximation with parallel hierarchical matrices for large spatia...Alexander Litvinenko The document discusses an efficient method for approximating likelihoods in Gaussian processes using parallel hierarchical matrices for large spatial datasets. It details the mathematical framework, including the parameters of the Matérn covariance function and the computational advantages of using hierarchical matrix approximations. Various numerical examples and comparisons of setup times, memory usage, and performance metrics are also provided, demonstrating significant reductions in resource usage.
A small introduction into H-matrices which I gave for my colleagues
A small introduction into H-matrices which I gave for my colleaguesAlexander Litvinenko The document introduces hierarchical matrix techniques by Alexander Litvinenko, focusing on motivations, the concept of low-rank matrices, and applications of hierarchical matrices in computational methods. It discusses iterative and direct solvers, the efficiency of handling structured matrices, and highlights storage advantages using rank-k representations. Additionally, it touches on a specific example involving a stiffness matrix for the Poisson problem and visual representations of h-matrices.
Low-rank methods for analysis of high-dimensional data (SIAM CSE talk 2017)
Low-rank methods for analysis of high-dimensional data (SIAM CSE talk 2017) Alexander Litvinenko The document discusses low-rank tensor methods for high-dimensional data analysis, detailing various techniques for efficient computation and approximation in tensor formats. It covers methodologies for evaluating specific parameters, finding maxima and minima, and computing level sets through tensor representations. The content also addresses the computational complexities of different tensor formats and their respective advantages and disadvantages.
Tensor Completion for PDEs with uncertain coefficients and Bayesian Update te...
Tensor Completion for PDEs with uncertain coefficients and Bayesian Update te...Alexander Litvinenko The document discusses tensor completion for partial differential equations (PDEs) with uncertain coefficients and Bayesian updates, emphasizing the importance of uncertainty quantification in simulations. It covers the stochastic forward problem, Bayesian updating methods, including Gaussian filters, and describes a tensor completion approach to handle these uncertainties efficiently. The document provides examples and techniques related to discretization, polynomial chaos expansions, and conditional expectations in the context of PDEs.
Scalable hierarchical algorithms for stochastic PDEs and UQ
Scalable hierarchical algorithms for stochastic PDEs and UQAlexander Litvinenko This document discusses scalable hierarchical algorithms for stochastic PDEs and uncertainty quantification using h-matrices and fast multipole methods, which significantly reduce computational costs and storage complexity for large-scale problems. It outlines advancements in parallel algorithms and communication complexities to enhance performance on modern multi-core architectures. The work aims to efficiently implement these algorithms across various platforms, facilitating applications in modeling large covariance matrices and solving complex mathematical equations.
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko 1) The document describes a method called Multilevel Monte Carlo (MLMC) to efficiently compute electromagnetic fields scattered from dielectric objects of uncertain shapes. MLMC balances statistical errors from random sampling and numerical errors from geometry discretization to reduce computational time.
2) A surface integral equation solver is used to model scattering from dielectric objects. Random geometries are generated by perturbing surfaces with random fields defined by spherical harmonics.
3) MLMC is shown to estimate scattering cross sections accurately while requiring fewer overall computations compared to traditional Monte Carlo methods. This is achieved by optimally allocating samples across discretization levels.
Disc seeding in conservation agriculture
Disc seeding in conservation agricultureJack McHugh This document discusses different types of disc openers and coulters used in no-till planters. It describes disc openers as being more variable than tine openers depending on soil conditions and settings. The main types of disc openers discussed are disc coulters, double discs, single discs, and disc/tine hybrids. For each type, the document provides details on their design characteristics and how they interact with soil and residue to perform functions like furrow opening, seed placement, and soil loosening.
Tensor train to solve stochastic PDEs
Tensor train to solve stochastic PDEsAlexander Litvinenko The document discusses numerical methods for solving stochastic partial differential equations using tensor train formats, emphasizing the need for uncertainty quantification in computational algorithms. It presents a framework for dealing with uncertain inputs and outputs, including the representation of these uncertainties in tensor formats to facilitate efficient computational analysis. The approach involves applying stochastic Galerkin methods and various numerical techniques to improve performance and accuracy in simulations.
add_2_diplom_main
add_2_diplom_mainAlexander Litvinenko This document is a dissertation submitted by Alexander Litvinenko to the Faculty of Mathematics and Computer Science at the University of Leipzig in partial fulfillment of the requirements for the degree of Doctor of Natural Sciences. The dissertation proposes the application of hierarchical matrices (H-matrices) to solve multiscale problems using the hierarchical domain decomposition (HDD) method. It begins with an introduction and literature review of multiscale problems and existing solution methods. It then describes the classical finite element method, the HDD method, and H-matrices. The main body of the dissertation focuses on applying H-matrices within the HDD method to efficiently solve problems involving multiple spatial and temporal scales. Numerical results demonstrate the effectiveness of the proposed approach.
My PhD on 4 pages
My PhD on 4 pagesAlexander Litvinenko The document summarizes a dissertation on applying hierarchical matrices to solve multiscale problems. The dissertation proposes a new hierarchical domain decomposition (HDD) method that combines hierarchical matrices and domain decomposition. HDD allows efficiently computing solution mappings and functionals, and solving problems on coarser grids or with multiple right-hand sides. Complexity analyses show HDD has lower complexity than other methods. Numerical tests on problems with oscillatory and jumping coefficients demonstrate HDD achieves the expected error bounds and is independent of frequency.
Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...
Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...Alexander Litvinenko The document discusses the application of low-rank tensors in statistical analysis and uncertainty quantification, focusing on various problems including predicting environmental variables like temperature and soil moisture. It outlines methodologies for kriging estimates, stochastic Galerkin operators, and the identification of uncertain parameters through log-likelihood maximization. Furthermore, the document addresses the multivariate characteristic function and its implications for understanding stable distributions.
RS
RSAlexander Litvinenko Alexander Litvinenko's research interests include developing efficient numerical methods for solving stochastic PDEs using low-rank tensor approximations. He has made contributions in areas such as fast techniques for solving stochastic PDEs using tensor approximations, inexpensive functional approximations of Bayesian updating formulas, and modeling uncertainties in parameters, coefficients, and computational geometry using probabilistic methods. His current research focuses on uncertainty quantification, Bayesian updating techniques, and developing scalable and parallel methods using hierarchical matrices.
Litvinenko low-rank kriging +FFT poster
Litvinenko low-rank kriging +FFT posterAlexander Litvinenko This document presents a study on accelerating kriging algorithms through the integration of low-rank tensor approximations and Fast Fourier Transform (FFT) techniques, achieving significant computational efficiency. The proposed method reduces complexity to O(d l log l) and enables the handling of large-scale kriging problems, with practical applications in geostatistical optimal design. The results demonstrate impressive time savings for massive datasets, highlighting advancements in both estimation variance and optimal sampling patterns.
Litvinenko nlbu2016
Litvinenko nlbu2016Alexander Litvinenko The document discusses a method for solving inverse problems through Bayesian updates of polynomial chaos expansion (PCE) coefficients in the context of uncertain physical systems modeled by PDEs or ODEs. It outlines a non-linear approximation approach that minimizes mean square error and computes conditional expectations to identify uncertain coefficients from noisy observations. The findings are supported by various examples and numerical implementations, highlighting the application of this methodology in uncertainty quantification and data assimilation.
Low-rank tensor methods for stochastic forward and inverse problems
Low-rank tensor methods for stochastic forward and inverse problemsAlexander Litvinenko The document discusses low-rank tensor methods for solving partial differential equations (PDEs) with uncertain coefficients. It covers two parts: (1) using the stochastic Galerkin method to discretize an elliptic PDE with uncertain diffusion coefficient represented by tensors, and (2) computing quantities of interest like the maximum value from the tensor solution in a efficient way. Specifically, it describes representing the diffusion coefficient, forcing term, and solution of the discretized PDE using tensors, and computing the maximum value and corresponding indices by solving an eigenvalue problem involving the tensor solution.
My PhD talk "Application of H-matrices for computing partial inverse"
My PhD talk "Application of H-matrices for computing partial inverse"Alexander Litvinenko This document describes a hierarchical domain decomposition (HDD) method for solving stochastic elliptic boundary value problems with oscillatory or jumping coefficients. HDD constructs mappings between boundary and interface values that allow the solution to be computed locally in each subdomain. These mappings are represented as H-matrices to reduce computational costs. The total storage cost of HDD is O(kn log2nh) and complexity is O(k2nh log3nh), where n is the number of degrees of freedom, k is the H-matrix rank, and h is the mesh size. HDD can also be used to compute solutions when the right-hand side is represented on a coarser grid.
My paper for Domain Decomposition Conference in Strobl, Austria, 2005
My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko This paper proposes a new h-Cholesky based preconditioner for the skin problem, focusing on a block diagonal matrix derived from modifying index ordering. The preconditioner is shown to be more efficient in memory and computational requirements compared to standard h-Cholesky methods. Additionally, the paper discusses numerical tests and outlines the implementation of this preconditioner in a conjugate gradient method.
Likelihood approximation with parallel hierarchical matrices for large spatia...
Likelihood approximation with parallel hierarchical matrices for large spatia...Alexander Litvinenko
Tensor Completion for PDEs with uncertain coefficients and Bayesian Update te...
Tensor Completion for PDEs with uncertain coefficients and Bayesian Update te...Alexander Litvinenko
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...
Computation of Electromagnetic Fields Scattered from Dielectric Objects of Un...Alexander Litvinenko
Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...
Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...Alexander Litvinenko
Ad
Similar to Hierarchical matrix approximation of large covariance matrices (20)
Hierarchical matrix techniques for maximum likelihood covariance estimation
Hierarchical matrix techniques for maximum likelihood covariance estimationAlexander Litvinenko The document discusses hierarchical matrix techniques for maximum likelihood covariance estimation, focusing on applications such as temperature and moisture prediction using high-resolution data. It outlines the motivation for using hierarchical matrices due to their efficient storage and computational benefits compared to dense matrices. Additionally, it covers the identification of uncertain covariance parameters through the maximization of log-likelihood functions.
Approximation of large covariance matrices in statistics
Approximation of large covariance matrices in statisticsAlexander Litvinenko The document discusses a hierarchical likelihood approximation method designed to enhance the estimation of unknown statistical parameters in large spatial datasets, specifically soil moisture fields. It focuses on reducing the computational cost of linear algebra from O(n^3) to O(n log n) using parallel hierarchical matrices, offering efficient approximations for Gaussian processes modeled by Matérn covariance functions. Various numerical examples and performance metrics are provided to illustrate the effectiveness of the approach in reducing computation time and storage costs while maintaining accuracy.
Poster litvinenko genton_ying_hpcsaudi17
Poster litvinenko genton_ying_hpcsaudi17Alexander Litvinenko The document discusses hierarchical likelihood approximation methods aimed at estimating unknown statistical parameters in spatial soil moisture fields by reducing computational costs from O(n^3) to O(n log n). It details the use of mean-zero, stationary Gaussian processes and presents a framework for approximating covariance matrices efficiently with parallel processing capabilities. Additionally, it showcases numerical experiments validating the performance of the proposed methods on large datasets, particularly in the context of soil moisture data analysis.
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Alexander Litvinenko The document discusses the use of parallel hierarchical matrices and low-rank tensors in spatial statistics and parameter identification, specifically in improving statistical models for moisture data. It covers techniques for estimating unknown parameters through maximizing the Gaussian log-likelihood function using hierarchical matrices and low-rank tensor methods. Key results involve approximations of covariance functions and their implications for agricultural and defense applications.
Data sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve ExpansionAlexander Litvinenko The document discusses the data sparse approximation of the Karhunen-Loève expansion in relation to stochastic partial differential equations (PDEs) and numerical techniques. Key topics include hierarchical matrices, sparse tensor approximations, and different covariance functions, emphasizing efficient computation methods using Fast Fourier Transforms (FFT) and matrix approximations. The applications of these techniques cover the covariance of solutions for stochastic PDEs and the computation of eigenpairs for stochastic systems.
Slides
SlidesAlexander Litvinenko The document discusses data sparse approximation techniques related to the Karhunen-Loève expansion, focusing on numerical methods, eigenvalue problems, and the covariance structure of random fields. It explores applications such as solving stochastic partial differential equations and higher-order moments, employing methods like FFT, hierarchical matrices, and tensor approximations to achieve computational efficiency. The analysis includes examples of covariance functions and the associated computational complexity of different approximation techniques.
Overview of sparse and low-rank matrix / tensor techniques
Overview of sparse and low-rank matrix / tensor techniques Alexander Litvinenko The document presents an overview of low-rank and sparse techniques in spatial statistics and parameter identification, highlighting their applications in predicting environmental variables. It addresses key problems such as predicting temperature and salinity, improving statistical models for moisture, and estimating covariance parameters. The document also outlines various statistical tasks and the techniques available, including sparse matrices and low-rank methods.
New data structures and algorithms for \\post-processing large data sets and ...
New data structures and algorithms for \\post-processing large data sets and ...Alexander Litvinenko The document discusses new data structures and algorithms for post-processing large datasets and multivariate functions in spatio-temporal statistics, focusing on tensor approximations. It outlines various statistical tasks involving low-rank tensor formats, including covariance approximation, kriging, and computing joint Gaussian log-likelihood functions. The research showcases methods for managing high-dimensional data effectively using tensor analysis and emphasizes their numerical efficiency compared to traditional matrix methods.
Identification of unknown parameters and prediction with hierarchical matrice...
Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko The document presents a method for identifying unknown parameters and making predictions using hierarchical matrices, highlighting a case study of reducing storage costs for a large covariance matrix. It demonstrates the maximum likelihood estimation of parameters through Gaussian log-likelihood functions and evaluates the effectiveness of this method in comparison to traditional machine learning approaches. The document also outlines the tools, error analysis, and structure of the discussion to improve statistical models and data analysis.
Application of parallel hierarchical matrices for parameter inference and pre...
Application of parallel hierarchical matrices for parameter inference and pre...Alexander Litvinenko The document presents a parallel hierarchical matrix technique to efficiently approximate large covariance matrices, likelihood functions, and to identify parameters for statistical models leveraging Gaussian random fields. The authors aim to significantly reduce storage costs and improve computational speed for matrix operations associated with large datasets, presenting a structured approach for parameter identification and prediction. They utilize tools such as hierarchical matrices and a Matérn covariance function to conduct their analysis and provide comparison with machine learning methods.
Hierarchical matrices for approximating large covariance matries and computin...
Hierarchical matrices for approximating large covariance matries and computin...Alexander Litvinenko This document discusses the application of data sparse approximation techniques for solving stochastic partial differential equations (SPDE). Key methods include Karhunen-Loève expansion, fast Fourier transformation, and hierarchical matrices, aimed at efficiently computing eigenvalues and approximating covariance functions. The conclusions highlight the potential of covariance matrices in enabling data sparse approximations and outline future research directions in this domain.
Identification of unknown parameters and prediction of missing values. Compar...
Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko The document discusses methods for identifying unknown parameters and predicting missing values in large datasets, focusing on maximizing the Gaussian log-likelihood using tools like hierarchical matrices. It showcases the ability to significantly reduce storage costs and approximate key matrix operations on modern computers. The presentation includes an analysis of the approach's effectiveness compared to machine learning methods for parameter estimation and predictions at new locations.
Bounded Approaches in Radio Labeling Square Grids -- Dev Ananda
Bounded Approaches in Radio Labeling Square Grids -- Dev AnandaDev Ananda This thesis presents research on determining the radio number of square grid graphs through bounded approaches. Section 1 provides background, defining key terms like radio labeling, span, and radio number. It also reviews previous work, including Jiang's proof of the radio number for all grids. Section 2 establishes an upper bound on the radio number for odd and even grids by constructing optimal radio labelings. Section 3 provides a lower bound through distance maximization and labeling techniques. Section 4 closes the gap between bounds, determining the radio number of even grids using bounded approaches. Future research directions are also discussed.
Application of hierarchical matrices for partial inverse
Application of hierarchical matrices for partial inverseAlexander Litvinenko The document discusses the application of hierarchical matrices for solving elliptic boundary value problems using hierarchical domain decomposition (HDD). It outlines problem setup, computational resources, and the methodology for constructing mappings necessary for the solution process, emphasizing the efficiency of the HDD approach. Additionally, it integrates the h-matrix arithmetic framework to facilitate the computations involved in solving such problems.
Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...
Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...Alexander Litvinenko The document discusses the application of parallel hierarchical matrices (h-matrices) in enhancing spatial statistics and parameter identification, particularly for moisture prediction in agricultural and defense contexts. It outlines the motivation, tools, implementation, and methodologies used to balance algebraic and statistical errors while estimating unknown parameters from large sets of moisture data. Key findings include the effectiveness of h-matrices in reducing computational costs and improving evaluation accuracy in statistical modeling.
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...Alexander Litvinenko The document discusses the application of hierarchical matrices (h-matrices) in solving multiscale problems, particularly in areas like integral equations and domain decomposition. It details methodologies including hierarchical domain decomposition (HDD) and the complexities involved in numerical solutions. Various mathematical frameworks and examples illustrate the effectiveness of h-matrices in addressing issues in multiscale modeling.
Tucker tensor analysis of Matern functions in spatial statistics
Tucker tensor analysis of Matern functions in spatial statistics Alexander Litvinenko The document presents a comprehensive analysis of Tucker tensor methods applied to Matérn covariance functions in spatial statistics. It discusses the motivation for low-rank tensor approximations, statistical operations using Tucker tensor format, and numerical experiments demonstrating these methodologies. Key statistical tasks and techniques, including Gaussian fields and approximation of covariance functions, are detailed to illustrate the effectiveness of tensor-based approaches in managing spatial data.
Approx
Approxguest0264d3b The document discusses approximation algorithms for NP-hard optimization problems. It provides examples of approximation algorithms for problems like set cover, vertex cover, traveling salesman problem (TSP), and knapsack. For set cover, it shows that a greedy algorithm provides a (1+ln n)-approximation. For vertex cover and TSP, it describes 2-approximation algorithms. It also presents a fully polynomial-time approximation scheme (FPTAS) for knapsack that provides a solution within (1-eps) of optimal.
Lecture5
Lecture5Atner Yegorov This lecture discusses dimensionality reduction techniques for big data, specifically the Johnson-Lindenstrauss lemma. It introduces linear sketching as a dimensionality reduction method from n dimensions to t dimensions (where t is logarithmic in n). It then proves the JL lemma, which shows that for t proportional to 1/ε^2, the l2 distances between points are preserved to within a 1±ε factor. As an application, it discusses locality sensitive hashing (LSH) for approximate nearest neighbor search, where points close in distance hash to the same bucket with high probability.
A Factor Graph Approach To Constrained Optimization
A Factor Graph Approach To Constrained OptimizationWendy Berg This thesis presents a framework for solving constrained optimization problems using factor graphs and the GTSAM toolbox. It demonstrates using factor graphs and the active set method to solve linear and quadratic programs with inequality constraints. It also implements a line search method for sequential quadratic programming to solve nonlinear equality constrained problems. Representing optimization problems as factor graphs allows them to be solved as a series of sparse least squares problems. This provides an open-source way to approach constrained optimization problems commonly found in robotics applications such as control and planning.
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Alexander Litvinenko
New data structures and algorithms for \\post-processing large data sets and ...
New data structures and algorithms for \\post-processing large data sets and ...Alexander Litvinenko
Identification of unknown parameters and prediction with hierarchical matrice...
Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko
Application of parallel hierarchical matrices for parameter inference and pre...
Application of parallel hierarchical matrices for parameter inference and pre...Alexander Litvinenko
Hierarchical matrices for approximating large covariance matries and computin...
Hierarchical matrices for approximating large covariance matries and computin...Alexander Litvinenko
Identification of unknown parameters and prediction of missing values. Compar...
Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko
Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...
Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...Alexander Litvinenko
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...Alexander Litvinenko
Ad
More from Alexander Litvinenko (20)
Poster_density_driven_with_fracture_MLMC.pdf
Poster_density_driven_with_fracture_MLMC.pdfAlexander Litvinenko The document explores density-driven flow in fractured porous media, particularly focusing on saltwater intrusion and its uncertainty quantification using a multi-level Monte Carlo method. It addresses key questions regarding the impact of fractures on flow, the sources and variations of uncertainty over time, and provides a detailed analysis of the governing equations and numerical methods used. Results include mean values, variances, and convergence rates related to the salt mass fraction over time in the context of the Henry-like problem.
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdfAlexander Litvinenko The document discusses uncertainty quantification in coastal aquifers using a Multi-Level Monte Carlo (MLMC) method to address Henry's problem involving saltwater intrusion. It covers the modeling of uncertainties in porosity, permeability, and recharge through random fields and outlines the efficiency of MLMC compared to traditional Monte Carlo methods, highlighting its significantly faster computation times. Future work aims to incorporate more complex scenarios and experimental data to improve accuracy in predictions.
litvinenko_Intrusion_Bari_2023.pdf
litvinenko_Intrusion_Bari_2023.pdfAlexander Litvinenko The document discusses uncertainty quantification in coastal aquifers using the multi-level Monte Carlo (MLMC) method to address saltwater intrusion problems, focusing on input uncertainties such as porosity, permeability, and recharge. It presents the mathematical modeling of two-phase subsurface flow and various computational methods employed, highlighting the efficiency of MLMC compared to the traditional Monte Carlo method. The results indicate that MLMC could be significantly faster, achieving time reductions by up to 3200 times while addressing the challenges of time dependence in modeling.
Density Driven Groundwater Flow with Uncertain Porosity and Permeability
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko The document discusses a model for density-driven groundwater flow that incorporates input uncertainties in porosity and permeability. It employs polynomial chaos expansion and a parallel multigrid solver to forecast pollution maps, assess risks, and analyze uncertainty over time. The findings are illustrated through numerical experiments and simulations that track the evolution of salt concentration in the groundwater system.
litvinenko_Gamm2023.pdf
litvinenko_Gamm2023.pdfAlexander Litvinenko The document presents an analysis of uncertainty quantification in coastal aquifers affected by saltwater intrusion using a multi-level Monte Carlo method. It addresses several key questions, including the longevity of wells, sources of uncertainty, and the time evolution of freshwater exceedance probabilities. The findings suggest that this approach can significantly enhance computational efficiency, demonstrating that the multi-level method can be up to 3200 times faster than traditional Monte Carlo methods.
Litvinenko_Poster_Henry_22May.pdf
Litvinenko_Poster_Henry_22May.pdfAlexander Litvinenko The document discusses the modeling of saltwater intrusion in coastal aquifers using a multi-level Monte Carlo method to quantify uncertainties in porosity, permeability, and recharge. It addresses key questions such as the lifespan of wells and the identification of the largest uncertainties while providing mathematical formulations of the problem. The methodology aims to forecast the evolution of salt mass fractions and freshwater availability under uncertain conditions.
Uncertain_Henry_problem-poster.pdf
Uncertain_Henry_problem-poster.pdfAlexander Litvinenko The document presents a study on saltwater intrusion in coastal aquifers using a multi-level Monte Carlo method to address uncertainties in porosity, permeability, and recharge. It explores key questions regarding the longevity of water wells, uncertainty locations, exceedance probabilities, and scenario variations under time-dependent conditions. The authors utilize advanced computational methods and numerical solvers to analyze the impacts of these uncertainties on salt mass fractions and freshwater availability.
Litvinenko_RWTH_UQ_Seminar_talk.pdf
Litvinenko_RWTH_UQ_Seminar_talk.pdfAlexander Litvinenko The document discusses methods for computing f-divergences and distances of high-dimensional probability density functions (PDFs), focusing on when PDFs are unknown. It includes the theoretical background, algorithms for computation, applications in stochastic partial differential equations, and various methods such as tensor formats and numerical techniques. The document also presents discrete approximations for divergences, computational algorithms, and iterative methods for evaluating functions of tensors.
Litv_Denmark_Weak_Supervised_Learning.pdf
Litv_Denmark_Weak_Supervised_Learning.pdfAlexander Litvinenko This document proposes a method for weakly supervised regression on uncertain datasets. It combines graph Laplacian regularization and cluster ensemble methodology. The method solves an auxiliary minimization problem to determine the optimal solution for predicting uncertain parameters. It is tested on artificial data to predict target values using a mixture of normal distributions with labeled, inaccurately labeled, and unlabeled samples. The method is shown to outperform a simplified version by reducing mean Wasserstein distance between predicted and true values.
Computing f-Divergences and Distances of High-Dimensional Probability Density...
Computing f-Divergences and Distances of High-Dimensional Probability Density...Alexander Litvinenko The document discusses methods for computing f-divergences and distances of high-dimensional probability density functions (pdfs), particularly when pdfs are not directly available. It presents a connection between probability characteristic functions and pdfs, alongside a low-rank tensor representation for efficient calculations. The research illustrates how numerical computations can reduce complexity and storage costs significantly, supported by various examples and acknowledgments for funding.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko This document discusses the computation of f-divergences and distances for high-dimensional probability density functions (PDFs) using techniques from stochastic analysis, such as Karhunen-Loève and polynomial chaos expansions. It highlights methods for calculating Kullback-Leibler divergence and other entropy measures when PDFs are either unknown or do not exist. Additionally, it presents various algorithms, tensor representations, and numerical examples to demonstrate the computational efficiency and effectiveness of the approaches presented.
Low rank tensor approximation of probability density and characteristic funct...
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko This document describes using the Continuation Multi-Level Monte Carlo (CMLMC) method to compute electromagnetic fields scattered from dielectric objects of uncertain shapes. CMLMC optimally balances statistical and discretization errors using fewer samples on fine meshes and more on coarse meshes. The method is tested by computing scattering cross sections for randomly perturbed spheres under plane wave excitation and comparing results to the unperturbed sphere. Computational costs and errors are analyzed to demonstrate the efficiency of CMLMC for this scattering problem with uncertain geometry.
Low-rank tensor approximation (Introduction)
Low-rank tensor approximation (Introduction)Alexander Litvinenko The document discusses low-rank tensor approximations aimed at reducing computational time and memory usage while extracting knowledge from high-dimensional datasets. It outlines historical developments, applications in fields like quantum mechanics and machine learning, and the importance of tensor formats and algorithms in handling complex data. Additionally, it highlights the challenges of high-dimensional data, such as the curse of dimensionality and presents various examples of tensor applications in scientific computing.
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko The document presents the application of the Coarse Multi-Level Monte Carlo (CMLMC) method for efficiently computing electromagnetic fields scattered from dielectric objects with uncertain shapes. It outlines the steps for solving the scattering problem, utilizing advanced statistical methods to reduce computational costs while maintaining accuracy. The study demonstrates that CMLMC can achieve significant speed improvements over traditional Monte Carlo methods, specifically in characterizing scattering phenomena in electromagnetics, optics, and photonics.
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko The document discusses the development of numerical methods for predicting electromagnetic scattering from dielectric objects of uncertain shapes using a computational approach called continuation multilevel Monte Carlo (CMLMC). This method optimizes the balance between statistical and discretization errors, demonstrating significant computational efficiency compared to traditional Monte Carlo methods. Through various numerical tests and methodologies outlined, the authors show that their CMLMC approach can achieve faster and more accurate results in electromagnetic wave scattering analysis.
Propagation of Uncertainties in Density Driven Groundwater Flow
Propagation of Uncertainties in Density Driven Groundwater FlowAlexander Litvinenko This document discusses the estimation of risks related to pollution in subsurface groundwater flow, emphasizing the impact of uncertainties in hydraulic parameters such as porosity and permeability. The study employs stochastic methods to model these uncertainties and analyzes their effects through numerical experiments in both 2D and 3D settings. Key findings highlight the propagation of uncertainty into pollution forecasts and the use of surrogate models for approximating concentration distributions over time.
Simulation of propagation of uncertainties in density-driven groundwater flow
Simulation of propagation of uncertainties in density-driven groundwater flowAlexander Litvinenko The document presents a study on the simulation of uncertainties in density-driven groundwater flow, highlighting stochastic modeling and numerical results. It discusses the mathematical framework for density-driven flow in porous media, emphasizing parameters like porosity and permeability. The study also explores the implications of these uncertainties on real-world applications such as seawater intrusion and contamination propagation.
Semi-Supervised Regression using Cluster Ensemble
Semi-Supervised Regression using Cluster EnsembleAlexander Litvinenko This document summarizes a semi-supervised regression method that combines graph Laplacian regularization with cluster ensemble methodology. It proposes using a weighted averaged co-association matrix from the cluster ensemble as the similarity matrix in graph Laplacian regularization. The method (SSR-LRCM) finds a low-rank approximation of the co-association matrix to efficiently solve the regression problem. Experimental results on synthetic and real-world datasets show SSR-LRCM achieves significantly better prediction accuracy than an alternative method, while also having lower computational costs for large datasets. Future work will explore using a hierarchical matrix approximation instead of low-rank.
Talk Alexander Litvinenko on SIAM GS Conference in Houston
Talk Alexander Litvinenko on SIAM GS Conference in HoustonAlexander Litvinenko This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.
Computing f-Divergences and Distances of High-Dimensional Probability Density...
Computing f-Divergences and Distances of High-Dimensional Probability Density...Alexander Litvinenko
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Low rank tensor approximation of probability density and characteristic funct...
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
Recently uploaded (20)
Romanticism in Love and Sacrifice An Analysis of Oscar Wilde’s The Nightingal...
Romanticism in Love and Sacrifice An Analysis of Oscar Wilde’s The Nightingal...KaryanaTantri21 The story revolves around a college student who despairs not having a red rose as a condition for dancing with the girl he loves. The nightingale hears his complaint and offers to create the red rose at the cost of his life. He sang a love song all night with his chest stuck to the thorns of the rose tree. Finally, the red rose grew, but his sacrifice was in vain. The girl rejected the flower because it didn’t match her outfit and preferred a jewellery gift. The student threw the flower on the street and returned to studying philosophy
THE PSYCHOANALYTIC OF THE BLACK CAT BY EDGAR ALLAN POE (1).pdf
THE PSYCHOANALYTIC OF THE BLACK CAT BY EDGAR ALLAN POE (1).pdfnabilahk908 Psychoanalytic Analysis of The Black Cat by Edgar Allan Poe explores the deep psychological dimensions of the narrator’s disturbed mind through the lens of Sigmund Freud’s psychoanalytic theory. According to Freud (1923), the human psyche is structured into three components: the Id, which contains primitive and unconscious desires; the Ego, which operates on the reality principle and mediates between the Id and the external world; and the Superego, which reflects internalized moral standards.
In this story, Poe presents a narrator who experiences a psychological breakdown triggered by repressed guilt, aggression, and internal conflict. This analysis focuses not only on the gothic horror elements of the narrative but also on the narrator’s mental instability and emotional repression, demonstrating how the imbalance of these three psychic forces contributes to his downfall.
How to Manage Different Customer Addresses in Odoo 18 Accounting
How to Manage Different Customer Addresses in Odoo 18 AccountingCeline George A business often have customers with multiple locations such as office, warehouse, home addresses and this feature allows us to associate with different addresses with each customer streamlining the process of creating sales order invoices and delivery orders.
Learning Styles Inventory for Senior High School Students
Learning Styles Inventory for Senior High School StudentsThelma Villaflores Learning styles inventory, learning strategies, visual learners, auditory learners, kinesthetic learners, tactile learners, read/write learners
June 2025 Progress Update With Board Call_In process.pptx
June 2025 Progress Update With Board Call_In process.pptxInternational Society of Service Innovation Professionals ---
June 25 ISSIP Event - slides in process
20250618 PPre-Event Presentation Summary - Progress Update with Board Series June 25
ISSIP Website Upcoming Events Description: https://issip.org/event/semi-annual-issip-progress-call/
Register here (even if you cannot attend live online, all who register will get link to recording and slides post-event): https://docs.google.com/forms/d/e/1FAIpQLSdThrop1rafOCo4PQkYiS2XApclJuMjYONEHRMGBsceRdcQqg/viewform
This pre-event presentation: https://www.slideshare.net/slideshow/june-2025-progress-update-with-board-call_in-process-pptx/280718770
This pre-event recording: https://youtu.be/Shjgd5o488o
---
University of Ghana Cracks Down on Misconduct: Over 100 Students Sanctioned
University of Ghana Cracks Down on Misconduct: Over 100 Students SanctionedKweku Zurek University of Ghana Cracks Down on Misconduct: Over 100 Students Sanctioned
How to use search fetch method in Odoo 18
How to use search fetch method in Odoo 18Celine George The search_fetch is a powerful ORM method used in Odoo for some specific addons to combine the functionality of search and read for more efficient data fetching. It might be used to search for records and fetch specific fields in a single call. It stores the result in the cache memory.
LDMMIA Shop & Student News Summer Solstice 25
LDMMIA Shop & Student News Summer Solstice 25LDM & Mia eStudios 6/18/25
Shop, Upcoming: Final Notes to Review as we Close Level One. Make sure to review the orientation and videos as well. There’s more to come and material to cover in Levels 2-3. The content will be a combination of Reiki and Yoga. Also energy topics of our spiritual collective.
Thanks again all future Practitioner Level Students. Our Levels so far are: Guest, Grad, and Practitioner. We have had over 5k Spring Views.
https://ldm-mia.creator-spring.com
IIT KGP Quiz Week 2024 Sports Quiz (Prelims + Finals)
IIT KGP Quiz Week 2024 Sports Quiz (Prelims + Finals)IIT Kharagpur Quiz Club The document outlines the format for the Sports Quiz at Quiz Week 2024, covering various sports & games and requiring participants to Answer without external sources. It includes specific details about question types, scoring, and examples of quiz questions. The document emphasizes fair play and enjoyment of the quiz experience.
SCHIZOPHRENIA OTHER PSYCHOTIC DISORDER LIKE Persistent delusion/Capgras syndr...
SCHIZOPHRENIA OTHER PSYCHOTIC DISORDER LIKE Persistent delusion/Capgras syndr...parmarjuli1412 SCHIZOPHRENIA INCLUDED TOPIC IS INTRODUCTION, DEFINITION OF GENERAL TERM IN PSYCHIATRIC, THEN DIFINITION OF SCHIZOPHRENIA, EPIDERMIOLOGY, ETIOLOGICAL FACTORS, CLINICAL FEATURE(SIGN AND SYMPTOMS OF SCHIZOPHRENIA), CLINICAL TYPES OF SCHIZOPHRENIA, DIAGNOSIS, INVESTIGATION, TREATMENT MODALITIES(PHARMACOLOGICAL MANAGEMENT, PSYCHOTHERAPY, ECT, PSYCHO-SOCIO-REHABILITATION), NURSING MANAGEMENT(ASSESSMENT,DIAGNOSIS,NURSING INTERVENTION,AND EVALUATION), OTHER PSYCHOTIC DISORDER LIKE Persistent delusion/Capgras syndrome(The Delusion of Doubles)/Acute and Transient Psychotic Disorders/Induced Delusional Disorders/Schizoaffective Disorder /CAPGRAS SYNDROME(DELUSION OF DOUBLE), GERIATRIC CONSIDERATION, FOLLOW UP, HOMECARE AND REHABILITATION OF THE PATIENT,
ECONOMICS, DISASTER MANAGEMENT, ROAD SAFETY - STUDY MATERIAL [10TH]
ECONOMICS, DISASTER MANAGEMENT, ROAD SAFETY - STUDY MATERIAL [10TH]SHERAZ AHMAD LONE This study material for Class 10th covers the core subjects of Economics, Disaster Management, and Road Safety Education, developed strictly in line with the JKBOSE textbook. It presents the content in a simplified, structured, and student-friendly format, ensuring clarity in concepts. The material includes reframed explanations, flowcharts, infographics, and key point summaries to support better understanding and retention. Designed for classroom teaching and exam preparation, it aims to enhance comprehension, critical thinking, and practical awareness among students.
How to Customize Quotation Layouts in Odoo 18
How to Customize Quotation Layouts in Odoo 18Celine George Customizing quotation layouts in Odoo 18 allows businesses to personalize their quotations to match branding or specific requirements. This can include adding logos, custom fields, or modifying headers and footers.
LAZY SUNDAY QUIZ "A GENERAL QUIZ" JUNE 2025 SMC QUIZ CLUB, SILCHAR MEDICAL CO...
LAZY SUNDAY QUIZ "A GENERAL QUIZ" JUNE 2025 SMC QUIZ CLUB, SILCHAR MEDICAL CO...Ultimatewinner0342 🧠 Lazy Sunday Quiz | General Knowledge Trivia by SMC Quiz Club – Silchar Medical College
Presenting the Lazy Sunday Quiz, a fun and thought-provoking general knowledge quiz created by the SMC Quiz Club of Silchar Medical College & Hospital (SMCH). This quiz is designed for casual learners, quiz enthusiasts, and competitive teams looking for a diverse, engaging set of questions with clean visuals and smart clues.
🎯 What is the Lazy Sunday Quiz?
The Lazy Sunday Quiz is a light-hearted yet intellectually rewarding quiz session held under the SMC Quiz Club banner. It’s a general quiz covering a mix of current affairs, pop culture, history, India, sports, medicine, science, and more.
Whether you’re hosting a quiz event, preparing a session for students, or just looking for quality trivia to enjoy with friends, this PowerPoint deck is perfect for you.
📋 Quiz Format & Structure
Total Questions: ~50
Types: MCQs, one-liners, image-based, visual connects, lateral thinking
Rounds: Warm-up, Main Quiz, Visual Round, Connects (optional bonus)
Design: Simple, clear slides with answer explanations included
Tools Needed: Just a projector or screen – ready to use!
🧠 Who Is It For?
College quiz clubs
School or medical students
Teachers or faculty for classroom engagement
Event organizers needing quiz content
Quizzers preparing for competitions
Freelancers building quiz portfolios
💡 Why Use This Quiz?
Ready-made, high-quality content
Curated with lateral thinking and storytelling in mind
Covers both academic and pop culture topics
Designed by a quizzer with real event experience
Usable in inter-college fests, informal quizzes, or Sunday brain workouts
📚 About the Creators
This quiz has been created by Rana Mayank Pratap, an MBBS student and quizmaster at SMC Quiz Club, Silchar Medical College. The club aims to promote a culture of curiosity and smart thinking through weekly and monthly quiz events.
🔍 SEO Tags:
quiz, general knowledge quiz, trivia quiz, SlideShare quiz, college quiz, fun quiz, medical college quiz, India quiz, pop culture quiz, visual quiz, MCQ quiz, connect quiz, science quiz, current affairs quiz, SMC Quiz Club, Silchar Medical College
📣 Reuse & Credit
You’re free to use or adapt this quiz for your own events or sessions with credit to:
SMC Quiz Club – Silchar Medical College & Hospital
Curated by: Rana Mayank Pratap
YSPH VMOC Special Report - Measles Outbreak Southwest US 6-14-2025.pptx
YSPH VMOC Special Report - Measles Outbreak Southwest US 6-14-2025.pptxYale School of Public Health - The Virtual Medical Operations Center (VMOC) BLUF:
The Texas outbreak has slowed down, but sporadic cases continue to emerge in Kansas, Oklahoma, and New Mexico.
Elsewhere in the US, we continue to see signs of acceleration due to outbreaks outside the Southwest (North Dakota, Montana, and Colorado) and travel-related cases. Measles exposures due to travel are expected to pose a significant challenge throughout the summer.
The U.S. is on track to exceed its 30-year high for measles cases (1,274) within the next two weeks.
Here is the latest update:
CURRENT CASE COUNT: 919
•Texas: 744 (+2) (55% of cases are in Gaines County).
•New Mexico: 81 (83% of cases are from Lea County).
•Oklahoma: 20 (+2)
•Kansas: 74 (+5) (38.89% of the cases are from Gray County).
HOSPITALIZATIONS: 104
• Texas: 96 (+2) – This accounts for 13% of all cases in Texas.
• New Mexico: 7 – This accounts for 9.47% of all cases in New Mexico.
• Kansas: 3 – This accounts for 5.08% of all cases in the state of Kansas.
DEATHS: 3
•Texas: 2 – This is 0.27% of all cases in Texas.
•New Mexico: 1 – This is 1.23% of all cases in New Mexico.
US NATIONAL CASE COUNT: 1,197
INTERNATIONAL SPREAD
•Mexico: 2337 (+257), 5 fatalities
‒Chihuahua, Mexico: 2,179 (+239) cases, 4 fatalities, 7 currently hospitalized.
•Canada: 3,207 (+208), 1 fatality
‒Ontario Outbreak, Canada: 2,115 (+74) cases, 158 hospitalizations, 1 fatality.
‒Alberta, Canada: 879(+118) cases, 5 currently hospitalized.
Peer Teaching Observations During School Internship
Peer Teaching Observations During School InternshipAjayaMohanty7 FOR B.ED,M.ED,M.A.EDUCATION AND ANY STUDENT OF TEACHER EDUCATION
OBSESSIVE COMPULSIVE DISORDER.pptx IN 5TH SEMESTER B.SC NURSING, 2ND YEAR GNM...
OBSESSIVE COMPULSIVE DISORDER.pptx IN 5TH SEMESTER B.SC NURSING, 2ND YEAR GNM...parmarjuli1412 OBSESSIVE COMPULSIVE DISORDER INCLUDED TOPICS ARE INTRODUCTION, DEFINITION OF OBSESSION, DEFINITION OF COMPULSION, MEANING OF OBSESSION AND COMPULSION, DEFINITION OF OBSESSIVE COMPULSIVE DISORDER, EPIDERMIOLOGY OF OCD, ETIOLOGICAL FACTORS OF OCD, CLINICAL SIGN AND SYMPTOMS OF OBSESSION AND COMPULSION, MANAGEMENT INCLUDED PHARMACOTHERAPY(ANTIDEPRESSANT DRUG+ANXIOLYTIC DRUGS), PSYCHOTHERAPY, NURSING MANAGEMENT(ASSESSMENT+DIAGNOSIS+NURSING INTERVENTION+EVALUATION))
K12 Tableau User Group virtual event June 18, 2025
K12 Tableau User Group virtual event June 18, 2025dogden2 National K12 Tableau User Group: June 2025 meeting slides
June 2025 Progress Update With Board Call_In process.pptx
June 2025 Progress Update With Board Call_In process.pptxInternational Society of Service Innovation Professionals
YSPH VMOC Special Report - Measles Outbreak Southwest US 6-14-2025.pptx
YSPH VMOC Special Report - Measles Outbreak Southwest US 6-14-2025.pptxYale School of Public Health - The Virtual Medical Operations Center (VMOC)
Hierarchical matrix approximation of large covariance matrices
- 1. Center for Uncertainty Quantification Hierarchical matrix approximation of large covariance matrices A. Litvinenko1 , M. Genton2 , Ying Sun2 , R. Tempone 1 SRI-UQ Center and 2 Spatio-Temporal Statistics & Data Analysis Group at KAUST [email protected] Center for Uncertainty Quantification Center for Uncertainty Quantification Abstract We approximate large non-structured covariance ma- trices in the H-matrix format with a log-linear com- putational cost and storage O(n log n). We compute inverse, Cholesky decomposition and determinant in H-format. As an example we consider the class of Matern covariance functions, which are very popu- lar in spatial statistics, geostatistics, machine learning and image analysis. Applications are: kriging and op- timal design 1. Matern covariance C(x, y) = C(|x−y|) = σ2 1 Γ(ν)2ν−1 √ 2ν r L ν Kν √ 2ν r L , where Γ is the gamma function, Kν is the modified Bessel function of the second kind, r = |x − y| and ν, L are non-negative parameters of the covariance. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 Matern covariance (nu=1) σ=0.5, l=0.5 σ=0.5, l=0.3 σ=0.5, l=0.2 σ=0.5, l=0.1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 nu=0.15 nu=0.3 nu=0.5 nu=1 nu=2 nu=30 As ν → ∞ [4], C(r) = σ2 exp(−r2 /2L2 ). When ν = 0.5, the Matern covariance is identical to the exponential covariance function. Cν=3/2(r) = 1 + √ 3r L exp − √ 3r L Cν=5/2(r) = 1 + √ 5r L + 5r2 3L2 exp − √ 5r L . Note: no need to assume neither C(x, y) = C(|x − y|) nor tensor grid. 2. H-matrix approximation 25 20 20 20 20 16 20 16 20 20 16 16 20 16 16 16 19 20 20 19 32 19 19 16 16 32 19 20 20 19 19 16 19 16 32 32 20 20 20 20 32 32 32 20 19 19 19 32 20 19 16 16 32 32 20 32 32 20 32 32 32 32 20 32 32 20 19 19 19 20 16 19 16 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 20 19 20 20 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 20 20 19 20 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 20 20 20 20 32 32 32 20 19 20 19 32 32 32 20 20 19 19 32 32 32 20 20 20 20 32 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 20 20 2019 20 20 20 32 32 32 32 20 32 32 20 32 32 32 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 20 20 20 20 20 20 20 20 19 20 20 20 32 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 20 20 20 20 20 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 19 19 20 20 32 32 32 32 20 32 32 20 32 32 32 32 20 32 32 19 20 19 20 32 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 32 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 32 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 32 32 32 20 32 20 32 20 32 32 32 20 32 20 32 32 20 32 32 32 20 32 32 32 20 20 32 32 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 25 9 9 20 9 9 20 7 7 16 9 9 20 9 9 20 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 Figure 2: Two approximation strategies [1]: fixed rank (left) and flexible rank (right) approximations, C ∈ Rn×n, n = 652. I I I I I I I I I I1 1 2 2 11 12 21 22 I11 I12 I21 I22 Q Qt S dist H= t s 1. Build cluster tree TI, I = {1, 2, ..., n} 2. Build block cluster tree TI×I 3. For each (t × s) ∈ TI×I, t, s ∈ TI, check admissibility condition min{diam(Qt), diam(Qs)} ≤ η·dist(Qt, Qs). if(adm=true) then M|t×s is a rank-k matrix block if(adm=false) then divide M|t×s further or define as a dense matrix block, if small enough. Grid → cluster tree (TI) + admissibility condition → block cluster tree (TI×I) → H-matrix → H-matrix arith- metics. Operation Sequential Complexity Parallel Complexity (Hackbusch et al. ’99-’06) (Kriemann ’05) storage(M) N = O(kn log n) N P Mx N = O(kn log n) N P M1 ⊕ M2 N = O(k2n log n) N P M1 M2, M−1 N = O(k2n log2 n) N P + O(n) H-LU N = O(k2n log2 n) N P + O(k2 n log2 n n1/d ) Table 1: Computational cost of H-matrix arithmetics, sequential and parallel. Let ε = (C−CH )z 2 C 2 z 2 , where z is a random vector. n rank k size, MB t, sec. ε max i=1..10 |λi − ˜λi|, i ε2 for ˜C C ˜C C ˜C 4.0 · 103 10 48 3 0.8 0.08 7 · 10−3 7.0 · 10−2 , 9 2.0 · 10−4 1.05 · 104 18 439 19 7.0 0.4 7 · 10−4 5.5 · 10−2 , 2 1.0 · 10−4 2.1 · 104 25 2054 64 45.0 1.4 1 · 10−5 5.0 · 10−2 , 9 4.4 · 10−6 Table 2: Accuracy of the H-matrix approx. exp. covariance function, l1 = l3 = 0.1, l2 = 0.5. l1 l2 ε 0.01 0.02 3 · 10−2 0.1 0.2 8 · 10−3 0.5 1 2.8 · 10−5 Table 3: Dependence of the H-matrix accuracy on the covari- ance lengths l1 and l2, n = 1292. The smaller cov. length the less accurate is H-approximation. 0 100 200 300 0 50 100 150 200 250 300 −1 0 1 2 −1 −0.5 0 0.5 1 1.5 2 2.5 3 0 100 200 300 0 50 100 150 200 250 300 −1 0 1 2 −3 −2 −1 0 1 2 Figure 4: Two realizations of random field generated via Cholesky decomposition of Matern covariance matrix, ν = 0.4. 3. Kullback-Leibler divergence Measure of the information lost when distribution Q is used to approximate P. DKL(P Q) = i P(i) ln P(i) Q(i) , DKL(P Q) = ∞ −∞ p(x) ln p(x) q(x) dx, where p, q densities of P and Q. For miltivariate normal distribu- tions (µ0, C) and (µ1, CH): 2DKL(N0 N1) = tr((CH )−1 C) + (µ1 − µ0)T (CH )−1 (µ1 − µ0) − n − ln det C det CH . 0 10 20 30 40 50 60 70 80 90 100 −16 −14 −12 −10 −8 −6 −4 −2 0 rank k log(rel.error) Spectral norm, L=0.1, nu=0.5 Frob. norm, L=0.1 Spectral norm, L=0.2 Frob. norm, L=0.2 Spectral norm, L=0.5 Frob. norm, L=0.5 0 10 20 30 40 50 60 70 80 90 100 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 rank k log(rel.error) Spectral norm, L=0.1, ν=1.5 Frob. norm, L=0.1 Spectral norm, L=0.2 Frob. norm, L=0.2 Spectral norm, L=0.5 Frob. norm, L=0.5 Figure 5: Relative H-matrix approx. error C−CH 2 for different cov. lengths L = {0.1, 0.2, 0.5} and ν = {0.5, 1.5} k KLD(C, CH) C − CH 2 C(CH)−1 − I 2 L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75 5 0.51 2.3 4.0e-2 0.1 4.8 63 6 0.34 1.6 9.4e-3 0.02 3.4 22 8 5.3e-2 0.4 1.9e-3 0.003 1.2 8 10 2.6e-3 0.2 7.7e-4 7.0e-4 6.0e-2 3.1 12 5.0e-4 2e-2 9.7e-5 5.6e-5 1.6e-2 0.5 15 1.0e-5 9e-4 2.0e-5 1.1e-5 8.0e-4 0.02 20 4.5e-7 4.8e-5 6.5e-7 2.8e-7 2.1e-5 1.2e-3 50 3.4e-13 5e-12 2.0e-13 2.4e-13 4e-11 2.7e-9 Table 4: Dependence of KLD on H-matrix rank k, Matern co- variance with L = {0.25, 0.75} and ν = 0.5, domain G = [0, 1]2, C(L=0.25,0.75) 2 = {212, 568}. For ν = 1.5 the KLD and the inverse (CH)−1 is hard to compute numerically. Results in Table 4 are better since covariance ma- trix with ν = 0.5 has smallest eigenvalues far enough from zero. The case ν = 1.5 is more smooth, the eigenvalues decay faster, but the smallest eigenvalues come much closer to zero than in ν = 0.5 case. 4. Other applications 4.1 Low-rank approximation of Kriging and geo- statistical optimal design Let ˆs ∈ Rn to be estimated, Css covariance matrix, y ∈ Rm is vector of measurements. The corresponding cross- and auto- covariance matrices are denoted by Csy and Cyy, respectively, sized n × m and m × m. Kriging estimate ˆs = CsyC−1 yy y . The estimation variance ˆσ is the diagonal of the cond. cov. ma- trix Css|y: ˆσs = diag(Css|y) = diag Css − CsyC−1 yy Cys Geostatistical optimal design: φA = n−1 trace Css|y φC = cT Css − CsyC−1 yy Cys c, c − a vector. 4.2 Weather forecast in Europa 180 240 30 60 Figure 6: Europa weather stations (≈ 2500). Collected data set M ∈ R2500×365 . 0 50 100 150 200 250 300 350 400 −20 −15 −10 −5 0 5 10 15 20 Figure 7: Truth temperature forecast and its low-rank approxi- mation (rank 50 approximation of matrix M) in one station, rel. error=25%. 5. Open question 1. Compute the whole spectrum of large covariance matrix 2. Compute KLD for large matrices (det Σ ?) 3. How sensible is KLD to H-matrix accuracy ? 4. Derive/estimate KLD for non-Gaussian distributions. Acknowledgements A. Litvinenko is a member of the KAUST SRI UQ Center. References 1. B. N. Khoromskij, A. Litvinenko, H. G. Matthies, Application of hierarchical matrices for computing the Karhunen?Lo´eve expan- sion, Computing, Vol. 84, Issue 1-2, pp 49-67, 2008 2. R. Furrer, M. Genton, D. Nychka, Covariance tapering for in- terpolation of large spatial datasets, J. Comp. & Graph. Stat., Vol.15, N3, pp502-523. 3. M. Stein, Limitations on low rank approximations for covari- ance matrices of spatial data, Spat. Statistics, 2013 4. J. Castrillion-Candis, M. Genton, R. Yokota, Multi-Level Re- stricted Maximum Likelihood Cov. Estim. and Kriging for Large Non-Gridded Datasets, 2014.