Application of QMC methods to PDEs with random coefficients
|
|
- Bryan Garrett
- 5 years ago
- Views:
Transcription
1 Application of QMC methods to PDEs with random coefficients a survey of analysis and implementation Frances Kuo f.kuo@unsw.edu.au University of New South Wales, Sydney, Australia joint work with Ivan Graham and Rob Scheichl (Bath), Dirk Nuyens (KU Leuven), Christoph Schwab (ETH Zurich), Elizabeth Ullmann (Hamburg), Josef Dick, Thong Le Gia, James Nichols, and Ian Sloan (UNSW) Based on a survey of the same title, written jointly with Dirk Nuyens Frances UNSW Australia p.1
2 Outline Motivating example a flow through a porous medium Quasi-Monte Carlo (QMC) methods Component-by-component (CBC) construction Application of QMC theory to PDEs with random coefficients estimate the norm choose the weights Software Concluding remarks Frances UNSW Australia p.2
3 Motivating example Uncertainty in groundwater flow eg. risk analysis of radwaste disposal or CO 2 sequestration Darcy s law q + a p = f in D R d, d = 1,2,3 mass conservation law q = 0 together with boundary conditions [Cliffe, et. al. (2000)] Uncertainty in a(x, ω) leads to uncertainty in q(x, ω) and p(x, ω) Frances UNSW Australia p.3
4 Expected values of quantities of interest To compute the expected value of some quantity of interest: 1. Generate a number of realizations of the random field (Some approximation may be required) 2. For each realization, solve the PDE using e.g. FEM / FVM / mfem 3. Take the average of all solutions from different realizations This describes Monte Carlo simulation. Example: particle dispersion release point p n = 0 p = 1 p = p n = Frances UNSW Australia p.4
5 Expected values of quantities of interest To compute the expected value of some quantity of interest: 1. Generate a number of realizations of the random field (Some approximation may be required) 2. For each realization, solve the PDE using e.g. FEM / FVM / mfem 3. Take the average of all solutions from different realizations This describes Monte Carlo simulation. NOTE: expected value = (high dimensional) integral 10 1 use quasi-monte Carlo methods 10 2 MC n 1/2 error n 1 or better n QMC s = stochastic dimension Frances UNSW Australia p.4
6 Frances UNSW Australia p.5 MC v.s. QMC in the unit cube [0,1] s F(y)dy 1 n n i=1 F(t i ) Monte Carlo method Quasi-Monte Carlo methods t i random uniform t i deterministic n 1/2 convergence close to n 1 convergence or better random points First 64 points of a 2D Sobol sequence A lattice rule with 64 points more effective for earlier variables and lower-order projections order of variables irrelevant order of variables very important use randomized QMC methods for error estimation
7 Frances UNSW Australia p.6 QMC Two main families of QMC methods: (t,m,s)-nets and (t,s)-sequences lattice rules First 64 points of a 2D Sobol sequence A lattice rule with 64 points (0,6,2)-net Having the right number of points in various sub-cubes A group under addition modulo Z and includes the integer points Niederreiter book (1992) Sloan and Joe book (1994) Important developments: component-by-component (CBC) construction, fast CBC higher order digital nets Dick and Pillichshammer book (2010) Dick, Kuo, Sloan Acta Numerica (2013) Nuyens and Cools (2006) Dick (2008)
8 Application of QMC to PDEs with random coefficients [0] Graham, K., Nuyens, Scheichl, Sloan (J. Comput. Physics, 2011) [1] K., Schwab, Sloan (SIAM J. Numer. Anal., 2012) [2] K., Schwab, Sloan (J. FoCM, 2015) [3] Graham, K., Nichols, Scheichl, Schwab, Sloan (Numer. Math., 2015) [4] K., Scheichl, Schwab, Sloan, Ullmann (in review) [5] Dick, K., Le Gia, Nuyens, Schwab (SIAM J. Numer. Anal., 2014) [6] Dick, K., Le Gia, Schwab (in review) [7] Graham, K., Nuyens, Scheichl, Sloan (in progress) Also Schwab (Proceedings of MCQMC 2012) Le Gia (Proceedings of MCQMC 2012) Dick, Le Gia, Schwab (in review) Harbrecht, Peters, Siebenmorgen (Math. Comp., 2015) Gantner, Schwab (Proceedings of MCQMC 2014) Robbe, Nuyens, Vandewalle (Master thesis of Robbe, 2015) Ganesh, Hawkings (SIAM J. Sci. Comput., 2015) Frances UNSW Australia p.7
9 Application of QMC to PDEs with random coefficients Three different QMC theoretical settings: [1,2] Weighted Sobolev space over [0,1] s and randomly shifted lattice rules [3,4] Weighted space setting in R s and randomly shifted lattice rules [5,6] Weighted space of smooth functions over [0,1] s and (deterministic) interlaced polynomial lattice rules AIM OF THIS TALK: to survey [1,2], [3,4], [5,6] in a unified view Uniform Lognormal Lognormal KL expansion KL expansion Circulant embedding Experiments only [0]* First order, single-level [1] [3]* [7]* First order, multi-level [2] [4]* Higher order, single-level [5] Higher order, multi-level [6]* The * indicates there are accompanying numerical results Frances UNSW Australia p.9
10 5-minute QMC primer Common features among all three QMC theoretical settings: Separation in error bound (rms) QMC error (rms) worst case error γ norm of integrand γ Weighted spaces Pairing of QMC rule with function space Optimal rate of convergence Rate and constant independent of dimension Fast component-by-component (CBC) construction Extensible or embedded rules Application of QMC theory Estimate the norm (critical step) Choose the weights Weights as input to the CBC construction Frances UNSW Australia p.10
11 Application of QMC to PDEs with random coefficients Three different QMC theoretical settings: [1,2] Weighted Sobolev space over [0,1] s and randomly shifted lattice rules [3,4] Weighted space setting in R s and randomly shifted lattice rules [5,6] Weighted space of smooth functions over [0,1] s and (deterministic) interlaced polynomial lattice rules Uniform Lognormal Lognormal KL expansion KL expansion Circulant embedding Experiments only [0]* First order, single-level [1] [3]* [7]* First order, multi-level [2] [4]* Higher order, single-level [5] Higher order, multi-level [6]* The * indicates there are accompanying numerical results Frances UNSW Australia p.11
12 Frances UNSW Australia p.12 Lattice rules Rank-1 lattice rules have points t i = frac ( i n z ), i = 1,2,...,n z Z s the generating vector, with all components coprime to n frac( ) means to take the fractional part of all components A lattice rule with 64 points
13 Lattice rules Rank-1 lattice rules have points ( ) i t i = frac n z, i = 1,2,...,n z Z s the generating vector, with all components coprime to n frac( ) means to take the fractional part of all components quality determined by the choice of z n = 64 z = (1,19) t i = frac ( ) i 64 (1,19) A lattice rule with 64 points Frances UNSW Australia p.12
14 Frances UNSW Australia p.13 Randomly shifted lattice rules Shifted rank-1 lattice rules have points t i = frac ( i n z + ), i = 1,2,...,n [0,1) s the shift shifted by = (0.1,0.3) A lattice rule with 64 points A shifted lattice rule with 64 points use a number of random shifts for error estimation
15 Component-by-component construction Want to find z with (shift-averaged) worst case error as small possible. Exhaustive search is practically impossible - too many choices! CBC algorithm [Sloan, K., Joe (2002);... ] 1. Set z 1 = With z 1 fixed, choose z 2 to minimize the worst case error in 2D. 3. With z 1,z 2 fixed, choose z 3 to minimize the worst case error in 3D. 4. etc. Cost of algorithm is only O(nlogns) using FFTs. [Nuyens, Cools (2006)] Optimal rate of convergence O(n 1+δ ) in weighted Sobolev space, with the implied constant independent of s under an appropriate condition on the weights. [K. (2003); Dick (2004)] Averaging argument: there is always one choice as good as average! Extensible/embedded variants. [Cools, K., Nuyens (2006); Dick, Pillichshammer, Waterhouse (2007)] Frances UNSW Australia p.14
16 Fast CBC construction [Nuyens, Cools (2006)] n = 53 Natural ordering of the indices 1 Generator ordering of the indices 53 Matrix-vector multiplication with a circulant matrix can be done using FFT Images by Dirk Nuyens, KU Leuven Frances UNSW Australia p.15
17 Fast CBC construction [Nuyens, Cools (2006)] n = 128 = Natural ordering of the indices 4 Symmetric reduction after application of B 2 kernel function Grouping on divisors 3 Generator ordering of the indices Images by Dirk Nuyens, KU Leuven Frances UNSW Australia p.16
18 Standard QMC theory today Worst case error bound F(y)dy 1 [0,1] s n n F(t i ) ewor γ (t 1,...,t n ) F γ i=1 Weighted Sobolev space [Sloan, Woźniakowski (1998)] F 2 γ = 1 u 2 F (y u ;0) dy y u u 2 s subsets u {1:s} γ u weights [0,1] u anchor at 0 (also unanchored ) Mixed first derivatives are square integrable Small weight γ u means that F depends weakly on the variables y u Choose weights to minimize the error bound [K., Schwab, Sloan (2012)] ( ) 1/(2λ) ( ) 1/2 2 ( ) γ λ u n a b 1/(1+λ) u bu u γ u = u {1:s} }{{} bound on worst case error (CBC) u {1:s} γ u } {{ } bound on norm Construct points (CBC) to minimize the worst case error a u POD weights Frances UNSW Australia p.17
19 Application of QMC to PDEs with random coefficients Three different QMC theoretical settings: [1,2] Weighted Sobolev space over [0,1] s and randomly shifted lattice rules [3,4] Weighted space setting in R s and randomly shifted lattice rules [5,6] Weighted space of smooth functions over [0,1] s and (deterministic) interlaced polynomial lattice rules Uniform Lognormal Lognormal KL expansion KL expansion Circulant embedding Experiments only [0]* First order, single-level [1] [3]* [7]* First order, multi-level [2] [4]* Higher order, single-level [5] Higher order, multi-level [6]* The * indicates there are accompanying numerical results Frances UNSW Australia p.18
20 QMC theory for integration over R s Change of variables s F(y) φ(y j )dy = R s j=1 [0,1] s F(Φ 1 (w))dw Norm with weight function [Nichols & K. (2014)] F 2 γ = 1 u F (y u ;0) y u u {1:s} γ u R u 2 j u ϕ 2 j (y j)dy u weight function Also unanchored variant Randomly shifted lattice rules CBC error bound for general weights γ u Convergence rate depends on the relationship between φ and ϕ j Fast CBC for POD weights Frances UNSW Australia p.19
21 Application of QMC to PDEs with random coefficients Three different QMC theoretical settings: [1,2] Weighted Sobolev space over [0,1] s and randomly shifted lattice rules [3,4] Weighted space setting in R s and randomly shifted lattice rules [5,6] Weighted space of smooth functions over [0,1] s and (deterministic) interlaced polynomial lattice rules Uniform Lognormal Lognormal KL expansion KL expansion Circulant embedding Experiments only [0]* First order, single-level [1] [3]* [7]* First order, multi-level [2] [4]* Higher order, single-level [5] Higher order, multi-level [6]* The * indicates there are accompanying numerical results Frances UNSW Australia p.20
22 Higher order digital nets [Dick (2008)] Classical polynomial lattice rule [Niederreiter (1992)] n = b m points with prime b An irreducible polynomial with degree m A generating vector of s polynomials with degree < m Interlaced polynomial lattice rule [Goda, Dick (2012)] Interlacing factor α An irreducible polynomial with degree m A generating vector of αs polynomials with degree < m Digit interlacing function D α : [0,1) α [0,1) (0.x 11 x 12 x 13 ) b,(0.x 21 x 22 x 23 ) b,..., (0.x α1 x α2 x α3 ) b becomes (0.x 11 x 21 x α1 x 12 x 22 x α2 x 13 x 23 x α3 ) b Weighted spaces with norm involving higher order mixed derivatives Variants of fast CBC construction available [Dick, K., Le Gia, Nuyens, Schwab (2014)] SPOD weights Frances UNSW Australia p.21
23 Properties of higher order digital nets 16 points obtained from a 4D Sobol sequence with interlacing factor 2: Each rectangle contains exactly 2 points. Frances UNSW Australia p.22
24 Properties of higher order digital nets 16 points obtained from a 4D Sobol sequence with interlacing factor 2: Each rectangle contains exactly 2 points. Frances UNSW Australia p.22
25 Properties of higher order digital nets 16 points obtained from a 4D Sobol sequence with interlacing factor 2: The shaded area contains exactly 2 points. Frances UNSW Australia p.22
26 Properties of higher order digital nets 16 points obtained from a 4D Sobol sequence with interlacing factor 2: The shaded area contains exactly 2 points. Frances UNSW Australia p.22
27 Frances UNSW Australia p.22 Properties of higher order digital nets 16 points obtained from a 4D Sobol sequence with interlacing factor 2: The shaded area contains exactly half of the points.
28 Projections of higher order digital nets Images by Dirk Nuyens, KU Leuven Frances UNSW Australia p.23
29 Outline Motivating example a flow through a porous medium Quasi-Monte Carlo (QMC) methods Component-by-component (CBC) construction Application of QMC theory to PDEs with random coefficients estimate the norm choose the weights Software Concluding remarks Frances UNSW Australia p.24
30 Uniform v.s. lognormal (a(x,y) u(x,y)) = f(x), x D R d, d = 1,2,3 u(x,y) = 0, x D Uniform a(x,y) = a 0 (x) + j 1y j ψ j (x), y j U[ 1 2, 1 2 ] Goal: ( 1 2,1 2 )N G(u(,y))dy G bounded linear functional Key assumption: 0 < a min a(x,y) a max < Lognormal a(x,y) = exp ( a 0 (x) + j 1y ) j ψ j (x), y j N(0,1) Goal: R N G(u(,y)) j 1 φ(y j )dy = (0,1) N G(u(,Φ -1 (w)))dw Key assumption: restrict to y such that j 1 y j ψ j L < Frances UNSW Australia p.25
31 Single-level v.s. multi-level (the uniform case) I = G(u(,y))dy ( 1 2,1 2 )N Single-level algorithm 1 n Deterministic: G(u s h n (,t i 1 2 )) i=1 (1) dimension truncation (2) FE discretization (3) QMC quadrature Multi-level algorithm L ( 1 Deterministic: n l=0 l n l i=1 I = (I I L ) + L l=0 (I l I l 1 ), I 1 := 0 [Heinrich (1998); Giles (2008)] G((u s l h l u s l 1 ) h l 1 )(,t l,i 1 2 )) Frances UNSW Australia p.26
32 Estimate the norm (the uniform case) Single-level algorithm G(u s h ) γ Multi-level algorithm G(u s l h l u s l 1 h l 1 ) γ Frances UNSW Australia p.27
33 Estimate the norm (the uniform case) Single-level algorithm G(u s h ) γ u ( G(u s h y (,y)) u ) = G u s h u y (,y) G H -1 u u s h u y (,y) u linearity and boundedness of G H 1 0 ν u(,y) H 1 0 = ν u(,y) L2 ν!b ν f H-1 a min, b j := ψ j L a min G(u s h ) γ f H-1 G H -1 a min ( u {1:s} ( u!) 2 j u b2 j γ u ) 1/2 Frances UNSW Australia p.27
34 Choose the weights (the uniform case) Single-level algorithm G(u s h ) γ b j := ψ j L a min ms-error ( 2 n u {1:s} γ u λ [ρ(λ)] u ) 1/λ( Minimize the bound POD weights γ u = If j 1 bp j <, take λ = p 2 p The convergence rate is n min(1 δ, 1 p 1 2 ). The implied constant is independent of s. u {1:s} ( u! j u ( u!) 2 j u b2 j Frances UNSW Australia p.28 γ u for all λ ( 1 2,1] ) ) b 2/(1+λ) j ρ(λ) 1 2 2δ for δ (0, 1 2 ) when p (0, 2 3 ], when p ( 2 3,1).
35 Estimate the norm (the uniform case) Single-level algorithm G(u s h ) γ b j := ψ j L a min Lemma 1 ν u(,y) L2 ν!b ν f H-1 a min Multi-level algorithm G(u s l h l u s l 1 h l 1 ) γ Lemma 2 b j := max( ψ j L, ψ j L ) a min ν u(,y) L2 ( ν + 1)!b ν f L2 Lemma 3 ν (u u h )(,y) L2 h( ν + 2)!b ν f L2 Lemma 4 Important: FE projection depends on y ν G((u u )(,y)) h h 2 ( ν + 5)!b ν f L2 G L2 duality argument Frances UNSW Australia p.29
36 Estimate the norm (the lognormal case) Single-level algorithm G(u s h ) γ β j := ψ j L = µ j ξ j L Lemma 5 ( ) β ν -1 ν f H -1 u(,y) L2 ν! ln2 a min (y) induction on a 1/2 (,y) ν u(,y) L2 Multi-level algorithm G(u s l h l u s l 1 h l 1 ) γ Lemma 6 β j := max( ψ j L, ψ j L ) induction on a 1/2 (,y) (a(,y) ν u(,y)) L2 ν u(,y) L2 T(y)( ν + 1)!(2β) ν f L2 Lemma 7 a 1/2 (,y) ν (u u h )(,y) L2 ht(y)a 1/2 max (y)( ν +2)!(2β)ν f L2 Lemma 8 ν G((u u )(,y)) h h 2 T 2 (y)a max (y)( ν +5)!(2β) ν f L2 G L2 Frances UNSW Australia p.30
37 Generic bound on mixed derivative Mixed derivative bound For all multi-indices ν {0,α} s we have ν F(y) ( ( ν + a 1 )! ) d 1 s (a 2 B j ) ν j exp(a 3 B j y j ) j=1 for integers α 1, a 1 0, real numbers a 2 > 0, a 3 0, d 1 0, B j > 0. Particular cases: G(u s h F(y) = ) single-level G(u s h l u s h l 1 ) multi-level B j is b j,b j,β j, or β j a 3 = 0 = uniform case, a 3 > 0 = lognormal case Specify these parameters as input to the construction scripts: Frances UNSW Australia p.31
38 MLQMC numerical results (the lognormal case) D = (0,1) 2, f 1, homogeneous Dirichlet boundary G(u(,y)) = 1 D D u(x,y)dx, D = ( 3 4, 7 8 ) (7 8,1) Piecewise linear finite elements, off-the-shelf randomly shifted lattice rule Matérn kernel: smoothness ν, variance σ 2, correlation length scale λ C 10 4 ν = 2.5, σ2 = 0.25, λ = 1 MC MLMC QMC 10 3 MLQMC 10 4 ν = 1.5, σ2 = 1, λ = 0.1 MC MLMC QMC 10 3 MLQMC Cost (in sec) Cost (in sec) ǫ [K., Scheichl, Schwab, Sloan, Ullmann (in review)] ǫ Frances UNSW Australia p.32
39 Application of QMC to PDEs with random coefficients [0] Graham, K., Nuyens, Scheichl, Sloan (J. Comput. Physics, 2011) [1] K., Schwab, Sloan (SIAM J. Numer. Anal., 2012) [2] K., Schwab, Sloan (J. FoCM, 2015) [3] Graham, K., Nichols, Scheichl, Schwab, Sloan (Numer. Math., 2015) [4] K., Scheichl, Schwab, Sloan, Ullmann (in review) [5] Dick, K., Le Gia, Nuyens, Schwab (SIAM J. Numer. Anal., 2014) [6] Dick, K., Le Gia, Schwab (in review) [7] Graham, K., Nuyens, Scheichl, Sloan (in progress) Uniform Lognormal Lognormal KL expansion KL expansion Circulant embedding Experiments only [0]* First order, single-level [1] [3]* [7]* First order, multi-level [2] [4]* Higher order, single-level [5] Higher order, multi-level [6]* The * indicates there are accompanying numerical results Frances UNSW Australia p.33
40 Concluding remarks QMC error bound can be independent of dimension Three weighted space settings, fast CBC construction Application of QMC theory: estimate the norm, choose the weights Pairing of method and setting: best rate with minimal assumption Theory versus practice Affine parametric operator equations [Schwab (2013)] Special orthogonality Cost reduction: single-level versus multi-level nsh d Circulant embedding nh d log(h d ) Fast QMC matrix-vector multiplication [Dick, K., Le Gia, Schwab (SIAM J. Sci. Comput., 2015)] nlognh d Multi-index methods [Haji-Ali, Nobile, Tempone (Numer. Math., 2015)] Frances UNSW Australia p.34
Hot new directions for QMC research in step with applications
Hot new directions for QMC research in step with applications Frances Kuo f.kuo@unsw.edu.au University of New South Wales, Sydney, Australia Frances Kuo @ UNSW Australia p.1 Outline Motivating example
More informationQuasi-Monte Carlo integration over the Euclidean space and applications
Quasi-Monte Carlo integration over the Euclidean space and applications f.kuo@unsw.edu.au University of New South Wales, Sydney, Australia joint work with James Nichols (UNSW) Journal of Complexity 30
More informationWhat s new in high-dimensional integration? designing for applications
What s new in high-dimensional integration? designing for applications Ian H. Sloan i.sloan@unsw.edu.au The University of New South Wales UNSW Australia ANZIAM NSW/ACT, November, 2015 The theme High dimensional
More informationNew Multilevel Algorithms Based on Quasi-Monte Carlo Point Sets
New Multilevel Based on Quasi-Monte Carlo Point Sets Michael Gnewuch Institut für Informatik Christian-Albrechts-Universität Kiel 1 February 13, 2012 Based on Joint Work with Jan Baldeaux (UTS Sydney)
More informationTutorial on quasi-monte Carlo methods
Tutorial on quasi-monte Carlo methods Josef Dick School of Mathematics and Statistics, UNSW, Sydney, Australia josef.dick@unsw.edu.au Comparison: MCMC, MC, QMC Roughly speaking: Markov chain Monte Carlo
More informationQuasi-Monte Carlo Methods for Applications in Statistics
Quasi-Monte Carlo Methods for Applications in Statistics Weights for QMC in Statistics Vasile Sinescu (UNSW) Weights for QMC in Statistics MCQMC February 2012 1 / 24 Quasi-Monte Carlo Methods for Applications
More informationarxiv: v1 [math.na] 7 Nov 2017
Combining Sparse Grids, Multilevel MC and QMC for Elliptic PDEs with Random Coefficients Michael B. Giles Frances Y. Kuo Ian H. Sloan arxiv:1711.02437v1 [math.na] 7 Nov 2017 Abstract Building on previous
More informationPart III. Quasi Monte Carlo methods 146/349
Part III Quasi Monte Carlo methods 46/349 Outline Quasi Monte Carlo methods 47/349 Quasi Monte Carlo methods Let y = (y,...,y N ) be a vector of independent uniform random variables in Γ = [0,] N, u(y)
More informationFAST QMC MATRIX-VECTOR MULTIPLICATION
FAST QMC MATRIX-VECTOR MULTIPLICATION JOSEF DICK, FRANCES Y. KUO, QUOC T. LE GIA, CHRISTOPH SCHWAB Abstract. Quasi-Monte Carlo (QMC) rules 1/N N 1 n=0 f(y na) can be used to approximate integrals of the
More informationOptimal Randomized Algorithms for Integration on Function Spaces with underlying ANOVA decomposition
Optimal Randomized on Function Spaces with underlying ANOVA decomposition Michael Gnewuch 1 University of Kaiserslautern, Germany October 16, 2013 Based on Joint Work with Jan Baldeaux (UTS Sydney) & Josef
More informationNumerical Analysis of Elliptic PDEs with Random Coefficients
Numerical Analysis of Elliptic PDEs with Random Coefficients (Lecture I) Robert Scheichl Department of Mathematical Sciences University of Bath Workshop on PDEs with Random Coefficients Weierstrass Institute,
More informationLattice rules and sequences for non periodic smooth integrands
Lattice rules and sequences for non periodic smooth integrands dirk.nuyens@cs.kuleuven.be Department of Computer Science KU Leuven, Belgium Joint work with Josef Dick (UNSW) and Friedrich Pillichshammer
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Mike Giles (Oxford) Monte Carlo methods 1 / 23 Lecture outline
More informationAPPLIED MATHEMATICS REPORT AMR04/16 FINITE-ORDER WEIGHTS IMPLY TRACTABILITY OF MULTIVARIATE INTEGRATION. I.H. Sloan, X. Wang and H.
APPLIED MATHEMATICS REPORT AMR04/16 FINITE-ORDER WEIGHTS IMPLY TRACTABILITY OF MULTIVARIATE INTEGRATION I.H. Sloan, X. Wang and H. Wozniakowski Published in Journal of Complexity, Volume 20, Number 1,
More informationTwo new developments in Multilevel Monte Carlo methods
Two new developments in Multilevel Monte Carlo methods Mike Giles Abdul-Lateef Haji-Ali Oliver Sheridan-Methven Mathematical Institute, University of Oxford Rob Scheichl s Farewell Conference University
More informationProgress in high-dimensional numerical integration and its application to stochastic optimization
Progress in high-dimensional numerical integration and its application to stochastic optimization W. Römisch Humboldt-University Berlin Department of Mathematics www.math.hu-berlin.de/~romisch Short course,
More informationFast evaluation of mixed derivatives and calculation of optimal weights for integration. Hernan Leovey
Fast evaluation of mixed derivatives and calculation of optimal weights for integration Humboldt Universität zu Berlin 02.14.2012 MCQMC2012 Tenth International Conference on Monte Carlo and Quasi Monte
More informationOn the existence of higher order polynomial. lattices with large figure of merit ϱ α.
On the existence of higher order polynomial lattices with large figure of merit Josef Dick a, Peter Kritzer b, Friedrich Pillichshammer c, Wolfgang Ch. Schmid b, a School of Mathematics, University of
More informationUNCERTAINTY QUANTIFICATION IN LIQUID COMPOSITE MOULDING PROCESSES
UNCERTAINTY QUANTIFICATION IN LIQUID COMPOSITE MOULDING PROCESSES Bart Verleye 1, Dirk Nuyens 1, Andrew Walbran 3,2, and Ming Gan 2 1 Dept. Computer Science, KU Leuven, Celestijnenlaan 200A, B-3001 Leuven,
More informationOn the Behavior of the Weighted Star Discrepancy Bounds for Shifted Lattice Rules
On the Behavior of the Weighted Star Discrepancy Bounds for Shifted Lattice Rules Vasile Sinescu and Pierre L Ecuyer Abstract We examine the question of constructing shifted lattice rules of rank one with
More informationConstruction Algorithms for Higher Order Polynomial Lattice Rules
Construction Algorithms for Higher Order Polynomial Lattice Rules Jan Baldeaux, Josef Dick, Julia Greslehner and Friedrich Pillichshammer Dedicated to Gerhard Larcher on the occasion of his 50th birthday
More informationLow Discrepancy Sequences in High Dimensions: How Well Are Their Projections Distributed?
Low Discrepancy Sequences in High Dimensions: How Well Are Their Projections Distributed? Xiaoqun Wang 1,2 and Ian H. Sloan 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
More informationApproximation of High-Dimensional Numerical Problems Algorithms, Analysis and Applications
Approximation of High-Dimensional Numerical Problems Algorithms, Analysis and Applications Christiane Lemieux (University of Waterloo), Ian Sloan (University of New South Wales), Henryk Woźniakowski (University
More informationIntegration of permutation-invariant. functions
permutation-invariant Markus Weimar Philipps-University Marburg Joint work with Dirk Nuyens and Gowri Suryanarayana (KU Leuven, Belgium) MCQMC2014, Leuven April 06-11, 2014 un Outline Permutation-invariant
More informationA Belgian view on lattice rules
A Belgian view on lattice rules Ronald Cools Dept. of Computer Science, KU Leuven Linz, Austria, October 14 18, 2013 Atomium Brussels built in 1958 height 103m figure = 2e coin 5 10 6 in circulation Body
More informationAPPLIED MATHEMATICS REPORT AMR04/2 DIAPHONY, DISCREPANCY, SPECTRAL TEST AND WORST-CASE ERROR. J. Dick and F. Pillichshammer
APPLIED MATHEMATICS REPORT AMR4/2 DIAPHONY, DISCREPANCY, SPECTRAL TEST AND WORST-CASE ERROR J. Dick and F. Pillichshammer January, 24 Diaphony, discrepancy, spectral test and worst-case error Josef Dick
More informationLifting the Curse of Dimensionality
Lifting the Curse of Dimensionality Frances Y. Kuo and Ian H. Sloan Introduction Richard Bellman [1] coined the phrase the curse of dimensionality to describe the extraordinarily rapid growth in the difficulty
More informationANOVA decomposition of convex piecewise linear functions
ANOVA decomposition of convex piecewise linear functions W. Römisch Abstract Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain,
More informationDigital lattice rules
Digital lattice rules Multivariate integration and discrepancy estimates A thesis presented to The University of New South Wales in fulfilment of the thesis requirement for the degree of Doctor of Philosophy
More informationIan Sloan and Lattice Rules
www.oeaw.ac.at Ian Sloan and Lattice Rules P. Kritzer, H. Niederreiter, F. Pillichshammer RICAM-Report 2016-34 www.ricam.oeaw.ac.at Ian Sloan and Lattice Rules Peter Kritzer, Harald Niederreiter, Friedrich
More informationUniform Distribution and Quasi-Monte Carlo Methods
Workshop on Uniform Distribution and Quasi-Monte Carlo Methods October 14-18, 2013 as part of the Radon Special Semester 2013 on Applications of Algebra and Number Theory Metric number theory, lacunary
More informationApproximation of Functions Using Digital Nets
Approximation of Functions Using Digital Nets Josef Dick 1, Peter Kritzer 2, and Frances Y. Kuo 3 1 UNSW Asia, 1 Kay Siang Road, Singapore 248922. E-mail: j.dick@unswasia.edu.sg 2 Fachbereich Mathematik,
More informationFOCM Workshop B5 Information Based Complexity
FOCM 2014 - Workshop B5 Information Based Complexity B5 - December 15, 14:30 15:00 The ANOVA decomposition of a non-smooth function of an infinite number of variables Ian Sloan University of New South
More informationarxiv: v1 [math.na] 13 Jul 2017
Richardson extrapolation of polynomial lattice rules Josef Dick, Takashi Goda, Takehito Yoshiki arxiv:707.03989v [math.na] 3 Jul 207 July 4, 207 Abstract We study multivariate numerical integration of
More informationBath Institute For Complex Systems
Quasi-Monte Carlo methods for computing flow in random porous media I.G.Graham,F.Y.Kuo,D.uyens,R.Scheichl,andI.H.Sloan Bath Institute For Complex Systems Preprint 04/0(200) http://www.bath.ac.uk/math-sci/bics
More informationExponential Convergence and Tractability of Multivariate Integration for Korobov Spaces
Exponential Convergence and Tractability of Multivariate Integration for Korobov Spaces Josef Dick, Gerhard Larcher, Friedrich Pillichshammer and Henryk Woźniakowski March 12, 2010 Abstract In this paper
More informationEFFICIENT ALGORITHMS FOR A CLASS OF SPACE-TIME STOCHASTIC MODELS
EFFICIENT ALGORITHMS FOR A CLASS OF SPACE-TIME STOCHASTIC MODELS by Brandon C. Reyes c Copyright by Brandon C. Reyes, 2018 All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees
More informationA Practical Guide to Quasi-Monte Carlo Methods
A Practical Guide to Quasi-Monte Carlo Methods Frances Y. Kuo and Dirk Nuyens Abstract These notes are prepared for the short course on High-dimensional Integration: the Quasi-Monte Carlo Way, to be held
More informationImplementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs
Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation
More informationQuasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications I. G. Graham,4, F. Y. Kuo 2, D. uyens 2,3, R. Scheichl, and I. H. Sloan 2 Dept of Mathematical Sciences, University
More informationGiovanni Migliorati. MATHICSE-CSQI, École Polytechnique Fédérale de Lausanne
Analysis of the stability and accuracy of multivariate polynomial approximation by discrete least squares with evaluations in random or low-discrepancy point sets Giovanni Migliorati MATHICSE-CSQI, École
More informationHigh dimensional integration of kinks and jumps smoothing by preintegration. Andreas Griewank, Frances Y. Kuo, Hernan Leövey and Ian H.
High dimensional integration of kinks and jumps smoothing by preintegration Andreas Griewank, Frances Y. Kuo, Hernan Leövey and Ian H. Sloan August 2017 arxiv:1712.00920v1 [math.na] 4 Dec 2017 Abstract
More informationKatholieke Universiteit Leuven Department of Computer Science
Extensions of Fibonacci lattice rules Ronald Cools Dirk Nuyens Report TW 545, August 2009 Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200A B-3001 Heverlee (Belgium Extensions
More informationStochastic optimization, multivariate numerical integration and Quasi-Monte Carlo methods. W. Römisch
Stochastic optimization, multivariate numerical integration and Quasi-Monte Carlo methods W. Römisch Humboldt-University Berlin Institute of Mathematics www.math.hu-berlin.de/~romisch Chemnitzer Mathematisches
More informationQMC designs and covering of spheres by spherical caps
QMC designs and covering of spheres by spherical caps Ian H. Sloan University of New South Wales, Sydney, Australia Joint work with Johann Brauchart, Josef Dick, Ed Saff (Vanderbilt), Rob Womersley and
More informationMultilevel Monte Carlo methods for highly heterogeneous media
Multilevel Monte Carlo methods for highly heterogeneous media Aretha L. Teckentrup University of Bath Claverton Down Bath, BA2 7AY, UK Abstract We discuss the application of multilevel Monte Carlo methods
More informationEffective dimension of some weighted pre-sobolev spaces with dominating mixed partial derivatives
Effective dimension of some weighted pre-sobolev spaces with dominating mixed partial derivatives Art B. Owen Stanford University August 2018 Abstract This paper considers two notions of effective dimension
More informationarxiv: v3 [math.na] 18 Sep 2016
Infinite-dimensional integration and the multivariate decomposition method F. Y. Kuo, D. Nuyens, L. Plaskota, I. H. Sloan, and G. W. Wasilkowski 9 September 2016 arxiv:1501.05445v3 [math.na] 18 Sep 2016
More informationarxiv: v2 [math.na] 15 Aug 2018
Noname manuscript No. (will be inserted by the editor) Lattice rules in non-periodic subspaces of Sobolev spaces Takashi Goda Kosuke Suzuki Takehito Yoshiki arxiv:72.2572v2 [math.na] 5 Aug 28 Received:
More informationMulti-Element Probabilistic Collocation Method in High Dimensions
Multi-Element Probabilistic Collocation Method in High Dimensions Jasmine Foo and George Em Karniadakis Division of Applied Mathematics, Brown University, Providence, RI 02912 USA Abstract We combine multi-element
More informationarxiv: v2 [math.na] 18 May 2011
Efficient calculation of the worst-case error and (fast component-by-component construction of higher order polynomial lattice rules arxiv:1105.2599v2 [math.na] 18 May 2011 Jan Baldeaux Josef Dick Gunther
More informationA NEW UPPER BOUND ON THE STAR DISCREPANCY OF (0,1)-SEQUENCES. Peter Kritzer 1. Abstract
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(3 (2005, #A11 A NEW UPPER BOUND ON THE STAR DISCREPANCY OF (0,1-SEQUENCES Peter Kritzer 1 Department of Mathematics, University of Salzburg,
More informationConstruction algorithms for plane nets in base b
Construction algorithms for plane nets in base b Gunther Leobacher, Friedrich Pillichshammer and Thomas Schell arxiv:1506.03201v1 [math.nt] 10 Jun 2015 Abstract The class of (0, m, s)-nets in base b has
More informationSparse polynomial chaos expansions in engineering applications
DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Sparse polynomial chaos expansions in engineering applications B. Sudret G. Blatman (EDF R&D,
More informationFast Numerical Methods for Stochastic Computations
Fast AreviewbyDongbinXiu May 16 th,2013 Outline Motivation 1 Motivation 2 3 4 5 Example: Burgers Equation Let us consider the Burger s equation: u t + uu x = νu xx, x [ 1, 1] u( 1) =1 u(1) = 1 Example:
More informationGeneralizing the Tolerance Function for Guaranteed Algorithms
Generalizing the Tolerance Function for Guaranteed Algorithms Lluís Antoni Jiménez Rugama Joint work with Professor Fred J. Hickernell Room 120, Bldg E1, Department of Applied Mathematics Email: lluisantoni@gmail.com
More information-Variate Integration
-Variate Integration G. W. Wasilkowski Department of Computer Science University of Kentucky Presentation based on papers co-authored with A. Gilbert, M. Gnewuch, M. Hefter, A. Hinrichs, P. Kritzer, F.
More informationKarhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques
Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,
More informationERROR ESTIMATES FOR THE ANOVA METHOD WITH POLYNOMIAL CHAOS INTERPOLATION: TENSOR PRODUCT FUNCTIONS
ERROR ESTIMATES FOR THE ANOVA METHOD WITH POLYNOMIAL CHAOS INTERPOLATION: TENSOR PRODUCT FUNCTIONS ZHONGQIANG ZHANG, MINSEOK CHOI AND GEORGE EM KARNIADAKIS Abstract. We focus on the analysis of variance
More informationWhat Should We Do with Radioactive Waste?
What Should We Do with Radioactive Waste? Andrew Cliffe School of Mathematical Sciences University of Nottingham Simulation of Flow in Porous Media and Applications in Waste Management and CO 2 Sequestration
More informationMonte Carlo Method for Numerical Integration based on Sobol s Sequences
Monte Carlo Method for Numerical Integration based on Sobol s Sequences Ivan Dimov and Rayna Georgieva Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia,
More informationTrivariate polynomial approximation on Lissajous curves 1
Trivariate polynomial approximation on Lissajous curves 1 Stefano De Marchi DMV2015, Minisymposium on Mathematical Methods for MPI 22 September 2015 1 Joint work with Len Bos (Verona) and Marco Vianello
More informationSolving the steady state diffusion equation with uncertainty Final Presentation
Solving the steady state diffusion equation with uncertainty Final Presentation Virginia Forstall vhfors@gmail.com Advisor: Howard Elman elman@cs.umd.edu Department of Computer Science May 6, 2012 Problem
More informationNEEDLET APPROXIMATION FOR ISOTROPIC RANDOM FIELDS ON THE SPHERE
NEEDLET APPROXIMATION FOR ISOTROPIC RANDOM FIELDS ON THE SPHERE Quoc T. Le Gia Ian H. Sloan Yu Guang Wang Robert S. Womersley School of Mathematics and Statistics University of New South Wales, Australia
More informationOrthogonal Polynomials, Quadratures & Sparse-Grid Methods for Probability Integrals
1/33 Orthogonal Polynomials, Quadratures & Sparse-Grid Methods for Probability Integrals Dr. Abebe Geletu May, 2010 Technische Universität Ilmenau, Institut für Automatisierungs- und Systemtechnik Fachgebiet
More informationSTOCHASTIC SAMPLING METHODS
STOCHASTIC SAMPLING METHODS APPROXIMATING QUANTITIES OF INTEREST USING SAMPLING METHODS Recall that quantities of interest often require the evaluation of stochastic integrals of functions of the solutions
More informationQuasi-optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient
Quasi-optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient F. Nobile, L. Tamellini, R. Tempone, F. Tesei CSQI - MATHICSE, EPFL, Switzerland Dipartimento di
More informationMultilevel accelerated quadrature for elliptic PDEs with random diffusion. Helmut Harbrecht Mathematisches Institut Universität Basel Switzerland
Multilevel accelerated quadrature for elliptic PDEs with random diffusion Mathematisches Institut Universität Basel Switzerland Overview Computation of the Karhunen-Loéve expansion Elliptic PDE with uniformly
More informationTractability of Multivariate Problems
Erich Novak University of Jena Chemnitz, Summer School 2010 1 Plan for the talk Example: approximation of C -functions What is tractability? Tractability by smoothness? Tractability by sparsity, finite
More informationarxiv: v2 [math.na] 17 Oct 2017
Modern Monte Carlo Variants for Uncertainty Quantification in Neutron Transport Ivan G. Graham, Matthew J. Parkinson, and Robert Scheichl arxiv:1702.03561v2 [math.na] 17 Oct 2017 Dedicated to Ian H. Sloan
More informationSTRONG TRACTABILITY OF MULTIVARIATE INTEGRATION USING QUASI MONTE CARLO ALGORITHMS
MATHEMATICS OF COMPUTATION Volume 72, Number 242, Pages 823 838 S 0025-5718(02)01440-0 Article electronically published on June 25, 2002 STRONG TRACTABILITY OF MULTIVARIATE INTEGRATION USING QUASI MONTE
More informationarxiv: v2 [math.na] 8 Sep 2017
arxiv:1704.06339v [math.na] 8 Sep 017 A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations Juan Galvis O. Andrés Cuervo September 3, 018 Abstract We use a Monte
More informationAn intractability result for multiple integration. I.H.Sloan y and H.Wozniakowski yy. University of New South Wales. Australia.
An intractability result for multiple integration I.H.Sloan y and H.Wozniakowski yy Columbia University Computer Science Technical Report CUCS-019-96 y School of Mathematics University of ew South Wales
More informationAdvanced Algorithms for Multidimensional Sensitivity Studies of Large-scale Air Pollution Models Based on Sobol Sequences
Advanced Algorithms for Multidimensional Sensitivity Studies of Large-scale Air Pollution Models Based on Sobol Sequences I. Dimov a,, R. Georgieva a, Tz. Ostromsky a, Z. Zlatev b a Department of Parallel
More informationarxiv: v1 [math.na] 3 Apr 2019
arxiv:1904.02017v1 [math.na] 3 Apr 2019 Poly-Sinc Solution of Stochastic Elliptic Differential Equations Maha Youssef and Roland Pulch Institute of Mathematics and Computer Science, University of Greifswald,
More informationLow-discrepancy point sets lifted to the unit sphere
Low-discrepancy point sets lifted to the unit sphere Johann S. Brauchart School of Mathematics and Statistics, UNSW MCQMC 2012 (UNSW) February 17 based on joint work with Ed Saff (Vanderbilt) and Josef
More informationLow-rank techniques applied to moment equations for the stochastic Darcy problem with lognormal permeability
Low-rank techniques applied to moment equations for the stochastic arcy problem with lognormal permeability Francesca Bonizzoni 1,2 and Fabio Nobile 1 1 CSQI-MATHICSE, EPFL, Switzerland 2 MOX, Politecnico
More informationApplicability of Quasi-Monte Carlo for lattice systems
Applicability of Quasi-Monte Carlo for lattice systems Andreas Ammon 1,2, Tobias Hartung 1,2,Karl Jansen 2, Hernan Leovey 3, Andreas Griewank 3, Michael Müller-Preussker 1 1 Humboldt-University Berlin,
More informationBath Institute For Complex Systems
BICS Bath Institute for Complex Systems Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and its Application to Multilevel Monte Carlo Methods Julia Charrier, Robert Scheichl and
More informationToward Approximate Moving Least Squares Approximation with Irregularly Spaced Centers
Toward Approximate Moving Least Squares Approximation with Irregularly Spaced Centers Gregory E. Fasshauer Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 6066, U.S.A.
More informationMath Day at the Beach 2017
Math Day at the Beach 017 Multiple Choice Write your name and school and mark your answers on the answer sheet. You have 0 minutes to work on these problems. No calculator is allowed. 1. How many integers
More informationThe ANOVA decomposition of a non-smooth function of infinitely many variables can have every term smooth
Wegelerstraße 6 53115 Bonn Germany phone +49 228 73-3427 fax +49 228 73-7527 www.ins.uni-bonn.de M. Griebel, F.Y. Kuo and I.H. Sloan The ANOVA decomposition of a non-smooth function of infinitely many
More informationReliable Error Estimation for Quasi-Monte Carlo Methods
Reliable Error Estimation for Quasi-Monte Carlo Methods Department of Applied Mathematics, Joint work with Fred J. Hickernell July 8, 2014 Outline 1 Motivation Reasons Problem 2 New approach Set-up of
More informationQuasi- Monte Carlo Multiple Integration
Chapter 6 Quasi- Monte Carlo Multiple Integration Introduction In some sense, this chapter fits within Chapter 4 on variance reduction; in some sense it is stratification run wild. Quasi-Monte Carlo methods
More informationScenario generation in stochastic programming with application to energy systems
Scenario generation in stochastic programming with application to energy systems W. Römisch Humboldt-University Berlin Institute of Mathematics www.math.hu-berlin.de/~romisch SESO 2017 International Thematic
More informationOn Infinite-Dimensional Integration in Weighted Hilbert Spaces
On Infinite-Dimensional Integration in Weighted Hilbert Spaces Sebastian Mayer Universität Bonn Joint work with M. Gnewuch (UNSW Sydney) and K. Ritter (TU Kaiserslautern). HDA 2013, Canberra, Australia.
More informationNumerical Approximation of Stochastic Elliptic Partial Differential Equations
Numerical Approximation of Stochastic Elliptic Partial Differential Equations Hermann G. Matthies, Andreas Keese Institut für Wissenschaftliches Rechnen Technische Universität Braunschweig wire@tu-bs.de
More informationLow-discrepancy sequences obtained from algebraic function fields over finite fields
ACTA ARITHMETICA LXXII.3 (1995) Low-discrepancy sequences obtained from algebraic function fields over finite fields by Harald Niederreiter (Wien) and Chaoping Xing (Hefei) 1. Introduction. We present
More informationOn ANOVA expansions and strategies for choosing the anchor point
On ANOVA expansions and strategies for choosing the anchor point Zhen Gao and Jan S. Hesthaven August 31, 2010 Abstract The classic Lebesgue ANOVA expansion offers an elegant way to represent functions
More informationApproximation numbers of Sobolev embeddings - Sharp constants and tractability
Approximation numbers of Sobolev embeddings - Sharp constants and tractability Thomas Kühn Universität Leipzig, Germany Workshop Uniform Distribution Theory and Applications Oberwolfach, 29 September -
More informationA Kernel-based Collocation Method for Elliptic Partial Differential Equations with Random Coefficients
A Kernel-based Collocation Method for Elliptic Partial Differential Equations with Random Coefficients Gregory E. Fasshauer and Qi Ye Abstract This paper is an extension of previous work where we laid
More informationBMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension
BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension arxiv:1411.5794v2 [math.fa] 9 Jul 2015 Dmitriy Bilyk a, Lev Markhasin b a School of Mathematics, University
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationSobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis
Sobol-Hoeffding decomposition Application to Global SA Computation of the SI Sobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis Olivier Le Maître with Colleague & Friend Omar
More informationAdvanced Algorithms and Complexity Course Project Report
Advanced Algorithms and Complexity Course Project Report Eklavya Sharma (2014A7PS0130P) 26 November 2017 Abstract This document explores the problem of primality testing. It includes an analysis of the
More informationImproved Discrepancy Bounds for Hybrid Sequences. Harald Niederreiter. RICAM Linz and University of Salzburg
Improved Discrepancy Bounds for Hybrid Sequences Harald Niederreiter RICAM Linz and University of Salzburg MC vs. QMC methods Hybrid sequences The basic sequences Deterministic discrepancy bounds The proof
More informationBoundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis
Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis T. Tran Q. T. Le Gia I. H. Sloan E. P. Stephan Abstract Radial basis functions are used to define approximate solutions
More informationIMPROVEMENT ON THE DISCREPANCY. 1. Introduction
t m Mathematical Publications DOI: 10.2478/tmmp-2014-0016 Tatra Mt. Math. Publ. 59 (2014 27 38 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences Shu Tezuka ABSTRACT. Recently a notion of (t es-sequences
More informationRandom and Quasi-Random Point Sets
Peter Hellekalek Gerhard Larcher (Editors) Random and Quasi-Random Point Sets With contributions by JözsefBeck Peter Hellekalek Fred J. Hickemell Gerhard Larcher Pierre L'Ecuyer Harald Niederreiter Shu
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract. Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree
More information