Application of QMC methods to PDEs with random coefficients Frances Kuo f.kuo@unsw.edu.au University of New South Wales, Sydney, Australia Based on joint works with Ivan Graham (Bath), Rob Scheichl (Bath), Dirk Nuyens (KU Leuven), Christoph Schwab (ETH Zurich), Ian Sloan (UNSW), Josef Dick (UNSW), Thong Le Gia (UNSW), James Nichols (UNSW). Frances Kuo
Motivating example Uncertainty in groundwater flow eg. risk analysis of radwaste disposal or CO 2 sequestration q + a � ∇ p = f Darcy’s law in D ⊂ R d , d = 1 , 2 , 3 ∇ · q = 0 mass conservation law together with boundary conditions [Cliffe, et. al. (2000)] x x x Uncertainty in a ( x x, ω ) leads to uncertainty in q ( x x, ω ) and p ( x x, ω ) Frances Kuo
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 ∂p n = 0 ∂� 1 0.8 p = 1 p = 0 0.6 release point → ◦ 0.4 • 0.2 0 0 0.2 0.4 0.6 0.8 1 ∂p n = 0 ∂� Frances Kuo
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 → use quasi-Monte Carlo methods −1 10 MC n − 1 / 2 −2 10 −3 10 error n − 1 or better −4 10 QMC s = stochastic dimension −5 10 3 4 5 6 10 10 10 10 n Frances Kuo
MC v.s. QMC in the unit cube n � 1 � [0 , 1] s F ( y y y ) d y y y ≈ F ( t t t i ) n i =1 Monte Carlo method Quasi-Monte Carlo methods t t t t i deterministic t t i random uniform close to n − 1 convergence (or better) n − 1 / 2 convergence more effective for earlier variables and lower -order projections order of variables irrelevant order of variables very important 1 1 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 1 0 1 0 1 First 64 points of a 64 random points A lattice rule with 64 points 2D Sobol ′ sequence use randomized QMC methods for error estimation Frances Kuo
QMC Two main families of QMC methods: (t,m,s)-nets and (t,s)-sequences lattice rules Now also higher order digital nets Niederreiter book (1992) Sloan and Joe book (1994) A group under addition modulo Z Dick and Pillichshammer book (2010) and includes the integer points K., Schwab, Sloan ANZIAM review (2011) • • • • • • • Dick, K., Sloan Acta Numerica (2013) • • • • • • • • 1 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Having the right number of • 0 1 0 1 • • • First 64 points of a • • A lattice rule with 64 points points in various sub-cubes 2D Sobol ′ sequence • (0,6,2)-net Frances Kuo
Lattice rules Rank-1 lattice rules have points � i � t t t i = frac n z z z , i = 1 , 2 , . . . , n z ∈ Z s – the generating vector, with all components coprime to n z z frac( · ) – means to take the fractional part of all components 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 1 A lattice rule with 64 points Frances Kuo
Lattice rules Rank-1 lattice rules have points � i � t t t i = frac n z z z , i = 1 , 2 , . . . , n z ∈ Z s – the generating vector, with all components coprime to n z z frac( · ) – means to take the fractional part of all components z ∼ quality determined by the choice of z z ∼ � i � 1 37 10 47 20 57 30 n = 64 z z = (1 , 19) z t t t i = frac 64 (1 , 19) 3 40 13 50 23 60 33 6 43 16 53 26 63 36 9 46 19 56 29 2 39 12 49 22 59 32 5 42 15 52 25 62 35 8 “ Component-by-component (CBC) construction ” 45 18 55 28 1 38 11 48 21 58 31 4 41 [Korobov (1959); Sloan and Reztsov (2002); · · · ] 14 51 24 61 34 7 44 17 54 27 64 0 1 A lattice rule with 64 points Frances Kuo
Randomly shifted lattice rules Shifted rank-1 lattice rules have points � i � t t t i = frac n z z z + ∆ ∆ ∆ , i = 1 , 2 , . . . , n ∆ ∈ [0 , 1) s – the shift ∆ ∆ • • • • • • • • • • • • • • • • • • 1 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • shifted by • • • • • • • • • • • • • • • • • • ∆ ∆ ∆ = (0 . 1 , 0 . 3) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 1 0 1 A lattice rule with 64 points A shifted lattice rule with 64 points Frances Kuo
Randomly shifted lattice rules Shifted rank-1 lattice rules have points � i � t t t i = frac n z z z + ∆ ∆ ∆ , i = 1 , 2 , . . . , n ∆ ∈ [0 , 1) s – the shift ∆ ∆ ∼ use a number of random shifts for error estimation ∼ 1 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • shifted by • • • • • • • • • • • • • • • • • • ∆ ∆ ∆ = (0 . 1 , 0 . 3) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 1 0 1 A lattice rule with 64 points A shifted lattice rule with 64 points Frances Kuo
Weighted spaces [Sloan and Wo´ zniakowski (1998); · · · ] IDEA: the variables are not equally important (cf. effective dimension) We assume that F belongs to a weighted Sobolev space , with norm � � 2 � ∂ | u | F 1 � � � � F � 2 � � y y γ = ( y y u ; 0) d y y u � � γ γ y γ u ∂y y u [0 , 1] | u | � � u ⊆{ 1 ,...,s } “anchor” at 0 2 s subsets “weights” Mixed first derivatives are square integrable y Small weight γ u means that F depends weakly on the variables y y u Frances Kuo
Weighted spaces [Sloan and Wo´ zniakowski (1998); · · · ] IDEA: the variables are not equally important (cf. effective dimension) We assume that F belongs to a weighted Sobolev space , with norm � � 2 � ∂ | u | F 1 � � � � F � 2 � � y y γ = ( y y u ; 0) d y y u � � γ γ y γ u ∂y y u [0 , 1] | u | � � u ⊆{ 1 ,...,s } “anchor” at 0 2 s subsets “weights” Mixed first derivatives are square integrable y Small weight γ u means that F depends weakly on the variables y y u Alternatively, we may use the “ unanchored ” norm � � 2 � � ∂ | u | F 1 � � � � F � 2 � � y y y y γ = ( y y u ; y y − u )d y y − u d y y u � � γ γ γ u ∂y y u y [0 , 1] | u | � [0 , 1] s −| u | � u ⊆{ 1 ,...,s } Frances Kuo
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