M. Croci (Oxford), M. B. Giles (Oxford), P. E. Farrell (Oxford), M. E. Rognes (Simula) MCQMC July 4, 2018

Transcription

1 Efficient wite noise sampling and coupling for multilevel Monte Carlo M. Croci (Oxford), M. B. Giles (Oxford), P. E. Farrell (Oxford), M. E. Rognes (Simula) MCQMC July 4, 2018

2 Overview Introduction Wite noise sampling Numerical results Conclusions and furter work

3 Overview Introduction Wite noise sampling Numerical results Conclusions and furter work

4 Motivation Te motivation of our researc is te sampling of lognormal Gaussian fields. A Matérn Gaussian field (approximately) satisfies a linear elliptic SPDE of te form Lu = W, x D, ω Ω + BCs, were u = u(x, ω) and W is spatial wite noise. Oter approaces can be used (wit pros and cons), but we will not discuss tem ere.

5 Motivation Te motivation of our researc is te sampling of lognormal Gaussian fields. A Matérn Gaussian field (approximately) satisfies a linear elliptic SPDE of te form Lu = W, x D, ω Ω + BCs, were u = u(x, ω) and W is spatial wite noise. Oter approaces can be used (wit pros and cons), but we will not discuss tem ere. Te same tecniques can be used to solve a more general class of SPDEs, e.g. N(u) + Lu = W, x D, ω Ω + BCs. In tis case solving means to compute E[P(u)] for some functional P of te solution.

6 Motivation Te motivation of our researc is te sampling of lognormal Gaussian fields. A Matérn Gaussian field (approximately) satisfies a linear elliptic SPDE of te form Lu = W, x D, ω Ω + BCs, were u = u(x, ω) and W is spatial wite noise. Oter approaces can be used (wit pros and cons), but we will not discuss tem ere. Te same tecniques can be used to solve a more general class of SPDEs, e.g. N(u) + Lu = W, x D, ω Ω + BCs. In tis case solving means to compute E[P(u)] for some functional P of te solution. Common applications: finance, geology, meteorology, biology...

7 Motivation Te motivation of our researc is te sampling of lognormal Gaussian fields. A Matérn Gaussian field (approximately) satisfies a linear elliptic SPDE of te form Lu = W, x D, ω Ω + BCs, were u = u(x, ω) and W is spatial wite noise. Oter approaces can be used (wit pros and cons), but we will not discuss tem ere. Te same tecniques can be used to solve a more general class of SPDEs, e.g. N(u) + Lu = W, x D, ω Ω + BCs. In tis case solving means to compute E[P(u)] for some functional P of te solution. Common applications: finance, geology, meteorology, biology... Main issue: sampling W is ard!

8 Motivation Te motivation of our researc is te sampling of lognormal Gaussian fields. A Matérn Gaussian field (approximately) satisfies a linear elliptic SPDE of te form Lu = W, x D, ω Ω + BCs, were u = u(x, ω) and W is spatial wite noise. Oter approaces can be used (wit pros and cons), but we will not discuss tem ere. Te same tecniques can be used to solve a more general class of SPDEs, e.g. N(u) + Lu = W, x D, ω Ω + BCs. In tis case solving means to compute E[P(u)] for some functional P of te solution. Common applications: finance, geology, meteorology, biology... Main issue: sampling Te efficient sampling of W is ard! W is te focus of tis talk.

9 Wite noise (1D) WARNING! Point evaluation not defined!

10 University of Oxford Matematical Institute Wite noise (2D) WARNING! Point evaluation not defined!

11 University of Oxford Matematical Institute Wite noise (2D) WARNING! Point evaluation not defined! IDEA! Avoid point evaluation by integrating W.

12 University of Oxford Matematical Institute Wite Noise (practical definition) Definition (Spatial Wite Noise W ) R For any φ L2 (D), define W, φi := D W φ dx. For any φi, φj L2 (D), bi = W, φi i, bj = W, φj i are zero-mean Gaussian random variables, wit, Z E[bi bj ] = φi φj dx =: Mij, b N (0, M). (1) D

13 Finite element (FEM) framework Wen solving SPDEs (see 1 st slide) wit FEM, we get (for linear problems) Discrete weak form: find u V s.t. for all v V, Were V = span({φ i } n i=0 ), (e.g. wit Lagrange elements). a(u, v ) = W, v, (2)

14 Finite element (FEM) framework Wen solving SPDEs (see 1 st slide) wit FEM, we get (for linear problems) Discrete weak form: find u V s.t. for all v V, Were V = span({φ i } n i=0 ), (e.g. wit Lagrange elements). FEM linear system: u = i u iφ i, u = [u 0,..., u n ] T, were te entries of b are given by, wit b N (0, M) as before. M is te mass matrix of V. a(u, v ) = W, v, (2) Au = b(ω), (3) W, φ i (ω) = b i (ω), (4)

15 Multilevel Monte Carlo [Giles 2008] + FEM framework For MLMC, we ave two approximation levels l and l 1. For any particular ω Ω, we need to solve: find u l V l, ul 1 V l 1 s.t. for all v l V l, v l 1 V l 1, a(u l 1 a(u l, v l ) = W, v l (ω), (5), v l 1 ) = W, v l 1 (ω). (6)

16 Multilevel Monte Carlo [Giles 2008] + FEM framework For MLMC, we ave two approximation levels l and l 1. For any particular ω Ω, we need to solve: find u l V l, ul 1 V l 1 s.t. for all v l V l, v l 1 V l 1, a(u l 1 Tis yields te linear system [ A l 0 0 A l 1 a(u l, v l ) = W, v l (ω), (5), v l 1 ) = W, v l 1 ] [ u l u l 1 ] = (ω). (6) [ b l b l 1 ] = b,

17 Multilevel Monte Carlo [Giles 2008] + FEM framework For MLMC, we ave two approximation levels l and l 1. For any particular ω Ω, we need to solve: find u l V l, ul 1 V l 1 s.t. for all v l V l, v l 1 V l 1, a(u l 1 Tis yields te linear system [ A l 0 0 A l 1 a(u l, v l ) = W, v l (ω), (5), v l 1 ) = W, v l 1 ] [ u l u l 1 ] = (ω). (6) [ b l b l 1 ] = b, were b N (0, M). Let V l = span({φl i }n l l 1 i=0 ) and V = span({φ l 1 i [ ] M l M l,l 1 M = (M l,l 1 ) T M l 1, M l,l 1 ij = φ l i φ l 1 j dx. } n l 1 i=0 ), ten

18 Multilevel Monte Carlo [Giles 2008] + FEM framework For MLMC, we ave two approximation levels l and l 1. For any particular ω Ω, we need to solve: find u l V l, ul 1 V l 1 s.t. for all v l V l, v l 1 V l 1, a(u l 1 Tis yields te linear system [ A l 0 0 A l 1 a(u l, v l ) = W, v l (ω), (5), v l 1 ) = W, v l 1 ] [ u l u l 1 ] = (ω). (6) [ b l b l 1 ] = b, were b N (0, M). Let V l = span({φl i }n l l 1 i=0 ) and V = span({φ l 1 i [ ] M l M l,l 1 M = (M l,l 1 ) T M l 1, M l,l 1 ij = φ l i φ l 1 j dx. } n l 1 i=0 NOTE: we do not require te FEM approximation subspaces to be nested! ), ten

19 Overview Introduction Wite noise sampling Numerical results Conclusions and furter work

20 Te callenge SAMPLING PROBLEM 1: single level realisations: sample b N (0, M), were M is te mass matrix of V. SAMPLING PROBLEM 2: coupled realisations: sample b N (0, M), were M is te block mass matrix given by V l and V l 1.

21 How to sample b? Sampling b is ard!

22 How to sample b? Sampling b is ard! Naïve approac - Factorise M = HH T (cubic complexity!) and set b = Hz, wit z N (0, I ). E[bb T ] = E[Hz(Hz) T ] = HE[zz T ]H T = HIH T = M.

23 How to sample b? Sampling b is ard! Naïve approac - Factorise M = HH T (cubic complexity!) and set b = Hz, wit z N (0, I ). E[bb T ] = E[Hz(Hz) T ] = HE[zz T ]H T = HIH T = M. - Works well if M diagonal: previous work used eiter mass-lumping [Lindgren, Rue and Lindström 2009], piecewise constant elements [Osborn, Vassilevski and Villa 2017] or a piecewise constant approximation of wite noise [Drzisga, et al. 2017, Du and Zang 2002]. - We do not require M to be diagonal (and we do not approximate wite noise). - We can sample b wit linear complexity.

24 How to sample b? Sampling b is ard! Naïve approac - Factorise M = HH T (cubic complexity!) and set b = Hz, wit z N (0, I ). E[bb T ] = E[Hz(Hz) T ] = HE[zz T ]H T = HIH T = M. - Works well if M diagonal: previous work used eiter mass-lumping [Lindgren, Rue and Lindström 2009], piecewise constant elements [Osborn, Vassilevski and Villa 2017] or a piecewise constant approximation of wite noise [Drzisga, et al. 2017, Du and Zang 2002]. - We do not require M to be diagonal (and we do not approximate wite noise). - We can sample b wit linear complexity. IDEA! H does not need to be square, maybe we can find a more efficient factorisation!

25 Wite noise sampling: single level realisations SAMPLING PROBLEM 1: need to sample b N (0, M). Exploit te FEM assembly M 1 0 M = L T. 0 M.. 2 L = LT diag e (M e )L. (7)

26 Wite noise sampling: single level realisations SAMPLING PROBLEM 1: need to sample b N (0, M). Exploit te FEM assembly N (0, M) b = L T b 1 b 2. = L T vstack e (b e ) (8)

27 Wite noise sampling: single level realisations SAMPLING PROBLEM 1: need to sample b N (0, M). Exploit te FEM assembly - Eac b e can be sampled as b e = H e z e wit z e N (0, I ) and H e H T e = M e. - b = L T vstack e (b e ) is N (0, M) since E[bb T ] = L T E[vstack e (b e )vstack e (b e ) T ]L = L T diag e (H e )diag e (H T e )L = L T diag e (M e )L = M. - If te mapping to te FEM reference element is affine (e.g. Lagrange elements on simplices) we ave tat M e / e = const on eac element and only one local factorisation is needed. Tis approac is trivially parallelisable!

28 Wite noise sampling: coupled realisations SAMPLING PROBLEM 2: need to sample b N (0, M), were M is now te block mixed mass matrix. Definition (Supermes, [Farrell 2009]) Let A and B be two (possibly non-nested) meses. Teir supermes S is one of teir common refinements. A and B are bot nested witin S.

29 Wite noise sampling: coupled realisations SAMPLING PROBLEM 2: need to sample b N (0, M), were M is now te block mixed mass matrix. - Factorise locally, tis time on eac supermes element. - Sample b on S, ten interpolate/project te result onto A and B (tis step can be performed locally). - Since A and B are nested witin S, tis operation is exact. Note tat A and B need not be nested. Previous work on wite noise coupling for MLMC used eiter a nested ierarcy [Drzisga et al. 2017, Osborn et al. 2017] or an algebraically constructed ierarcy of agglomerated meses [Osborn, Vassilevski and Villa 2017].

30 Complexity overview offline cost online cost (per sample) memory storage single level 0 (or O(m 3 N)) O(m 3 N) (or O(m 2 N)) O(m 2 ) (or O(m 2 N)) single l. (affine) O(m 3 ) O(m 2 N) O(m 2 ) coupled 0 (or O(m 3 N S )) O(m 3 N S ) (or O(m 2 N S )) O(m 2 ) (or O(m 2 N S )) coupled (affine) O(m 3 ) O(m 2 N S ) O(m 2 ) Table: Memory and cost complexity of our wite noise sampling strategy. In te non-affine case te cost per sample can be lowered by precomputing and storing te local factorisations (see entries in blue). N S is te number of supermes elements. In our experience wit MLMC, N S c d N l and c d = 2 (1D), c d = 2.5 (2D), c d = 45 (3D).

31 Overview Introduction Wite noise sampling Numerical results Conclusions and furter work

32 Numerical results: convergence of P(u) = u 2 L 2 (D) Consider te linear elliptic SPDE [Lindgren, Rue and Lindström 2009], [Bolin, Kircner and Kovács 2017], ( I κ 2 ) k u(x, ω) = η W, x D R d, ω Ω, ν = 2k d/2 > 0. We compute FEM solutions {u l }l=8 l=1 wit a non-nested ierarcy of subspaces {V l }l=8 l=

33 Numerical results: covariance convergence C(r) = E[u(x)u(y)] = ν 2 ν 1 Γ(ν) (κr)ν K ν (κr), r = x y 2, κ =, x, y D, λ

34 Overview Introduction Wite noise sampling Numerical results Conclusions and furter work

35 Conclusions and furter work Outlook - Wite noise is an extremely non-smoot object and is defined troug its integral. - We can sample single level wite noise realisations efficiently. - We can couple wite noise between different FEM approximation subspaces. A supermes construction is not needed in te nested case. - Te overall order of complexity is linear in te number of elements of te supermes and it can be trivially parallelised. Standard tecniques usually ave cubic complexity. Furter work: extensions to QMC and MLQMC. Paper: ttps://arxiv.org/abs/

36 References - Tank you for listening! [1] M. Croci, M. B. Giles, M. E. Rognes, and P. E. Farrell. Efficient wite noise sampling and coupling for multilevel Monte Carlo wit non-nested meses. Preprint, URL ttps://arxiv.org/abs/ [2] D. Bolin, K. Kircner, and M. Kovács. Numerical solutions of fractional elliptic stocastic PDEs wit spatial wite noise. Preprint, [3] D. Drzisga, B. Gmeiner, U. Ude, R. Sceicl, and B. Wolmut. Sceduling massively parallel multigrid for multilevel Monte Carlo metods. SIAM Journal of Scientific Computing, 39(5): , [4] Q. Du and T. Zang. Numerical approximation of some linear stocastic partial differential equations driven by special additive noises. SIAM Journal on Numerical Analysis, 40(4): , [5] M. B. Giles. Multilevel Monte Carlo pat simulation. Operations Researc, 56(3): , [6] F. Lindgren, H. Rue, and J. Lindström. An explicit link between Gaussian fields and Gaussian Markov random fields: te stocastic partial differential equation approac. Journal of te Royal Statistical Society: Series B (Statistical Metodology), 73(4): , [7] S. Osborn, P. S. Vassilevski, and U. Villa. A multilevel, ierarcical sampling tecnique for spatially correlated random fields. Preprint, [8] S. Osborn, P. Zulian, T. Benson, U. Villa, R. Krause, and P. S. Vassilevski. Scalable ierarcical PDE sampler for generating spatially correlated random fields using non-matcing meses. Preprint, [9] A. J. Waten. Realistic eigenvalue bounds for te Galerkin mass matrix. IMA Journal of Numerical Analysis, 7: , 1987.