Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis.

Transcription

1 Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis Houman Owhadi Joint work with Clint Scovel IPAM Apr 3, 2017 DARPA EQUiPS / AFOSR award no FA (Computational Information Games)

2 Question Can we design a scalable solver that could be applied to nearly all linear operators? Answer Yes under two minor conditions: (1) The operator must be bounded and invertible (2) Its image space must have a regular multiresolution decomposition

3 Problem: Solve (1) as fast as possible to a given accuracy (1) div(a u) =g, x Ω, u =0, x Ω, Ω R d Ω is piec. Lip. a unif. ell. a i,j L (Ω) log 10 (a)

4 Multigrid Methods Multigrid: [Fedorenko, 1961, Brandt, 1973, Hackbusch, 1978] Multiresolution/Wavelet based methods [Brewster and Beylkin, 1995, Beylkin and Coult, 1998, Averbuch et al., 1998] Linear complexity with smooth coefficients Problem Severely affected by lack of smoothness

5 Robust/Algebraic multigrid [Mandel et al., 1999,Wan-Chan-Smith, 1999, Xu and Zikatanov, 2004, Xu and Zhu, 2008], [Ruge-Stüben, 1987] [Panayot ] Stabilized Hierarchical bases, Multilevel preconditioners [Vassilevski - Wang, 1997, 1998] [Panayot - Vassilevski, 1997] [Chow - Vassilevski, 2003] [Aksoylu- Holst, 2010] Some degree of robustness but problem remains open with rough coefficients Why? Interpolation operators are unknown Don t know how to bridge scales with rough coefficients!

6 Low Rank Matrix Decomposition methods Fast Multipole Method: [Greengard and Rokhlin, 1987] Hierarchical Matrix Method: [Hackbusch et al., 2002] [Bebendorf, 2008]: N ln 2d+8 N complexity To achieve grid-size accuracy in L 2 -norm

7 Common theme between these methods Their process of discovery is based on intuition, brilliant insight, and guesswork Can we turn this process of discovery into an algorithm?

8 ( div(a u) =g in Ω, u =0on Ω, H 1 0 (Ω) =W (1) a W (2) a a W (k) a Gamblet O(N ln 3d N) u = w (1) + w (2) + + w (k) + Linear Solve Transform O(N ln d+1 N) u = 0.14 w (1) 0.03 w (2) w (3) + + w (4) w (5) w (6)

9 Hierarchy of nested Measurement functions φ (1) i φ (2) i φ (3) i Example φ (k) i φ (4) i φ (5) i : Indicator functions of a hierarchical nested partition of Ω of resolution H k =2 k φ (6) i τ (1) 2 τ (2) 2,3 τ (3) 2,3,1

10 Formulation of the hierarchy of games ( div(a u) =g in Ω, Player I u =0on Ω, Player II Sees { uφ (k),i I Ω i k } Must predict u and { uφ (k+1),j I Ω j k+1 }

11 ( div(a u) =g in Ω, Player II s best strategy u =0on Ω, Player II s bets u (1) u(2) u (3) u (4) u (5) u (6)

12 Gamblets Elementary gambles form a hierarchy of deterministic basis functions for player II s hierarchy of bets Theorem u (k) (x) = P i ψ(k) i R (x) Ω u(y)φ(k) (y) dy i ψ (k) i : Elementary gambles/bets at resolution H k =2 k Your best bet on the value of u given the information that Ω

13 Gamblets

14 χ (k) i = ψ (k) i ψ (k) i

15 Multiresolution decomposition of the solution space V (k) := span{ψ (k) i,i I k } W (k) := span{χ (k),i} i W (k+1) : Orthogonal complement of V (k) in V (k+1) with respect to < ψ, χ > a := R Ω ( ψ)t a χ Theorem H 1 0 (Ω) =V (1) a W (2) a a W (k) a

16 Multiresolution decomposition of the solution Theorem u (k+1) u (k) = F.E. sol. of PDE in W (k+1) u = 0.14 u (1) 0.03 u (2) u (1) u (3) u (2) + + u (4) u (3) u (5) u (4) u (6) u (5) Subband solutions u (k+1) u (k) can be computed independently

17 Energy content ( div(a u) =g, x Ω, u =0, x Ω, If r.h.s. is regular we don t need to compute all subbands

18 Numerical Homogenization Harmonic Coordinates Babuska, Caloz, Osborn, 1994 Kozlov, 1979 Allaire Brizzi 2005; Owhadi, Zhang 2005 MsFEM [Hou, Wu: 1997]; [Efendiev, Hou, Wu: 1999] [Fish - Wagiman, 1993] Variational Multiscale Method, Orthogonal decomposition Projection based method HMM Nolen, Papanicolaou, Pironneau, 2008 Engquist, E, Abdulle, Runborg, Schwab, et Al Flux norm Berlyand, Owhadi 2010; Symes 2012 Bayesian Numerical Homogenization Gamblets Operator compression

19 Energy content ( div(a u) =g, x Ω, u =0, x Ω, Beyond numerical homogenization (gamblet mesh refinement)

20 A (k) i,j Uniformly bounded condition numbers := ψ (k) i, ψ (k) j a B (k) i,j := χ (k), χ (k) i j a Theorem λ max (B (k) ) λ min (B (k) ) C log 10 ( λ max(a (k) ) λ min (A (k) ) ) log 10 ( λ max(b (k) ) λ min (B (k) ) ) Just relax! In v W (k) to get u (k) u (k 1)

21 ψ (1) i χ (2) i χ (3) i χ (4) i χ (5) i χ (6) i Gamblets are not only localized in space and their linear combinations remain localized in frequency They behave like wavelets and Wannier functions infψ V kψk 2 a kψk 2 L 2, sup ψ V infψ W (k) kψk 2 a kψk 2 L 2 kψk 2 a kψk 2 L 2, sup ψ W (k) kψk 2 a kψk 2 L 2

22 Wannier functions

23 Operator adapted wavelets First Generation Wavelets: Signal and imaging processing First Generation Operator Adapted Wavelets (shift and scale invariant) Lazy wavelets (Multiresolution decomposition of solution space)

24 Operator adapted wavelets Second Generation Operator Adapted Wavelets We want (open problem solved here) 1. Scale-orthogonal wavelets with respect to operator scalar product (leads to block-diagonalization) 2. Operator to be well conditioned within each subband 3. Wavelets need to be localized (compact support or exp. decay)

25 Gamblet Transform

26 Fast/Localized Gamblet Transform

27 Theorem The number of operations to compute gamblets and achieve accuracy ² is O N ln 3d max( 1 ²,N1/d ) (and O N ln d (N 1/d )ln 1 ² for subsequent solves) Complexity Gamblet Transform Linear Solve O(N ln 3d N) O(N ln d+1 N)

28 Can we design a universal scalable solver? Sparse matrix Laplacians Structured sparse matrices (SDD matrices)

29 The problem

30 Example ( div(a u) =g, x Ω, u =0, x Ω,

31 Example L

32 Example L

33 Example

34 Example

35 Hierarchy of measurement functions Hierarchy of gamblets

36 Theorem Biorthogonal system

37 Measurement functions are nested Gamblets are nested Orthogonalized gamblets

38 Theorem Operator adapted MRA

39 u g Theorem

40 Theorem Eigenspace adapted MRA

41 The method Regularity Conditions

42 The method φ (1) i φ (2) i φ (3) i φ (4) i φ (5) i φ (6) i ψ (1) i χ (2) i χ (3) i χ (4) i χ (5) i χ (6) i

43 Regularity Conditions

44 Example Regularity Conditions

45 Example φ (1) i φ (2) i φ (3) i φ (4) i φ (5) i φ (6) i τ (1) 2 τ (2) 2,3 τ (3) 2,3,1

46 Example τ (1) 2 τ (2) 2,3 τ (3) 2,3,1

47 Example τ (1) 2 τ (2) 2,3 τ (3) 2,3,1

48 Example Regularity Conditions

49 Gamblet Transform/Solve

50 Fast Gamblet Transform obtained by truncation/localization Complexity Theorem Based on exponential decay of gamblets and locality of the operator

51 Localization of Gamblets Localization problem in Numerical Homogenization [Chu-Graham-Hou-2010] (limited inclusions) [Efendiev-Galvis-Wu-2010] (limited inclusions or mask) [Babuska-Lipton 2010] (local boundary eigenvectors) [Owhadi-Zhang 2011] (localized transfer property) Subspace decomposition/correction and Schwarz iterative methods

52 Example L

53 Condition for localization

54 Examples

55 Theorem

56 Straightforward generalization L

57 Localization of gamblets Operator connectivity distance

58 Theorem

59 Condition for localization Measurement functions

60 Game theoretic origin/interpreation To compute fast we need to compute with partial information Restriction Interpolation Missing information Problem

61 Repeated adversarial information games Player I Player II Max Min

62 Theorem

63 Theorem Universal Optimal Measure

64 Theorem Gamblets

65 Theorem

66 Optimal recovery splines

67 Theorem Theorem

68 Link between numerical analysis and statistical inference Coupling numerical approximation error with model uncertainty

69 Statistical inference approaches to numerical approximation Pioneering work

70 Statistical inference approaches to numerical approximation Information based complexity

71 Statistical inference approaches to numerical approximation Bayesian Numerical Analysis

72 Probabilistic Numerics

73 Probabilistic Numerics

74 Game Theoretic approach to Numerical Analysis

75 Thank you DARPA EQUiPS / AFOSR award no FA (Computational Information Games)