Computational Information Games A minitutorial Part II Houman Owhadi ICERM June 5, 2017

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1 Computational Information Games A minitutorial Part II Houman Owhadi ICERM June 5, 2017 DARPA EQUiPS / AFOSR award no FA (Computational Information Games)

2 Question Can we design a linear solver with some degree of universality? (that could be applied to a large class of linear operators) Motivation There are (nearly) as many linear solvers as linear systems. Number of google scholar references to linear solvers : 447,000 Not clear that this can be done Of course no one method of approximation of a linear operator can be universal. Arthur Sard ( )

3 ( div(a u) =g, x Ω, u =0, x Ω, 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

4 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

5 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 Hierarchical numerical homogenization method O(N ln 3d N) O(N ln d+1 N)

6 Sparse matrix Laplacians Structured sparse matrices (SDD matrices)

7 The problem

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

9 Example L

10 Example L

11 Example

12 Example

13 Hierarchy of measurement functions

14 Example φ (1) i φ (2) i φ (3) i φ (4) i φ (5) i φ (6) i

15 Example

16 Example

17 Player I Player II Must predict

18 Example Player I Player II Sees { uφ (k),i I Ω i k } Must predict u and { uφ (k+1),j I Ω j k+1 }

19 Player II s bets

20 Example ( div(a u) =g, x Ω, u =0, x Ω, u (1) u(2) u (3) u (4) u (5) u (6)

21 Accuracy of the recovery Theorem τ (k) i φ (k) i =1 τ (k) i log 10 ku u (k) k a kuk a log 10 ku u (k) k a kuk a

22 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

23 Energy content ( div(a u) =g, x Ω, u =0, x Ω,

24 Gamblets

25 Example

26 Gamblets

27 Gamblets are nested Interpolation/Prolongation operator

28 Player I Player II Must predict Optimal bet of Player II

29 Example ( div(a u) =g, x Ω, u =0, x Ω, R (k) i,j τ (k) i R Your best bet on the value of τ (k+1) j given the information that R τ (k) i 1 u =1and R τ l u =0forl 6= i R (k) i,j u τ (k+1) j

30 Hierarchy of measurement functions Hierarchy of gamblets

31 Theorem Biorthogonal system

32 Measurement functions are nested Gamblets are nested Orthogonalized gamblets

33

34 Theorem Operator adapted MRA

35 u g Theorem

36 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

37 Energy content ( div(a u) =g, x Ω, u =0, x Ω,

38 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)

39 Operator adapted wavelets Second Generation Operator Adapted Wavelets We want 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)

40 Theorem Eigenspace adapted MRA

41 log 10 ( λ max(a (k) ) λ min (A (k) ) ) log 10 ( λ max(b (k) ) λ min (B (k) ) ) log 10 ( λ max(a (k) ) λ min (A (k) ) ) log 10 ( λ max(b (k) ) λ min (B (k) ) )

42 Wannier functions

43 Regularity Conditions

44 Example L 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

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

48 Example Regularity Conditions

49 Regularity Conditions on Primal Space

50 Gamblet Transform/Solve

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

52 Localization of Gamblets

53 Sparsity of the precision matrix

54 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

55 Example L

56 Examples

57 Theorem

58 Condition for localization

59 Theorem

60 Banach space setting Condition for localization

61 Theorem Operator connectivity distance

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