Computational Information Games A minitutorial Part II Houman - PowerPoint PPT Presentation

Computational Information Games A minitutorial Part II Houman Owhadi ICERM June 5, 2017 DARPA EQUiPS / AFOSR award no FA9550-16-1-0054 (Computational Information Games)

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 course no one method of approximation of a ‘linear operator ’ can be universal. ” of a ‘linear operator can be universal. Arthur Sard (1909-1980)

( − div( a ∇ u ) = g, x ∈ Ω , x ∈ ∂ Ω , u = 0 , 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

Robust/Algebraic multigrid [Mandel et al., 1999,Wan-Chan-Smith, 1999, Xu and Zikatanov, 2004, Xu and Zhu, 2008], [Ruge-St¨ uben, 1987] [Panayot - 2010] Stabilized Hierarchical bases, Multilevel preconditioners [Vassilevski - Wang, 1997, 1998] [Panayot - Vassilevski, 1997] [Chow - Vassilevski, 2003] [Aksoylu- Holst, 2010] • Some degree of robustness

Low Rank Matrix Decomposition methods Fast Multipole Method: [Greengard and Rokhlin, 1987] Hierarchical Matrix Method: [Hackbusch et al., 2002] [Bebendorf, 2008]: N ln 2 d +8 N complexity To achieve grid-size accuracy in L 2 -norm Hierarchical numerical homogenization method O ( N ln 3 d N ) O ( N ln d +1 N )

Sparse matrix Laplacians Structured sparse matrices (SDD matrices)

The problem

Example ( − div( a ∇ u ) = g, x ∈ Ω , x ∈ ∂ Ω , u = 0 ,

Example L

Example L

Example

Example

Hierarchy of measurement functions

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

Example

Example

Player I Player II Must predict

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

Player II’s bets

Example ( − div( a ∇ u ) = g, x ∈ Ω , x ∈ ∂ Ω , u = 0 , u (2) u (3) u (1) u (5) u (4) u (6)

Accuracy of the recovery Theorem τ ( k ) i φ ( k ) = 1 τ ( k ) i i k u − u ( k ) k a log 10 k u k a k u − u ( k ) k a log 10 k u k a − 3 . 5 − 12

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

Energy content ( − div( a ∇ u ) = g, x ∈ Ω , u = 0 , x ∈ ∂ Ω ,

Gamblets

Example

Gamblets

Gamblets are nested Interpolation/Prolongation operator

Player I Player II Must predict Optimal bet of Player II

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

Hierarchy of measurement functions Hierarchy of gamblets

Biorthogonal system Theorem

Measurement functions are nested Gamblets are nested Orthogonalized gamblets

Operator adapted MRA Theorem

g u Theorem

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

Energy content ( − div( a ∇ u ) = g, x ∈ Ω , u = 0 , x ∈ ∂ Ω ,

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)

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)

Eigenspace adapted MRA Theorem

log 10 ( λ max ( A ( k ) ) log 10 ( λ max ( A ( k ) ) λ min ( A ( k ) ) ) λ min ( A ( k ) ) ) log 10 ( λ max ( B ( k ) ) log 10 ( λ max ( B ( k ) ) λ min ( B ( k ) ) ) λ min ( B ( k ) ) )

Wannier functions

Regularity Conditions

Example L Regularity Conditions

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

Example

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

Example Regularity Conditions

Regularity Conditions on Primal Space

Gamblet Transform/Solve

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

Localization of Gamblets

Sparsity of the precision matrix

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

Example L

Examples

Theorem

Condition for localization

Theorem

Banach space setting Condition for localization

Operator connectivity distance Theorem

Thank you DARPA EQUiPS / AFOSR award no FA9550-16-1-0054 (Computational Information Games)