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

Houman Owhadi Computational Information Games A minitutorial Part II

ICERM June 5, 2017

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

Motivation Can we design a linear solver with some degree of universality? (that could be applied to a large class of linear

• perators)

Not clear that this can be done

Arthur Sard (1909-1980)

“Of course no one method of approximation Of course no one method of approximation

• f a ‘linear operator
• f a ‘linear operator’ can be

can be universal. universal.” There are (nearly) as many linear solvers as linear systems. Question

Number of google scholar references to “linear solvers”: 447,000

Multigrid Methods Multiresolution/Wavelet based methods

[Brewster and Beylkin, 1995, Beylkin and Coult, 1998, Averbuch et al., 1998

Multigrid: [Fedorenko, 1961, Brandt, 1973, Hackbusch, 1978]

• Linear complexity with smooth coefficients

Severely affected by lack of smoothness

Problem

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

[Mandel et al., 1999,Wan-Chan-Smith, 1999, Xu and Zikatanov, 2004, Xu and Zhu, 2008], [Ruge-St¨ uben, 1987]

Robust/Algebraic multigrid

• Some degree of robustness

[Vassilevski - Wang, 1997, 1998] Stabilized Hierarchical bases, Multilevel preconditioners [Panayot - Vassilevski, 1997] [Chow - Vassilevski, 2003] [Panayot - 2010] [Aksoylu- Holst, 2010]

Low Rank Matrix Decomposition methods

Fast Multipole Method: [Greengard and Rokhlin, 1987]

Hierarchical Matrix Method: [Hackbusch et al., 2002]

[Bebendorf, 2008]:

N ln2d+8 N complexity To achieve grid-size accuracy in L2-norm

Hierarchical numerical homogenization method

O(N ln3d N)

O(N lnd+1 N)

Sparse matrix Laplacians Structured sparse matrices (SDD matrices)

The problem

Example

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

L

Example

L

Example

Example

Example

Hierarchy of measurement functions

Example

φ(1)

i

φ(2)

i

φ(3)

i

φ(4)

i

φ(5)

i

φ(6)

i

Example

Example

Player I Player II

Must predict

Player I Player II

Example

Must predict

Sees { Ω uφ(k)

i

, i ∈ Ik} u and { Ω uφ(k+1)

j

, j ∈ Ik+1}

Player II’s bets

u(1)

u(2)

u(3)

u(4)

u(5)

u(6)

Example

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

Accuracy of the recovery

Theorem

φ(k)

i

= 1τ(k)

i

τ (k)

i log10

ku−u(k)ka kuka

log10

ku−u(k)ka kuka

−3.5 −12

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

Energy content

If r.h.s. is regular we don’t need to compute all subbands

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

Energy content

Gamblets

Example

Gamblets

Gamblets are nested

Interpolation/Prolongation operator

Player I Player II

Must predict

Optimal bet of Player II

1

R(k)

i,j

Your best bet on the value of R

τ (k+1)

j

u

given the information that R

τ (k)

i

u = 1 and R

τl u = 0 for l 6= i

τ (k)

i

R(k)

i,j

τ (k+1)

j

Example

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

Hierarchy of measurement functions Hierarchy of gamblets

Biorthogonal system Theorem

Gamblets are nested Orthogonalized gamblets

Measurement functions are nested

Operator adapted MRA Theorem

u g

Theorem

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

Energy content

If r.h.s. is regular we don’t need to compute all subbands

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

Energy content

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

• 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) We want

Theorem Eigenspace adapted MRA

log10( λmax(A(k))

λmin(A(k)) )

log10( λmax(B(k))

λmin(B(k)) )

log10( λmax(A(k))

λmin(A(k)) )

log10( λmax(B(k))

λmin(B(k)) )

Wannier functions

Regularity Conditions

Example

L

Regularity Conditions

τ (1)

2

τ (2)

2,3

τ (3)

2,3,1

Example

φ(1)

i

φ(2)

i

φ(3)

i

φ(4)

i

φ(5)

i

φ(6)

i

Example

Example

τ (1)

2

τ (2)

2,3

τ (3)

2,3,1

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) [Babuska-Lipton 2010] (local boundary eigenvectors) [Efendiev-Galvis-Wu-2010] (limited inclusions or mask) [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)