Recent advances in compressed sensing techniques for the numerical approximation of PDEs Simone Brugiapaglia Simon Fraser University, Canada simone_brugiapaglia@sfu.ca Joint work with Ben Adcock (SFU), Stefano Micheletti (MOX), Fabio Nobile (EPFL), Simona Perotto (MOX), Clayton G. Webster (ONL). QUIET 2017 SISSA. Trieste, Italy – July 20, 2017
Compressed sensing CS for (parametric) PDEs Inside the black box Outside the black box Conclusions 0
Compressed Sensing (CS) Pioneering papers: [Donoho, 2006; Candès, Romberg, & Tao, 2006] Main ingredients: ◮ Sparsity / Compressibility; ◮ Random measurements (sensing); ◮ Sparse recovery. Sparsity: Let s ∈ C N be an s -sparse w.r.t. a basis Ψ : s = { z ∈ C N : � z � 0 ≤ s } , x ∈ Σ N s = Ψ x and where � x � 0 := # { i : x i � = 0 } and s ≪ N . Compressibility: fast decay of the best s -term approximation error � x − z � p ≤ Cs − α , σ s ( x ) p = inf z ∈ Σ N s for some C, α > 0 , where . 1
Sensing In order to acquire s , we perform m ∼ s · polylog ( N ) linear nonadaptive random measurements � s , ϕ i � =: y i , for i = 1 , . . . , m. If we consider the matrix Φ = [ ϕ i ] ∈ C N × m , we have Ax = y , where A = Φ ∗ Ψ ∈ C m × N and y ∈ C m . This system is highly underdetermined . Φ ∗ y Ψ x * f Φ Ψ u = sensing matrix measurements vector sparsity basis unknown sparse signal 2
Sparse recovery Thanks to the sparsity / compressibility of s , we can resort to sparse recovery techniques . We aim at approximating the solution to (P 0 ) z ∈ C N � z � 0 , min s.t. Az = y . � In general, (P 0 ) is a NP-hard problem... � There are computationally tractable strategies to approximate it! In particular, it is possible to employ ◮ greedy strategies , e.g. Orthogonal Matching Pursuit ( OMP ); ◮ convex relaxation , e.g., the quadratically-constrained basis pursuit (QCBP) program: z ∈ C N � z � 1 , min s.t. � Az − y � 2 ≤ η, referred to as Basis pursuit (BP) when η = 0 . 3
Restricted isometry property Many important recovery results in CS are based on the Restricted Isometry Property ( RIP ). Definition (RIP) A matrix A ∈ C m × N satisfies the RIP ( s, δ ) with δ ∈ [0 , 1) if (1 − δ ) � z � 2 2 ≤ � Az � 2 2 ≤ (1 + δ ) � z � 2 ∀ z ∈ Σ N 2 , s . The RIP implies recovery results for: ◮ OMP [Zhang, 2011; Cohen, Dahmen, DeVore, 2015]; ◮ QCBP [Candés, Romberg, Tao, 2006], [Foucart, Rauhut; 2013]; Optimal recovery error estimates (without noise) for a decoder ∆ look like [Cohen, Dahmen, DeVore, 2009] � x − ∆( Ax ) � 2 � σ s ( x ) 1 ∀ x ∈ C N , √ s , and hold with high probability. 4
Compressed sensing CS for (parametric) PDEs Inside the black box Outside the black box Conclusions 4
CS as a tool to solve PDEs Parametric PDEs’ setting: ◮ z ∈ D ⊆ R d : parametric domain, d ≫ 1 ; ◮ L z u z = g : PDE; ◮ z �→ u z : solution map (the “black box”); ◮ u z �→ Q ( u z ) : quantity of interest. Can we take advantage of the CS paradigm in this setting? 5
CS as a tool to solve PDEs Parametric PDEs’ setting: ◮ z ∈ D ⊆ R d : parametric domain, d ≫ 1 ; ◮ L z u z = g : PDE; ◮ z �→ u z : solution map (the “black box”); ◮ u z �→ Q ( u z ) : quantity of interest. Can we take advantage of the CS paradigm in this setting? YES! At least in two ways, addressed in this talk: 1. Inside the black box, to approximate z �→ u z 2. Outside the black box, to approximate z �→ f ( z ) = Q ( u z ) 5
Compressed sensing CS for (parametric) PDEs Inside the black box Outside the black box Conclusions 5
CS inside the black box Consider the weak formulation of a PDE find u ∈ U : a ( u, v ) = F ( v ) , ∀ v ∈ V, and its Petrov-Galerkin ( PG ) discretization [Aziz, Babuška, 1972]. Motivation to apply CS: ◮ reduce the computational cost associated with a classical PG discretization; ◮ situations with a limited budget of evaluations of F ( · ) ; ◮ deeper theoretical understanding of the PG method. Case study: ✎ Advection-diffusion-reaction (ADR) equation , with U = V = H 1 0 (Ω) , Ω = [0 , 1] d , and a ( u, v ) = ( η ∇ u, ∇ v ) + ( b · ∇ u, v ) + ( ρu, v ) , F ( v ) = ( f, v ) . 6
Related literature Ancestors: PDE solvers based on ℓ 1 -minimization 1988 [J. Lavery, 1988; J. Lavery, 1989] Inviscid Burgers’ equation, conservation laws 2004 [J.-L. Guermond, 2004; J.-L. Guermond and B. Popov, 2009] Hamilton-Jacobi, transport equation CS techniques for PDEs 2010 [S. Jokar, V. Mehrmann, M. Pfetsch, and H. Yserentant, 2010] Recursive mesh refinement based on CS (Poisson equation) 2015 [S. B., S. Micheletti, S. Perotto, 2015; S. B., F. Nobile, S. Micheletti, S. Perotto, 2017] CORSING for ADR problems 7
The Petrov-Galerkin method Choose U N ⊆ H 1 0 (Ω) and V M ⊆ H 1 0 (Ω) with U N = span { ψ 1 , . . . , ψ N V M = span { ϕ 1 , . . . , ϕ M } , } � �� � � �� � tests trials Then we can discretize the weak problem as Ax = y , A ij = a ( ψ j , ϕ i ) , y i = F ( ϕ i ) with A ∈ C M × N , y ∈ C M . 8
The Petrov-Galerkin method Choose U N ⊆ H 1 0 (Ω) and V M ⊆ H 1 0 (Ω) with U N = span { ψ 1 , . . . , ψ N V M = span { ϕ 1 , . . . , ϕ M } , } � �� � � �� � tests trials Then we can discretize the weak problem as Ax = y , A ij = a ( ψ j , ϕ i ) , y i = F ( ϕ i ) with A ∈ C M × N , y ∈ C M . We can establish the following analogy: Petrov-Galerkin method: Sampling: solution of a PDE ⇐ ⇒ signal tests (bilinear form) measurements (inner product) 8
Classical case: square matrices When dealing with Petrov-Galerkin discretizations, one usually ends up with a big square matrix. ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ϕ 1 → × × × × × × × u 1 F ( ϕ 1 ) ϕ 2 → × × × × × × × u 2 F ( ϕ 2 ) ϕ 3 → × × × × × × × F ( ϕ 3 ) u 3 ϕ 4 → × × × × × × × u 4 = F ( ϕ 4 ) ϕ 5 → × × × × × × × F ( ϕ 5 ) u 5 ϕ 6 → × × × × × × × u 6 F ( ϕ 6 ) ϕ 7 → × × × × × × × F ( ϕ 7 ) u 7 � �� � a ( ψ j ,ϕ i ) 9
“Compressing” the discretization We would like to use only m random tests instead of N , with m ≪ N ... ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ϕ 1 → × × × × × × × u 1 F ( ϕ 1 ) ϕ 2 → × × × × × × × u 2 F ( ϕ 2 ) ϕ 3 → × × × × × × × F ( ϕ 3 ) u 3 ϕ 4 → × × × × × × × u 4 = F ( ϕ 4 ) ϕ 5 → × × × × × × × F ( ϕ 5 ) u 5 ϕ 6 → × × × × × × × u 6 F ( ϕ 6 ) ϕ 7 → × × × × × × × F ( ϕ 7 ) u 7 � �� � a ( ψ j ,ϕ i ) 10
Sparse recovery ...in order to obtain a reduced discretization . ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ↓ ↓ ↓ ↓ ↓ ↓ ↓ � × � F ( ϕ 2 ) � � u 1 ϕ 2 → × × × × × × = u 2 ϕ 5 → × × × × × × × F ( ϕ 5 ) u 3 � �� � a ( ψ j ,ϕ i ) u 4 u 5 u 6 u 7 The solution is then computed using sparse recovery techniques. 11
CORSING (COmpRessed SolvING) First, we define the local a -coherence [Krahmer, Ward, 2014; B., Nobile, Micheletti, Perotto, 2017]: µ N | a ( ψ j , ϕ q ) | 2 , q := sup ∀ q ∈ N . j ∈ [ N ] COSRING algorithm: 1. Define a truncation level M and a number of measurements m ; 2. Draw τ 1 , . . . , τ m independently at random from [ M ] according to the probability p ∼ ( µ N 1 , . . . , µ N M ) (up to rescaling). 3. Build A ∈ R m × N , y ∈ R m and D ∈ R m × m , defined as: δ ik A ij := a ( ψ j , ϕ τ i ) , f i := F ( ϕ τ i ) , D ik := √ mp τ i . z ∈ R N � D ( Az − y ) � 2 4. Use OMP to solve min 2 , s.t. � z � 0 ≤ s ; 12
Sparsity + Sensing: How to choose { ψ j } and { ϕ i } ? Heuristic criterion commonly used in CS: space vs. frequency . Hierarchical hat functions Sine functions [Smoliak, Dahmen, Griebel, Yserentant, Zienkiewicz, ...] 0.5 H 0,0 0.5 0.4 0.3 S 2 0.4 H 1,0 H 1,1 S 1 0.2 S 5 0.3 0.1 S 3 H 2,0 H 2,1 H 2,2 H 2,3 0 S 4 0.2 −0.1 0.1 −0.2 −0.3 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 S H We name the corresponding strategies CORSING HS and SH . 13
Homogeneous 1D Poisson problem CORSING HS TS := N − m · 100% ≈ 85% N = 8191 , s = 50 , m = 1200 . � Test Savings : N 3.5 3.5 exact 3.45 3 corsing 3.4 2.5 3.35 2 3.3 1.5 3.25 1 3.2 exact 0.5 3.15 corsing 0 3.1 0 0.2 0.4 0.6 0.8 1 0.38 0.39 0.4 0.41 0.42 × = hat functions selected by OMP Level-based ordering ( log 10 | � u ℓ,k | ) 14
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