Constructive tractability of the Helmholtz problem Arthur G. - PowerPoint PPT Presentation

Constructive tractability of the Helmholtz problem Arthur G. Werschulz Henryk Wo´ zniakowski Fordham University Department of Computer and Information Sciences Columbia University Department of Computer Science University of Warsaw Department of Mathematics ICERM HDA-IBC Providence, RI 15 September 2014 1 / 19

Helmholtz problem For f ∈ F d , q ∈ Q d , find an approximation of the solution u = u f , q to in I d := (0 , 1) d , L q u := − ∆ u + qu = f with either Dirichlet on ∂ I d u = g or Neumann on ∂ I d ∂ ν u = g boundary conditions. Actually consider variational form of this problem ( H 1 ( I d )-error). Here, d can be huge!!! 2 / 19

Our previous work ◮ “Tractability of quasilinear problems I: general results”, J. Approx. Theory , 2007. ◮ “Tractability of quasilinear problems II: second-order elliptic problems”, Math. Comp. , 2007. ◮ “Tractability of multivariate approximation over a weighted unanchored Sobolev space”, Constr. Approx. , 2009. ◮ “Tractability of the Helmholtz equation with non-homogeneous Neumann boundary conditions: relation to L 2 -approximation”, J. Comp. , 2009 (only W). ◮ “Tight tractability results for a model second-order Neumann problem”, J. FoCM. , 2014. 3 / 19

What we really want ◮ General q . ◮ General Dirichlet or Neumann boundary conditions. ◮ Wide range of weighted spaces. ◮ Necessary and sufficient weight conditions for various flavors of tractability. ◮ Explicit optimal tractability algorithms. 4 / 19

What you’re getting today ◮ General q , but sometimes specializing to q ≡ 1. ◮ Homogeneous Neumann boundary conditions. ◮ Variety of F d and Q d . ◮ Tractability results, but sometimes optimal tractability algorithms. 5 / 19

Problem definition ◮ Let F d = unit ball of H d , γ = H ( K d , γ ) , Q d = { q ∈ F d : q ( · ) ≥ q 0 } where q 0 ∈ (0 , 1). ◮ Let � ∀ v , w ∈ H 1 ( I d ) , q ∈ Q d . B d ( v , w ; q ) = I d [ ∇ v ·∇ w + qvw ] ◮ Seek u = S d ( f , q ) ∈ H 1 ( I d ): ∀ w ∈ H 1 ( I d ) , B d ( u , w ; q ) = � f , w � L 2 ( I d ) for ( f , q ) ∈ F d × Q d , i.e., the variational solution of in I d , − ∆ u + qu = f on ∂ I d . ∂ ν u = 0 6 / 19

Quasilinearity ◮ S d is quasilinear : ◮ Linear in first argument. ◮ Lipschitz in both arguments. ◮ See WW, 2007a. 7 / 19

“Usual IBC stuff” ◮ Continuous linear information. ◮ Error of algorithm A n using at most n info evals: e ( A n , S d ) = sup � S d ( f , q ) − A n ( f , q ) � H 1 ( I d ) [ f , q ] ∈ F d × Q d ◮ n th minimal error: e ( n , S d ) = inf A n e ( A n , S d ) ◮ Info complexity: n ( ε, S d ) = inf { n ∈ N 0 : e ( n , S d ) ≤ CRI d ε } ABS : CRI d ≡ 1 NOR : CRI d = e (0 , S d ) 8 / 19

Reduction to approximation problem ◮ Easy lower bound: S d � APP H d , γ → [ H 1 ( I d )] ∗ ◮ Known upper bound: ◮ S d ( · , q ) is [ H 1 ( I d )] ∗ -Lipschitz ◮ S d ( f , · ) is L 2 ( I d )-Lipschitz ◮ S d � APP H d , γ → L 2 ( I d ) ◮ New upper bound: S d is [ H 1 ( I d )] ∗ -Lipschitz . . . vile constants. ◮ Interpolatory algorithm A n ( f , q ) = S d (˜ f , ˜ q ) ◮ yields sharp info complexity bounds ◮ no joy implementation-wise when q �≡ const ◮ How to find easily-implementable “good” algorithms? We are currently investigating Galerkin algorithms, but we are not yet done . . . 9 / 19

Tractability results for arbitrary q ∈ Q d [WW 2007] ◮ H d , γ = H ( K d , γ ), where � � K d , γ ( x , y ) = γ d , u K ( x j , y j ) , j ∈ u u ⊆{ 1 , 2 ,..., d } | u |≤ ω � with [0 , 1] 2 K ( x , y ) dx dy ∈ (0 , ∞ ) ◮ Finite-order weights γ d , u = 0 for all | u | > ω 10 / 19

Tractability results for arbitrary q ∈ Q d [WW 2007] ◮ ABS + FOW = ⇒ PT: n ( ε, S d ) ≤ C ε − 2 d 2 ω ◮ ABS + FOW + sup d � | u |≤ ω γ d , u < ∞ = ⇒ SPT: n ( ε, S d ) ≤ C ε − 2 ◮ NOR + FOW = ⇒ PT: n ( ε, S d ) ≤ C ε − 2 d ω ◮ NOR + FOW + sup d � | u |≤ ω γ d , u < ∞ = ⇒ SPT: n ( ε, S d ) ≤ C ε − 2 ◮ Also have results for Λ std , as well as for Dirichlet problems. ◮ Based on Wasilkowski+W, 2004. 11 / 19

Tractability results for q ≡ 1 [WW 2014] ◮ Now S d is linear! ◮ Here, F d is unit ball of H d , γ . ◮ Choose general weights (not necessarily FOW) γ = { γ d , u ≥ 0 : u ⊆ [ d ] := { 1 , 2 , . . . , d } , d ∈ N } with γ d , ∅ = 1. ◮ Then H d , γ = { w ∈ [ H 1 ( I )] ⊗ d : ∂ u w ≡ 0 whenever γ d , u = 0 } , with ∂ u = � j ∈ u ∂ j , where � γ − 1 � v , w � H d , γ = d , u � ∂ u v , ∂ u w � L 2 ( I d ) ∀ v , w ∈ H d , γ . u ⊆ [ d ] ◮ ABS=NOR, since e (0 , S d ) = 1. 12 / 19

Tractability results for q ≡ 1 [WW 2014] Everything depends on eigenpairs ( β k , γ , e k ) of S ∗ d S d for k ∈ N d : ◮ Eigenvalues: 1 1 β k , γ = j =1 ( k j − 1) 2 · ∅� = u ⊆ [ d ] γ − 1 1 + π 2 � d 1 + � � j ∈ u [ π 2 ( k j − 1) 2 ] d , u ◮ Eigenvectors: d � e k ( x ) = cos[ π ( k j − 1) x i ] j =1 ◮ Optimal algorithm: Let n = | M ( ε, d , γ ) | , where M ( ε, d , γ ) = { k ∈ N d : β k , γ > ε 2 } . Then � f , e k � H d , γ � A n ( f , 1) = S d e k � e k � 2 H d , γ k ∈ M ( ε, d , γ ) 13 / 19

Tractability results for q ≡ 1 [WW 2014] ◮ Relation to L 2 -approximation: Let APP d , γ = APP H d , γ → L 2 ( I d ) and c d = min { 1 , M − 1 M d = j =1 , 2 ,..., d γ d , { j } < ∞ max and d } . Then n ( √ ε c − 1 / 4 , APP d , γ ) ≤ n ( ε, S d ) ≤ n ( ε, APP d , γ ) . d ◮ APP d , γ was studied in WW, 2009. 14 / 19

Tractability for q ≡ 1 + product weights [WW 2014] ◮ Let 1 ≥ γ 1 ≥ γ 2 ≥ · · · > 0. Then γ d , u = � j ∈ u γ j . ◮ Quasi-polynomial tractability t (1 + ln ε − 1 )(1 + ln d ) � � n ( ε, S d ) ≤ C exp , holds for all product weights, with 2 . t = = 0 . 838233 ln(1 + π 2 ) ◮ Polynomial tractability holds iff ∃ τ > 1 2 : d B τ = ζ (2 τ ) 1 � γ τ lim sup j < ∞ , π 2 τ ln d d →∞ j =1 in which case n ( ε, S d ) ≤ C τ d B τ ε − 2 τ . 15 / 19

Tractability for q ≡ 1 + product weights [WW 2014] ◮ Strong polynomial tractability iff ∃ τ > 1 2 : d � γ τ A τ := sup d , j < ∞ , d ∈ N j =1 When this holds, let τ ∗ = inf { τ > 1 2 : A τ < ∞ } . Then for all τ > τ ∗ , we have n ( ε, S d ) ≤ ε − 2 τ exp ζ (2 τ ) π − 2 τ A τ � � 16 / 19

Future work ◮ Study general q . ◮ We believe that results for q ∈ Q d will be analogous to those for q ≡ 1, but work is still in progress. 17 / 19

A new approximation problem ◮ For s ∈ N 0 , let � v � 2 � γ − 1 d , u � ∂ u v � 2 H s ( H d , γ ) = H s ( I d ) . u ⊆ [ d ] ◮ APP r , s : Approximate H s ( H d , γ )-functions in the H r ( I d )-norm, where r , s ∈ N 0 and r ≤ s . ◮ Then Galerkin error is bounded by e (APP 1 , 2 ), modulo a constant, with still-unknown dependence on d . 18 / 19

Spectral results for APP r , s ◮ Let W r , s = APP ∗ � r , s APP r , s . Then e ( n , APP r , s ) = β n +1 , r , s , where β 1 , r , s ≥ β 2 , r , s ≥ · · · > 0 are eigenvalues of W r , s . ◮ For k ∈ N d , let e k ( x ) = � d j =1 cos[ π ( k j − 1) x j ]. Then ∀ k ∈ N d , W r , s e k = β r , s , γ , k e k where β r , s , γ , k = ζ r , k α k , γ , ζ s , k d � � [ π ( k j − 1)] 2 m j , ζ s , k = j =1 0 ≤| m |≤ s � − 1 � � γ − 1 � [ π ( k j − 1)] 2 α k , γ = . d , u j ∈ u u ⊆ [ d ] 19 / 19