Adaptive wavelet methods for space-time variational formulations of - PowerPoint PPT Presentation

Adaptive wavelet methods for space-time variational formulations of evolutionary PDEs Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam

Motivation Consider heat eq., implicit time discretization, AFEM for solving the seq. of elliptic problems. Issues: • How to distribute ‘grid points’ optimally over space and time? • Often the class of possible space-time grids doesn’t include ones that are suited for singularities that are local in space and time. • Inherently sequential. • When whole time evolution is needed, as with problems of optimal control, huge memory requirements. Aim at adaptive solver for the problem as a whole , which is quasi-optimal within a class of trial spaces that contains one that gives best possible rate allowed by the order. Furthermore, using that the space-time cylinder is a product domain, we apply (adaptive) tensor product approximation (cf. sparse grids). So we solve time-evolution at a complexity of an optimal solver for the stationary problem. 1/33

Topics • Optimal adaptive wavelet method for solving well-posed (non-) linear operator eqs. • New approximate residual evaluation scheme. • To avoid C 1 wavelets: FOSLS formulations. • Applications: elliptic; stationary NSE; parabolic; instat. NSE. • Some numerical results. 2/33

Well-posed op. eqs. R ). Let B ∈ L is( X , Y ′ ) . Given Let X , Y be sep. Hilbert spaces (over I f ∈ Y ′ , we seek u ∈ X s.t. Bu = f Ex.: � Ω ∇ w · ∇ v dx , X = Y := H 1 • ( Bw )( v ) := 0 (Ω) (Poisson problem), � � � • ( B ( � Ω ∇ � w : ∇ � v − v − w, p ))( � v, q ) := Ω p div � Ω q div � w dx , 0 (Ω) n × L 2 (Ω) /I X = Y := H 1 R (stat. Stokes problem), � � ( w ( y ) − w ( x ))( v ( y ) − v ( x )) 1 • ( Bw )( v ) := dxdy , | x − y | 3 4 π ∂ Ω ∂ Ω 1 X = Y := H 2 ( ∂ Ω) / R (hypersingular boundary integral equation). � T � Ω − w ∂v • ( Bw )( v ) := ∂t + ∇ w · ∇ v dx dt , (heat eq.) 0 X := L 2 ( I ; H 1 0 (Ω)) , Y := L 2 ( I ; H 1 0 (Ω)) ∩ H 1 0 , { T } ( I, H − 1 (Ω)) . 3/33

Reformulation as well-posed bi-infinite MV eq Let Ψ X = { ψ X X } , Ψ Y = { ψ Y λ : λ ∈ ∨ λ : λ ∈ ∨ Y } Riesz bases for X , Y (we have wavelet bases in mind) . That is, the synthesis operator , � F X : c �→ c ⊤ Ψ X := c λ ψ X λ ∈ L is( ℓ 2 ( ∨ X ) , X ) , λ ∈∨ X and so its adjoint, the analysis operator , F ′ X : g �→ g (Ψ X ) := [ g ( ψ X X ∈ L is( X ′ , ℓ 2 ( ∨ λ )] λ ∈∨ X )) . (analogously for F Y ) . F − 1 ⇒ F ′ = F ′ Bu = f ⇐ Y B F X X u Y f , � �� � � �� � ���� u B f where B = ( B Ψ X )(Ψ Y ) ∈ L is( ℓ 2 ( ∨ X ) , ℓ 2 ( ∨ Y )) (infinite “stiffness” matrix) , f = f (Ψ Y ) ∈ ℓ 2 ( ∨ Y ) (infinite “load” vector). 4/33

Least squares problems Let B ∈ L ( X , Y ′ ) with � B · � Y ′ � � · � X Then 2 � Bu − f � 2 1 2 � Bu − f � 2 1 ⇒ B ⊤ ( Bu − f ) = 0 , argmin Y ′ ⇐ ⇒ argmin Y ) ⇐ ℓ 2 ( ∨ u ∈X u ∈ ℓ 2 ( ∨ X ) where B ⊤ B ∈ L is( ℓ 2 ( ∨ X ) , ℓ 2 ( ∨ X )) . We will apply these normal equations also when B ∈ L is( X , Y ′ ) and thus Bu = f is well-posed ( B ∈ L is( ℓ 2 ( ∨ X ) , ℓ 2 ( ∨ Y ) ), but not B = B ⊤ > 0 , because not X = Y , Ψ X = Ψ Y and B = B ′ > 0 . 5/33

Adaptive Wavelet-Galerkin scheme ( Bu = f ) ([CDD01]) Let B = B ⊤ > 0 . Otherwise apply to normal equations. Goal: To generate sequence of approx. to u that, whenever for some s > 0 , � u � A s := sup N N s � u − u N � < ∞ , converges with this rate s , at linear cost. (Here u N is a best approx. to u with # supp u N ≤ N .) Notations: Λ ⊆ ∨ , I Λ : ℓ 2 (Λ) → ℓ 2 ( ∨ ) , R Λ = I ⊤ Λ : ℓ 2 ( ∨ ) → ℓ 2 (Λ) , 1 u Λ := B − 1 B Λ := R Λ BI Λ , Λ R Λ f , ||| · ||| := � B · , ·� 2 (we will identify u Λ with I Λ u Λ ). awgm: • Solve B Λ i u Λ i = R Λ i f ; • Λ i +1 ⊃ Λ i by bulk chasing on f − Bu Λ i ; • repeat with i := i + 1 (Familiar solve-estimate-mark-refine loop known from AFEM, with role a posteriori estimator played by the residual vector. Convergence and optimality not restricted to elliptic problems) 6/33

awgm Prop 1 ([CDD01]) . Let θ ∈ (0 , 1] , Λ ⊂ Ξ ⊂ ∨ , s.t. � R Ξ ( f − Bu Λ ) � ≥ θ � f − Bu Λ � . Then ||| u − u Ξ ||| ≤ [1 − κ ( B ) − 1 θ 2 � 1 2 ||| u − u Λ ||| . Prop 2 ([GHS07]) . If θ < κ ( B ) − 1 2 and Ξ is the smallest set satisfying bulk chasing criterium, then #(Ξ \ Λ) ≤ N for smallest N s.t. 1 ||| u − u N ||| ≤ [1 − θ 2 κ ( B )] 2 ||| u − u Λ ||| . Corol 1. awgm realizes optimal rate s ( N � � u − u Λ � − 1 /s & lin. conv. ) , but, in this form, it is not implementable. of residual f − Bu Λ and approx. Thm 1. awgm with approx. eval. solution of B Λ u Λ = R Λ f within suff. small, but fixed rel. tolerance δ , also converges with optimal rate s , and, if such approx. eval. of f − Bw for w ∈ ℓ 2 (Λ) takes O ( � u − w � − 1 /s + #Λ) operations ( cost condition ) , then scheme has optimal comput. compl. 7/33

Nonlinear operator equations ([XZ03, Ste14]) Theorem 1 generalizes to equations F ( u ) = 0 for F : X ⊃ dom( F ) → Y ′ , written as F ( u ) = 0 , where F := F ′ Y F F X , assuming that X = Y , DF ( u ) ∈ L is( X , X ′ ) and DF ( u ) = DF ( u ) ′ > 0 , or to 1 2 � F ( u ) � 2 argmin Y ′ , u ∈ dom( F ) written as D F ( u ) ⊤ F ( u ) = 0 , only assuming that � DF ( u )( · ) � Y ′ � � · � X . 8/33

Verification of cost condition Valid approx. eval. of f − Bw from [CDD01] exploits near-sparsity of f , B , and w ( u ∈ A s ❀ w ∈ A s ). Depends non-linearly on w . A quantitatively better scheme is obtained by not splitting the residual: Ex 1. Poisson 1D. w := w ⊤ Ψ , to approx. � � 1 � � � 1 � 0 w ′ ψ ′ f − Bw = 0 fψ λ λ ∈∇ − λ dx λ ∈∇ within δ � u − w � H 1 (0 , 1) . Assuming Ψ ⊂ H 2 (0 , 1) , � � 1 � 0 ( f + w ′′ ) ψ λ dx f − Bw = λ ∈∇ , and f + w ′′ is piecewise polynomial w.r.t some mesh 1 , with �·� H − 1 (0 , 1) -norm � � u − w � H 1 (0 , 1) . By dropping all ψ λ whose levels exceed level of local mesh by fixed increment, resulting linear approx. res. eval. scheme meets acc. req. By putting tree constraint on wavelet index sets, multi- to locally single scale transforms in lin. complex, and so approx res. eval in O (# supp w ) ops., thus meets cost condition. 1 modulo data oscillation 9/33

Verification of cost condition • Scheme applies whenever operator applies to any wavelet in mild sense , and also applies to semi-linear PDEs. One vanishing moment suffices. • Instead of applying C 1 -wavelets, we advocate to write second order PDE as (well-posed) first order system least squares . Always possible. 10/33

FOSLS for semi-linear 2nd order PDEs Let F : X ⊃ dom( F ) → Y ′ . For some sep. H. space P , let F = F 0 + F 1 F 2 where F 2 ∈ L ( X , P ) , F 1 ∈ L ( P , Y ′ ) . Then ⇒ � H ( u, θ ) := ( F 0 ( u ) + F 1 θ, θ − F 2 u ) = � F ( u ) = 0 ⇐ 0 . Thm 2 ([RS16]) . ⇒ � D � � DF ( u ) v � Y ′ � � v � X = H ( u, θ )( v, η ) � Y ′ ×P � � ( v, η ) � X×P . So F ( u ) = 0 can be found as 2 � � H ( u, θ ) � 2 1 argmin Y ′ ×P , ( u,θ ) ∈ dom( F ) ×P i.e. as solution of D � H ( u , θ ) ⊤ � H ( u , θ ) = 0 . 11/33

Application: Semi-linear elliptic eq. � R n −△ u + N ( u ) = f on Ω ⊂ I u = 0 at ∂ Ω . ( −△ u may read as ∇ · A ∇ u + b · ∇ u + cu ; other (inhom.) b.c. can be used). Let standard var. form. with X = Y = H 1 0 (Ω) be well-posed (i.e. solution u exists, linearized operator at u in L is( X , Y ′ ) ). � Take P = L 2 (Ω) n . F 2 u := ∇ u , ( F 1 � Ω � θ )( v ) := θ · ∇ v dx . Well-posed FOSLS: � � � � Y ′ + � � 1 θ · ∇ v dx + ( N ( u ) − f ) v dx � 2 θ − ∇ u � 2 � v �→ argmin . P 2 ( u,� Ω θ ) ∈X×P 12/33

Application: Semi-linear elliptic eq. Equip X , Y , P , with wavelet Riesz bases Ψ X , Ψ X , Ψ P . To solve � � � ∇ Ψ X � , ∇ u − � � 0 = θ L 2 (Ω) n + − Ψ P � � �� � DN ( u )Ψ X , Ψ Y � L 2 (Ω) � � L 2 (Ω) + �∇ Ψ Y , � Ψ Y , N ( u ) − f θ � L 2 (Ω) n � Ψ P , ∇ Ψ Y � L 2 (Ω) n which rhs under the additional condition Ψ P ⊂ H (div; Ω) , and u , � θ from the span of the wavelets, reads as � � � � � ∇ Ψ X � DN ( u )Ψ X , Ψ Y � L 2 (Ω) � � � � , ∇ u − � Ψ Y , N ( u ) − f − div � θ + θ L 2 (Ω) . − Ψ P � Ψ P , ∇ Ψ Y � L 2 (Ω) n L 2 (Ω) n 13/33

Application: Semi-linear elliptic eq. Numerical experiment (Nikolaos Rekatsinas (KdVI, Amsterdam)) � −△ u + u 3 = f R 2 on Ω ⊂ I u = 0 at ∂ Ω . L-shaped domain. FOSLS. Finite element wavelets. Continuous piecewise linears for Ψ Y and Ψ P , continuous piecewise quadratics for Ψ X . Figure 1: Approximate solution of −△ u + u 3 = 1 on L-shaped, u = 0 on bdr., as lin. combi of 202 wavelets. 14/33

Application: Semi-linear elliptic eq. Figure 2: Norm of residual vector, vs. number of wavelets, and optimal slope 3 − 1 = 1 2 15/33

Application: Semi-linear elliptic eq. Figure 3: Centers of supports wavelets ( # = 5447 ) that were selected 16/33