Space-Time Adaptive Wavelet Methods for Parabolic Evolution Problems

Transcription

1 Space-Time Adaptive Wavelet Methods for Parabolic Evolution Problems Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam joint work with Christoph Schwab (ETH, Zürich), Nabi Chegini (A dam)

2 Contents Adaptive wavelet methods. Tensor product approximation Application to parabolic problems 1/39

3 Adaptive wavelet methods 2/39

4 Well-posed lin. op. eqs. Let X, Y be sep. Hilbert spaces. Let B L(X, Y ) boundedly invertible. Given f Y, we seek u X s.t. Ex.: Bu = f. (Bw)(v) = Ω w v, X = Y = H1 0(Ω) (Poisson problem), (B( w, p))( v, q) = Ω w : v Ω p div v q div w, X = Y = Ω H0(Ω) 1 n L 2,0 (Ω) for a domain Ω IR n (stat. Stokes problem), (Bw)(v) = 1 (w(y) w(x))(v(y) v(x)) 4π dxdy, Ω IR 3, X = Y = Ω Ω x y 3 H 2( Ω)/R 1 (hypersingular boundary integral equation). Parabolic problems and ODE s. X Y. 3/39

5 Reformulation as well-posed bi-infinite MV eqs Let Ψ X = {ψλ X : λ }, ΨY = {ψ Y λ : λ } Riesz bases for X, Y. That is, the synthesis operator, s X : l 2 ( ) X : c c Ψ X := λ c λ ψ X λ and so its adjoint, the analysis operator, s X : X l 2 ( ) : g [g(ψ X λ )] λ. are boundedly invertible (anal. for s Y ). Bu = f s YBs X }{{} B s 1 X u = s }{{} Yf, }{{} u f where B = [(Bψµ X )(ψ Y λ )] λ,µ L(l 2 ( ), l 2 ( )) (infinite stiffness matrix) is boundedly invertible, f = s Ψ Y f = [f(ψ Y λ )] λ l 2 ( ) (infinite load vector). As Riesz bases we have wavelet bases in mind. 4/39

6 Adaptive Wavelet-Galerkin scheme (Bu = f) [Cohen, Dahmen, DeVore 00] (CDD1). Goal: Whenever u A s := sup N N s u u N <, to generate sequence of approx. to u that converge with rate s at linear cost. [If u A s, then to get u u N ε, sufficient and generally necessary is N := ε 1/s u 1/s A s.] Thanks to vanishing moments and smoothness of wavelets, for large class of partial differential and integral operators, B is close to a sparse matrix, and f is close to sparse. We pretend that f is finite and B is sparse. Let B SPD (otherwise apply to B Bu = B f). Notations: Λ, I Λ : l 2 (Λ) l 2 ( ), P Λ = I Λ : l 2( ) l 2 (Λ), B Λ := P Λ BI Λ, u Λ = B 1 Λ P Λf, := B, 1 2 5/39

7 Adaptive Wavelet-Galerkin scheme Prop 1 (Cohen, Dahmen, DeVore 00). Let θ (0, 1], Λ Ξ, s.t. P Ξ (f Bu Λ ) θ f Bu Λ. (1) Then u u Ξ [1 κ(b) 1 θ 2] 1 2 u u Λ. Prop 2 (Gantumur, Harbrecht, St. 07). If in (1), θ < κ(b) 1 2 and Ξ is the smallest set satisfying (1), then #(Ξ\Λ) N for smallest N s.t. u u N [1 θ 2 κ(b)] 1 2 u uλ (2) Corol 1. If u A s, #(Ξ\Λ) ([ ] 1 2 u u Λ ) 1/s u 1/s A s. 6/39

8 Adaptive Wavelet-Galerkin (CDD1) Majorized linear convergence + upper bound on sizes of expansions gives (quasi-) optimal support lengths: Let Λ 1 Λ 2, u Λ1, u Λ2,..., be produced by Adaptive Wavelet-Galerkin scheme.then, when u A s, #Λ l = l #(Λ k \Λ k 1 ) k=1 l k=1 u u Λk 1 1/s u 1/s A s u u Λl 1 1/s u 1/s A s u u Λl 1/s u 1/s A s, i.e., sup(#λ l ) s u u Λl u A s. l 7/39

9 Adaptive Wavelet-Galerkin (CDD1) awgm[ε, ε 1 ] u ε : % Input: ε, ε 1 > 0. % Parameters: θ, δ, γ, ω such that δ (0, θ), θ+δ 1 δ < κ(b) 1 2, ω > 0 and % γ ( 0, (1 δ)(θ δ) 1+δ κ(b) 1). i := 0, u (i) := 0, Λ i := do ζ := ωε i 1 do ζ := ζ/2, r (i) := rhs[ζ/2] apply[u (i), ζ/2] if ε i := r (i) + ζ ε then u ε := u (i) stop endif until ζ δ r (i) Λ i+1 := expand[λ i, r (i), θ] u (i+1) := galsolve[λ i+1, u (i), γ r (i) ] i := i + 1 enddo Th 1. f Bu ε ε. If u A s, then #supp u ε ε 1/s u 1/s A s s < s, cost ε 1/s u 1/s A s. and when 8/39

10 Tensor product approximation 9/39

11 Isotropic vs. anisotropic wavelets Construction of wavelets on (0, 1) 2 (general: Ω 1 Ω n ): 1D: Let V 0 V 1 L 2 (0, 1), Ṽ 0 Ṽ1 L 2 (0, 1) v satisfy Jackson & Bernstein estimates and inf j,ṽ j vj supṽj v j ṽ j 1. Let Ψ j unif. L 2 -Riesz basis for V j Ṽ j 1 (wavelets). Then, normalized in the corresponding norm, Ψ := j Ψ j = {ψ λ : λ } is a Riesz basis for H s (0, 1) (possibly incorporating b.c. s) for a range of s around 0. λ denotes the level (j) of ψ λ. 2D isotropic: (V j V j ) j and (Ṽj Ṽj) j biorth mult. resol. analyses of L 2 ((0, 1) 2 ) that satisfy J & B and inf-sup cond. With Φ j unif. L 2 -Riesz basis for V j (scaling functions), Φ j 1 Ψ j Ψ j Φ j 1 Ψ j Ψ j unif. L 2 -Riesz basis for V j V j (Ṽj 1 Ṽj 1). normalized in corr. norm, union over j is Riesz basis for H s ((0, 1) 2 ) for range of s around 0. 2D anisotropic: Ψ Ψ = j,k Ψ j Ψ k = {ψ λ : λ = 2 } is Riesz basis L 2 ((0, 1) 2 (as well as, renormalized, for H s ((0, 1) 2 ) for range of s around 0). 10/39

12 General domains: Domain decomposition Ω κq Ω q Ω q κ q Two approaches Connect multiresolution analyses ( composite wavelets [Dahmen, Schneider 99], WEM [Canuto, Tabacco, Urban 99]). ([Harbrecht & St. 06]) Bound to the isotropic wavelet construction on subdomains. 11/39

13 General domains: Domain decomposition Apply extensions operators ([Dahmen, Schneider 99], [Kunoth, Sahner 06]). η 2 Ω 2 Ω 1 E 1 If E 1 : H m (Ω 1 ) H m (Ω) bounded, then with η 2 the extension with zero, [E 1 η 2 ] : H m (Ω 1 ) H m 0, Ω 1 Ω 2 (Ω 2 ) H m (Ω) is boundedly invertible (inverse being [ R 1 R 2 (I E 1 R 1 ) ] ). So if Ψ 1, Ψ 2 are some Riesz bases for H m (Ω 1 ), H m 0, Ω 1 Ω 2 (Ω 2 ), then E 1 Ψ 1 η 2 Ψ 2 is Riesz basis for H m (Ω). Current work with Dahlke and Friedrich to improve quantitative properties, in particular in combination with tensor product wavelet bases on the subdomains. 12/39

14 Linear approximation rates iso- vs. anisotropic wavelets = (0, 1) n. Univariate wavs of order d, dyadic refs, m [0, d] inf u v H v V J V m ( ) 2 j(d m) u H d ( ), J [V J V J is spanned by all isotropic wavelets up to level J, but also by all ψ λ with λ J,] inf u v H m ( ) 2 j(d m) u H v span{ψ λ : λ 1 J} d (0,1) H d (0,1), (sparse grids), and for m > 0 and σ (1, d d m ) even inf u v H m ( ) 2 j(d m) u H v span{ψ λ :σ λ 1 (σ 1) λ J} d (0,1) H d (0,1) (optimized sparse grids) ([DeVore, Konyagin, Temlyakov 98], [Griebel & Knapek 00])). 13/39

15 iso- vs. anisotropic wavelets λ 2 J V J V J J n Dims are 2 Jn, J n 1 2 Jn and 2 J, and so rates are d m n,... and d m, respectively. No curse of dimensionality J (n 1)σ+1 λ 1 Argument: u, ψ λ1 φ λ2 φ λn L2 ( ) 2 λ 1 d u H d (0,1) L 2 (0,1) L 2 (0,1) u, ψ λ1 ψ λ2 ψ λn L2 ( ) 2 ( λ 1 + λ λ n )d u H d (0,1) H d (0,1) H d (0,1) 14/39

16 Application: Poisson on Even with smooth f, generally solution does not satisfy the reg. u H d (0, 1) H d (0, 1). cond Instead of rate d 1 in H 1 ( ), generally only n ([Dauge & St. 09]). [ If f vanishes to a sufficiently high order at all corners, then d 1 is guaranteed. ] 15/39

17 Best N term approximation from tensor product basis [Nitsche 06, Hansen & Sickel 09]: Suff. for rate s < d m in H m ( ) is where τ = (s ) 1. u Bτ,τ s+m (0, 1) τ Bτ,τ(0, s 1) τ τ Bτ,τ(0, s 1). B s τ,τ(0, 1) τ τ B s τ,τ(0, 1) τ B s+m τ,τ (0, 1) [Dauge & St. 09]: Suff. for rate d m in H m ( ) is u in H d θ 1(0, 1) H d θ (0, 1) H d θ (0, 1) H d θ (0, 1) H d θ (0, 1) H d θ 1(0, 1), for some θ [0, d), with weighted Sobolev norm reading as v H d ω (0,1) := [ d i=0 (cf. also [Nitsche 05]), and 1 0 x ω (1 x) ω v (i) (x) 2 dx ] /39

18 Application: Poisson on Ω n for sufficiently smooth rhs, the solution of Poisson s problem on, i.e., m = 1, satisfies this regularity condition for θ (d 1 n, d) for arbitrary n and arbitrary d. [ With best N-term isotropic approximation, not only d m n n > 2 also limitations by regularity. ] instead of d m, but for 17/39

19 Adaptive approximation from tensor product basis Apply adaptive wavelet scheme. Rate of best N-term approximation is realized in linear complexity when stiffness matrix w.r.t. tensor product wavelet coordinates is s -computable for some s > s max = d m. [Schwab & St 08]: Valid for elliptic bvp with smooth coefficients and univariate spline wavelets with sufficiently many vanishing moments. 18/39

20 Numerical results [Dijkema, Schwab & St 08]: Poisson on, f = 1, hom Dirichlet at all faces that contain origin, n = 1,..., 10 and d = 2. f Ãgu N f N /39

21 Numerical results By careful choice of univariate wavelets (L 2 -orthogonal ones), error CN 1 u A 1 with C independent of n. Apparently, u A 1 grows exponentially with n. [Novak & Woźniakowski 08]: The number of evaluations of, possibly adaptively chosen, functionals of a C ((0, 1) n )-function, with all derivatives being uniformly bounded, that is generally needed to approximate it in H m ((0, 1) n )-norm within tolerance ε grows exponentially with n. I.e. multivariate approximation is intractable. 20/39

22 Numerical results f Ãgu N f N /39

23 Parabolic problems 22/39

24 Overview With time marching methods, an optimal distribution of grid points over space and time is hard to realize. We apply an adaptive method to a simultaneously space-time variational formulation of the parabolic problem. While keeping discrete solutions on all time levels is prohibitive for time marching methods, thanks to the use of tensorized multi-level bases our method produces approximations simultaneously in space and time without penalty in complexity because of the additional time dimension. 23/39

25 Parabolic problems Let V H V, I := (0, T ). Consider parabolic problem u t (t, ) + A(t)u(t, ) = g(t, ) in V, u(0, ) = u 0 in H, where a(t; η, ζ) := (A(t)(η))(ζ) satisfies for a.e. t I, a(t; η, ζ) M a η V ζ V (η, ζ V ) (boundedness), Ra(t; η, η) + λ η 2 H α η 2 V (η V ) (Gårding inequality). E.g., A(t) differential or integrodifferential operator of order 2m 0, H = L 2 (Ω), V = H m (Ω) (H m 0 (Ω)). 24/39

26 Weak formulations Parabolic problems Multiplication with smooth v with v(t, ) = 0, integration over space and time, and int. by parts w.r.t. time Find u X := L 2 (I; V ) s.t. b(u, v) = f(v) (v Y := L 2 (I; V ) H 1 0,{T } (I; V )) where b(u, v) := f(v) := I I u(t, ), v t (t, ) H + a(t; u(t, ), v(t ))dt, g(t, ), v(t, ) H dt + u 0, v(0, ) H. Th 2 (Dautray & Lions 92, Wloka 82, Schwab & St. 08). B : X Y defined by (Bu)(v) = b(u, v) is boundedly invertible. Th 3. Let W V, A( ) C(Ī, L(W, H)), and A(t) + λi : W H is boundedly invertible. Then with X := L 2 (I; H) and Y := L 2 (I; W ) H 1 0,{T } (I; H), B L(X, Y ) is boundedly invertible. 25/39

27 Tensor product bases Let Θ X, Θ Y, and Σ X, Σ Y be collections of temporal or spatial functions such that, normalized in the corresponding norms, Θ X is a Riesz basis for L 2 (I), Θ Y L 2 (I) and for H0,{T 1 } (I), Σ X H, Σ Y W H. Then, with X = L 2 (I; H) and Y = L 2 (I; W ) H0,{T 1 }(I; H), normalized in the corresponding norms, Θ X Σ X is a Riesz basis for X, Θ Y Σ Y L 2 (I; W ), H0,{T 1 }(I; H), and so for Y, 26/39

28 Structure of system matrix With, for Z {X, Y}, D Z := diag{ θ Z σ Z Z : θ Z Θ Z, σ Z Σ Z }, f = D 1 Y [f(θy σ Y )] θ Y Θ Y,σ Y Σ Y and B = D 1 [ Y Θ X, Θ Y L2 (I) Σ X, Σ Y H + = D 1 Y I a(t, Θ X Σ X, Θ Y Σ Y )dt ] D 1 X [ Θ X, Θ Y L2 (I) Σ X, Σ Y H + Θ X, Θ Y L2 (I) a(σ X, Σ Y ) ] D 1 X when a(t,, ) = a(, ). 27/39

29 Best possible rates H = L 2 (Ω), Ω IR n, so X = L 2 (I; L 2 (Ω)) s max = { min(dt, d x n ) min(d t, d x ) isotropic spatial wavelets anisotropic spatial wavelets up to log-factors when d t = d x n or d t = d x. With linear approx, these rates require boundedness of certain mixed derivatives in L 2. Relaxed regularity conditions with best N-term approx. Realization of rates of best N-term approximation: Galerkin scheme to B Bu = B f. Adaptive wavelet 28/39

30 First test on an ODE Variational formulation where b(w, v) := I { u(t) + νu(t) = g(t) (t I), u(0) = u 0. b(u, v) = f(v), w(t) v(t) + ν w(t)v(t)dt, f(v) := I g(t)v(t)dt + u 0 v(0). Th 4. With Y := H0,{T 1 } (I) equipped with Y := 2 L 2 (I) ν 2 H 1 (I) and X := L 2 (I), B L(X, Y ), defined by (Bw)(v) = b(w, v), is boundedly invertible, uniformly in ν > 0. B = D 1 [ Y Θ X, Θ Y L2 (I) + ν Θ X, Θ Y L2 (I)] D 1 X. We applied adaptive scheme to B Bu = B f. 29/39

31 Custom designed wavelets Consider biorthogonal multires. analyses for trial and test spaces with (Vj X ) j H0,{0} 1 Y (I), (Vj ) j H0,{T 1 } (I) V Y j V X j + V Y j + V X j Ṽ X j+1, Ṽ Y j+1. and Then θµ X, θ Y λ L 2 (I) = 0 θµ X, θ Y λ L 2 (I) = 0 for µ > λ + 1 θµ X, θ Y λ L 2 (I) = 0 θµ X, θ Y λ L 2 (I) = θ µ X, θ Y λ L 2 (I) = 0 for λ > µ + 1, and so B is sparse. 30/39

32 Custom designed wavelets A realization: 2 j+1 { 1 H Vj Z := P 4 (k2 (j+1) T, (k+1)2 (j+1) T ) C 0 1 (I) 0,{0} (I) when Z = X, H0,{T 1 }(I) when Z = Y k=0 having dimension 8 2 j. V Z j + V Z j having dimension 5 2 j+1. 2 j+1 1 k=0 P 4 (k2 (j+1) T, (k + 1)2 (j+1) T ) (3) We selected Ṽ j+1 Z as an extension of latter space to a space of dimension 8 2 j+1, and constructed locally supported biorthogonal wavelets in test and trial space. 31/39

33 Custom designed wavelets Θ Χ, L Θ Χ, R Θ Χ, A Θ Χ, S Figure 1: Boundary wavelets θ X,L, θ X,R (left picture), and interface wavelets θ X,S, θ X,A (right picture). Θ Χ, S,2 Θ Χ, S,3 Χ, A,3 Θ Χ, S,1 Θ Θ Χ, A,1 Χ, A,2 Θ Figure 2: Symmetric interior wavelets θ X,S,1, θ X,S,2, θ X,S,3 (left picture), 32/39 and anti-symmetric interior wavelets θ X,A,1, θ X,A,2, θ X,A,3 (right picture).

34 By Nabi Chegini (UvA, A dam): Numerics First test on an ODE J κ((b B) J ) κ(b J B J) Let g(x) = { 1 x (0, 1 3 ), 2 x ( 1 3, 1), T = 1, ν = 1, u 0 = 0 or u 0 = Figure 3: For u 0 = 1 and #u ε = 202, the non-zero coefficients of u ε. Non-zero coefficients corr. to the left boundary wavelet run to level 51 33/39

35 Numerics ODE !5 10!10 10!15 10!20 10! Figure 4: Bu ε f / f and (Bu ε f) X \supp u ε / f vs. #supp u ε for u 0 = 1 (solid lines) and u 0 = 0 (dashed lines). Wavelets of order 5 34/39

36 Numerics heat eq., one spatial dimension Heat equation t u 2 x 2 u = g on (0, T ) (0, 1), u = 0 on (0, T ) {0, 1} u(0, ) = u 0. Custom designed temporal and spatial wavelets of order 5 (B is sparse). g = 1 and u 0 = 0 or u 0 = 1. 35/39

37 Figure 5: Positions of the non-zeros coefficients. Right u 0 = 1 and #u ε = Left u 0 = 0 and #u ε = /39

38 !2 10!4 10!6 10!8 10!10 10!12 10! Figure 6: Bu ε f vs. #supp u ε for u(0) = 0 and full grid, sparse grid and AWGM. 37/39

39 !2 10!4 10!6 10! Figure 7: Bu ε f vs. #supp u ε for u(0) = 1 and AWGM. 38/39

40 Conclusions Adaptive wavelet schemes for solving well-posed operator equations converge with the best possible rate from the basis being applied. Tensor product approximations give dimension independent rates. Promising applications for parabolic problems. Future work Improved quantitative properties (tree approximations and orthonormal bases) Nonlinear equations 39/39