Space-time sparse discretization of linear parabolic equations

Transcription

1 Space-time sparse discretization of linear parabolic equations Roman Andreev August 2, 200 Seminar for Applied Mathematics, ETH Zürich, Switzerland Support by SNF Grant No. PDFMP / Part of PhD thesis under the supervision of Prof. Ch. Schwab

2 TOC of the Talk Intro Variational formulation Abstract stability results Example of stable FE spaces Outro

3 Introduction Let V H = H V be a Gelfand triple of separable Hilbert spaces. Consider the abstract parabolic equation in t J R t u + Au = g, u(0) = h H (PDE) where g : J V A : J L(V, V ) u : J V (PDE) models heat conduction, e.g. H0 (D) = V H = L2 (D) = H V = H (D) option pricing, etc

4 Introduction Let V H = H V be a Gelfand triple of separable Hilbert spaces. Consider the abstract parabolic equation in t J R t u + Au = g, u(0) = h H (PDE) where g : J V A : J L(V, V ) u : J V (PDE) models parametric heat conduction, e.g. L 2 π(u, H0 (D)) L2 π(u, L 2 (D)) L 2 π(u, H (D)) option pricing, etc

5 Introduction Numerical solution of (PDE) many (adaptive, multi-level) Galerkin methods exist essentially variations on the method of lines Issues compression of u as a function of space-time efficient algorithms for computing the compressed u provable error and complexity bounds design of stable finite element spaces does not arise in the adaptive wavelet method [SS09]

6 Introduction Numerical solution of (PDE) many (adaptive, multi-level) Galerkin methods exist essentially variations on the method of lines Issues compression of u as a function of space-time efficient algorithms for computing the compressed u provable error and complexity bounds design of stable finite element spaces does not arise in the adaptive wavelet method [SS09]

7 References I hp-dg and -cg time-stepping [BJ90] Ivo Babuška and Tadeusz Janik, The h-p Version of the Finite Element Method for Parabolic Equations. II, Numerical Methods for PDEs, 6, no. 4, , 990 [Sch99] Dominik Schötzau, hp-dgfem for Parabolic Evolution Problems, Diss. ETH No. 304, 999 [SW0] Dominik Schötzau and Thomas P. Wihler, A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations, Numerische Mathematik, 5, no. 3, , 200 [HS0] Viet Ha Hoang and Christoph Schwab, Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs. I: Analytic regularity and gpc-approximation, SAM Report 200-, 200

8 References II Truly space-time wavelet methods [GO07] Michael Griebel and Daniel Oeltz, A sparse grid space-time discretization scheme for parabolic problems, Computing, 8, no., 34, 2007 [SS09] Christoph Schwab and Rob Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Mathematics of Computation, 78, no. 267, , 2009 [CS0] Rob Stevenson and Nabi Chegini, Adaptive wavelet methods for solving operator equations: parabolic problems, KdV report, 200 [A0] R.A., Space-time wavelet FEM for parabolic problems, submitted

9 Introduction continued Class of stable FE spaces: key steps variational formulation of (PDE) in space-time test space suitably finer than ansatz space (generic) anisotropic (tensor product) wavelets in space-time sparse tensor product spaces in space-time derivation of a non-square (overdetermined) linear system corresponding normal equations are well-conditioned iterative solution (= residual minimization)

10 Variational formulation The variational formulation, based on the bilinear form b : L 2 (J; V) H (J; V ) L 2 (J; V) H R, }{{}}{{} U V=V V 2 b(u, v) = t u + Au, v V V dt + u(0), v 2 H and the linear functional f : L 2 (J; V) H }{{} V=V V 2 R, f(v) = reads: find u U s.t. J b(u, v) = f(v) v V. J g, v V V dt + h, v 2 H,

11 Variational formulation Assume α > 0, c R s.t. ζ,η V : and sup t J A(t) L(V,V ) < J t A(t)ζ,η V V measurable A(t)η,η V V + c η 2 H α η 2 V g L 2 (J; V ) h H. for all t J Then b(u, ) = f( ) is well-posed [SS09]. In particular, inf sup u U\{0} v V\{0} b(u, v) u U v V > 0. Below assume: c = 0 and A = A but not A (t) K(V, V)

12 Abstract stability result Theorem Let R U and S = S S 2 V V 2 = V be subspaces. Assume there exists K(S ) > 0 such that z, z z S : z V S := sup V K(S ) z z S \{0} z V V R S and t R S R t=0 = {u(t = 0) H : u R} S 2 Then: γ > 0, only dependent on K(S ) and A, such that inf sup u R\{0} v S\{0} b(u, v) u U v V γ > 0.

13 Abstract stability result Btw: z, z z S : z V S := sup V K(S ) z z S \{0} z V V is equivalent to inf sup z S \{0} z S \{0} z, z V V z V z V K(S )

14 Abstract stability result Proof, step I Let u R. Set Z := V = L 2 (J; V) with Z =, /2 J Z, z, z Z = Az, z V V z, z Z. Note: Z L 2 (J;V) =: V and u R S Z. Define v 2 := u(0) H and v S by v, ṽ Z = t u + Au, ṽ V V ṽ S. J

15 Abstract stability result Proof, step II choosing ṽ = v yields fact b(u,(v, v 2 )) = t u + Au, v V V + u(0), v 2 H J = v, v Z + u(0), v 2 H = v 2 Z + v 2 2 V 2 choosing ṽ = u yields fact 2 v, u Z = t u + Au, u V V J = Au, u V V + t u, u V V J = u 2 Z + 2 J ( ) u(t) 2 H u(0) 2 H

16 Abstract stability result Proof, step III Using t u t R S we obtain fact 3 t u, ṽ V K(S ) t u V t u S = sup V ṽ S \{0} ṽ V = sup ṽ S \{0} J t u + Au Au, ṽ V V ṽ V v, ṽ Z u, ṽ Z = sup ṽ S \{0} ṽ V v = sup u, ṽ Z ṽ S \{0} ṽ Z v u Z Id L(V,Z) with Id L(V,Z) = ess sup t J A(t) /2 L(V,V ). ṽ Z ṽ V

17 Abstract stability result Proof, wrap up We obtain b(u,(v, v 2 )) = v 2 Z + v 2 2 V 2 v 2 Z + u(0) 2 H u(t) 2 H 2 = v u 2 Z + u 2 Z 3 γ 2( t u 2 V + u 2 Z with γ = min{k(s ) Id L(V,Z), }, and thus b(u,(v, v 2 )) γ t u 2 L 2 (J;V ) + u 2 Z v 2 Z + v 2 2 V 2 with γ γ. ) γ u L 2 (J;V) H (J;V ) (v, v 2 ) L 2 (J;V) H

18 Abstract stability result Assumptions on R U and S = S S 2 V V 2 = V there exists K(S ) > 0 such that inf sup z S \{0} z S \{0} R S and t R S R t=0 = {u(t = 0) H : u R} S 2 can be relaxed (numerics below). z, z V V z V z V K(S )

19 On condition K(S ) > 0 Lemma Assume S V satisfies inf sup s S \{0} s S \{0} similarly κ( S ) > 0 for S V s, s V V s V s V κ(s ) > 0 subspaces E, Ẽ L2 (J) are orthogonal in L 2 (J) Then S := (E S )+(Ẽ S ) L 2 (J; V) satisfies K(S ) inf{κ(s ),κ( S )} > 0. Note: generalizes to E, Ẽ,... L2 (J).

20 On condition K(S ) > 0 Proof of Lemma: dim E = dim Ẽ = Let v S and ṽ S (subscripts omitted). Let E = span{e} and Ẽ = span{ẽ}, v = e s and ṽ = ẽ s. By assumption, s S: κ(s) s V s V s, s V V w.l.o.g. s V = s V and similarly for some s S. Therefore, v := e s and ṽ := ẽ s are orthogonal in V and V, and moreover v + ṽ V v + ṽ V = v 2V + ṽ 2V v 2 V + ṽ 2 V = s 2V + s 2V s 2 V + s 2 V = s V s V + s V s V κ(s) v, v V V +κ( S) ṽ, ṽ V V max{κ(s),κ( S) } v + ṽ, v + ṽ V V

21 On condition K(S ) > 0 Application of Lemma Assume closed subspaces E (0) E ()... L 2 (J) closed subspaces S (0) S ()... V κ(s (l) ) κ > 0 for all l 0. Then for each L 0 there holds K ( L l=0 E (l) S (L l) ) κ > 0. Proof: using F (l) := ( E (l )) L 2 (J) E (l), l and F (0) := E (0), L l=0 E (l) S (L l) = L l=0 ( E (l ) + F (l)) S (L l) = L l=0 F (l) S (L l).

22 Example List of symbols Time interval J = (0, T) R Open Lipschitz domain D R d, d H 0 (D) = V H = L2 (D) = H V = H (D) Initial datum h L 2 (D) Source term g L 2 (J, H (D)) Conductivity q L (D J) uniformly positive, 0 < a min ess inf D J q ess sup q a max < D J

23 Example (PDE) in strong form Find u : D J R such that t u(x, t) (q(x, t) u(x, t)) = g(x, t), (x, t) D J with initial condition u(x, 0) = h(x), x D and boundary condition u(x, t) = 0, (x, t) D J

24 Example (PDE) in weak form Find u U = L 2 (J, H0 (D)) H (J, H (D)) such that (v t u + q u v ) dxdt = gv dxdt and J D D u(0)v 2 dx = D J hv 2 dx for all v = (v, v 2 ) V = L 2 (J, H 0 (D)) L2 (D), where q, u, v, g depend on (x, t), and u(0), h, v 2 on x D

25 Example (PDE) in weak form With bilin. form b : U V R and lin. functional f : V R, b(u, v) = (v t u + q u v ) dxdt + u(0)v 2 dx and J D f(v) = the variational formulation reads: find J D gv dxdt + hv 2 dx, D u U : b(u, v) = f(v) v V D

26 Example Choice of temporal basis Let Θ L = {θ λ : λ I Θ : λ L} L 2 (J) be a Riesz basis for its span uniformly in level L 0: Θ L c L 2 (J) c 2 c R #Θ L defined w.r.t. an equidistant partition of J = (0, T) with width h 2 L piecewise linear continuous s.t. l 0 Θ l rescales to a Riesz basis for H (J)

27 Θ Θ \ Θ Θ 2 \ Θ Figure: Piecewise linear continuous biorthogonal B-spline wavelets

28 Example Choice of spatial basis Let Σ L = {σ µ : µ I Σ : µ L} L 2 (D) be a Riesz basis for its span uniformly in level L 0 defined w.r.t. a sequence of quasi-uniform nested triangulations of D R d with mesh width h 2 L s.t. l 0 Σ l rescales to a Riesz basis for H 0 (D) (s.t. l 0 Σ l rescales to a Riesz basis for H (D))

29 Example Choice of ansatz/test spaces I Define full tensor product ansatz space R L U spanned by {θ λ σ µ : λ L, µ ρl} test space S L V spanned by {θ λ σ µ : λ L+, µ ρl} {σ µ : µ ρl} where ρ > 0 is the degree of anisotropy in the refinement

30 Example Choice of ansatz/test spaces II Define sparse tensor product ansatz space R L U spanned by {θ λ σ µ : λ +ρ µ L} test space S L V spanned by {θ λ σ µ : min{ λ, 0}+ρ µ L} {σ µ : µ ρl} where ρ > 0 is the degree of anisotropy in the refinement

31 Example Formulation via residual minimization (FTP case) The discrete version of the variational formulation: find ũ L := arg min w R L max v S L \{0} b(w, v) f(v) v V motivates, via Riesz bases l 0 Θ l and l 0 Σ l, the definition u L := arg min B L w f L 2 w R dim R L (argmin) which defines u L = [Θ L Σ L ] u L R L. In general u L ũ L.

32 Example Formulation via residual minimization (FTP case) The discrete version of the variational formulation: find ũ L := arg min w R L max v S L \{0} b(w, v) f(v) v V motivates, via Riesz bases l 0 Θ l and l 0 Σ l, the definition u L := arg min B L w f L 2 w R dim R L (argmin) which defines u L = [Θ L Σ L ] u L R L. In general u L ũ L.

33 Example Formulation via residual minimization (FTP case) The discrete version of the variational formulation: find ũ L := arg min w R L max v S L \{0} b(w, v) f(v) v V motivates, via Riesz bases l 0 Θ l and l 0 Σ l, the definition u L := arg min B L w f L 2 w R dim R L (argmin) which defines u L = [Θ L Σ L ] u L R L. In general u L ũ L.

34 Example Formulation via residual minimization (FTP case) The discrete version of the variational formulation: find ũ L := arg min w R L max v S L \{0} b(w, v) f(v) v V motivates, via Riesz bases l 0 Θ l and l 0 Σ l, the definition u L := arg min B L w f L 2 w R dim R L (argmin) which defines u L = [Θ L Σ L ] u L R L. In general u L ũ L.

35 Example Quasi-optimality Theorem The least-squares Galerkin solution u L exists, is unique and converges quasi-optimally uniformly in L 0: there exists C > 0 such that L 0 u L u U C inf w L u U. w L R L

36 Example Numerics I smooth solution, FTP Experimental set-up D = (, ) R, J = (0, T) with T = 2 conductivity in (PDE) is q exact solution u(x, t) = e t cos xπ 2, (x, t) D J, piecewise linear continuous B-spline wavelet basis degree of anisotropy in refinement ρ =

37 Example Numerics I smooth solution, FTP Error in L 2 (J, H 0(D)) Rate /2 Error in L 2 (J, L 2 (D)) Initial condition error in L 2 (D) Rate Total number of degrees of freedom Figure: Convergence of the Galerkin solution u L to a smooth u

38 Example Numerics I smooth solution, FTP The system (argmin) is sparse well-conditioned which suggests the use of an iterative least squares algorithm Level L ndof # iterations log 0 rel. res Table: The number of MATLAB s lsqr iterations with tolerance 0 0, the corresponding number of degrees of freedom, and the relative residual on various discretization levels L for u L Paige and Saunders, SIAM J. Numer. Anal, 2, 975

39 Example Numerics II rough solution, FTP Experimental set-up D = (, ) R, J = (0, T) with T = 2 source g, initial condition h 0 conductivity in (PDE) is q(x, t) = 2 sign(2+x 2t) piecewise linear continuous B-spline wavelet basis degree of anisotropy in refinement ρ = reference solution on level L = 7

40 q(t, x) 2 x q(t, x) t Figure: Conductivity q

41 Example Numerics II rough solution, FTP Error in X Error in L 2 (J, H 0(D)) Error in L 2 (J, L 2 (D)) Initial condition error in L 2 (D) Total number of degrees of freedom Figure: Convergence 2 of the Galerkin solution u L to u 7 2 X U

42 Example Numerics III FTP vs STP Experimental set-up D = (, ) R, J = (0, T) with T = 2 conductivity q, initial condition h 0 source g(x, t) = cos(x + cos πt πt 2 ) sin2 2 piecewise linear continuous B-spline wavelet basis degree of anisotropy in refinement ρ = reference FTP solution on level L = 7

43 x t x t Figure: Source g (left) and solution u (right)

44 Example Numerics III FTP vs STP 0 0 Rate /2 FTP, error in X Rate STP, error in X Total number of degrees of freedom Figure: Convergence 3 of the Galerkin solution: STP vs FTP 3 X U

45 Outro Review sparse tensor formulation of (PDE) in space-time inf-sup condition for the associated bilinear form quasi-optimality of the Galerkin least squares solution examples In progress sparse discretization of parametric parabolic equations weighted spaces for corners of the space-time cylinder

46 Thank You!