Adaptive Wavelet Methods for Elliptic SPDEs

Transcription

1 Adaptive Wavelet Methods for Elliptic SPDEs Klaus Ritter Computational Stochastics TU Kaiserslautern 1/1

2 Introduction In most papers on approximation of SPDEs non-adaptive discretization discretization in space and time is fixed in advance; typically uniform. 2/3

3 Introduction In most papers on approximation of SPDEs non-adaptive discretization In this talk adaptive discretization discretization is constructed during the computation. 2/2

4 Introduction In most papers on approximation of SPDEs non-adaptive discretization In this talk adaptive discretization for a very simple model problem. Given a bounded Lipschitz domaind R d, a random functionx : Ω L 2 (D). Approximate the solution U of the Poisson equation U = X U = 0 ond, on D. 2/1

5 OUTLINE I. Non-linear Approximation of X II. Adaptive Wavelet Algorithms for U Joint work with P. Cioica, S. Dahlke, S. Kinzel (Marburg), F. Lindner, R. Schilling (Dresden), N. Döhring (Kaiserslautern), and T. Raasch (Mainz) Supported by the DFG with Priority Programme /1

6 I. Non-linear Approximation of Random Functions OUTLINE 1. The Stochastic Model 2. Linear and Non-linear Approximation 3. Besov Regularity 4/1

7 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, 5/6

8 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, i.e., for it holds x = j=0 k j c j,k ψ j,k L 2 (D) x 2 2 j=0 k j c 2 j,k. 5/5

9 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, parametersα > 0 and0 β 1 withα+β > 1 as well asγ R. LetY j,k,z j,k be independent with Y j,k B(1,2 βjd ), Z j,k N(0,j γd 2 αjd ). 5/4

10 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, parametersα > 0 and0 β 1 withα+β > 1 as well asγ R. LetY j,k,z j,k be independent with Y j,k B(1,2 βjd ), Z j,k N(0,j γd 2 αjd ). Define X = j=0 k j Y j,k Z j,k ψ j,k. ( ) 5/3

11 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, parametersα > 0 and0 β 1 withα+β > 1 as well asγ R. LetY j,k,z j,k be independent with Y j,k B(1,2 βjd ), Z j,k N(0,j γd 2 αjd ). Define X = j=0 k j Y j,k Z j,k ψ j,k. ( ) Note thatx is Gaussian iffβ = 0. 5/2

12 I.1 The Stochastic Model Given a bounded Lipschitz domaind R d, a Riesz basis(ψ j,k ) j N0, k j forl 2 (D) with# j 2 jd, parametersα > 0 and0 β 1 withα+β > 1 as well asγ R. LetY j,k,z j,k be independent with Y j,k B(1,2 βjd ), Z j,k N(0,j γd 2 αjd ). Define X = j=0 k j Y j,k Z j,k ψ j,k. ( ) Note thatx is Gaussian iffβ = 0. Moreover,( ) is the Karhunen-Loève expansion iff(ψ j,k ) j N0, k j is an ONB. 5/1

13 A realization forα = 2.0,β = 0.0, andγ = level j k 6 6/1

14 A realization forα = 2.0,β = 0.1, andγ = level j k 6 7/1

15 A realization forα = 2.0,β = 0.25, andγ = level j k 6 8/1

16 A realization forα = 2.0,β = 0.5, andγ = level j k 5 6 9/1

17 A realization forα = 2.0,β = 0.75, andγ = level j k 6 10/1

18 A realization forα = 2.0,β = 0.9, andγ = level j k 6 11/1

19 A realization forα = 1.0,β = 0.1, andγ = level j k /1

20 A realization forα = 1.0,β = 0.9, andγ = level j k 6 13/1

21 A realization forα = 3.0,β = 0.1, andγ = level j k 6 14/1

22 A realization forα = 3.0,β = 0.9, andγ = level j k 6 15/1

23 I.2 Linear and Non-linear Approximation Put e( X) = ( E X X 2 L 2 (D)) 1/2. 16/2

24 I.2 Linear and Non-linear Approximation Put e( X) = ( E X X 2 L 2 (D)) 1/2. Linear approximation errors with infimum over all XN such that see Mathé (1990). e lin (N) = infe( X N ) dimspan( X N (Ω))) N, 16/1

25 Theorem 1 C-D-D-K-L-R-R-S (2010) For linear approximation e lin (N) (lnn) γd 2 N α+β /4

26 Theorem 1 C-D-D-K-L-R-R-S (2010) For linear approximation e lin (N) (lnn) γd 2 N α+β 1 2. Examples of classical Gaussian random functionx ond = [0,1] d, i.e.,β = 0. For the Brownian sheet α = 2, γ = 2(d 1). d 17/3

27 Theorem 1 C-D-D-K-L-R-R-S (2010) For linear approximation e lin (N) (lnn) γd 2 N α+β 1 2. Examples of classical Gaussian random functionx ond = [0,1] d, i.e.,β = 0. For the Brownian sheet α = 2, For Lévy s Brownian motion γ = 2(d 1). d α = d+1 d, γ = 0. See Papageorgiou, Wasilkowski (1990), Wasilkowski (1993), and R (2000). 17/2

28 Theorem 1 C-D-D-K-L-R-R-S (2010) For linear approximation e lin (N) (lnn) γd 2 N α+β 1 2. Examples of classical Gaussian random functionx ond = [0,1] d, i.e.,β = 0. For the Brownian sheet α = 2, γ = 2(d 1). d For Lévy s Brownian motion α = d+1 d, γ = 0. See Papageorgiou, Wasilkowski (1990), Wasilkowski (1993), and R (2000). Withβ > 0 we get sparse counterparts thereof. 17/1

29 Forx = j=0 k j c j,k ψ j,k L 2 (D) η(x) = #{(j,k) : c j,k 0}. 18/3

30 Forx = j=0 k j c j,k ψ j,k L 2 (D) η(x) = #{(j,k) : c j,k 0}. Two variants of best N -term approximation with infimum over all XN such that e(n) = infe( X N ) η( X N ) N a.s. 18/2

31 Forx = j=0 k j c j,k ψ j,k L 2 (D) η(x) = #{(j,k) : c j,k 0}. Two variants of best N -term approximation with infimum over all XN such that e(n) = infe( X N ) η( X N ) N a.s. and e avg (N) = infe( X N ) with infimum over all XN such that E(η( X N )) N. 18/1

32 Theorem 2 C-D-D-K-L-R-R-S (2010) For average N -term approximation e avg (N) (lnn) γd N γd 2 N α+β 1 2(1 β), ifβ < αdn/2, ifβ = 1. 19/4

33 Theorem 2 C-D-D-K-L-R-R-S (2010) For average N -term approximation e avg (N) (lnn) γd N γd 2 N α+β 1 ForN -term approximation, with anyε > 0, e(n) 2(1 β), ifβ < αdn/2, ifβ = 1. N α+β 1 2(1 β) +ε, ifβ < 1 N 1/ε, ifβ = 1. 19/3

34 Theorem 2 C-D-D-K-L-R-R-S (2010) For average N -term approximation e avg (N) (lnn) γd N γd 2 N α+β 1 2(1 β), ifβ < αdn/2, ifβ = 1. Remarks Forβ < 1 e avg (N 1 β ) e lin (N). 19/2

35 Theorem 2 C-D-D-K-L-R-R-S (2010) For average N -term approximation e avg (N) (lnn) γd N γd 2 N α+β 1 2(1 β), ifβ < αdn/2, ifβ = 1. Remarks Forβ < 1 e avg (N 1 β ) e lin (N). Simulation of XN that yield the upper bound fore avg (N) is possible at costn. 19/1

36 Nonlinear approximation of deterministic functions..., DeVore (1998),... 20/3

37 Nonlinear approximation of deterministic functions..., DeVore (1998),... Nonlinear approximation of stochastic processes, D = [0, 1], wavelet methods for piecewise stationary processes, Cohen, d Ales (1997), Cohen, Daubechies, Guleryuz, Orchard (2002), 20/2

38 Nonlinear approximation of deterministic functions..., DeVore (1998),... Nonlinear approximation of stochastic processes, D = [0, 1], wavelet methods for piecewise stationary processes, Cohen, d Ales (1997), Cohen, Daubechies, Guleryuz, Orchard (2002), free Knot splines for Brownian motion, SDEs, Kon, Plaskota (2005), Creutzig, Müller-Gronbach, R (2007), Slassi (2010), free Knot splines for Lévy driven SDEs, Dereich (2010), Dereich, Heidenreich (2010). 20/1

39 I.3 Besov Regularity In the sequel0 < p,q,s <. Assumption For allx = j=0 k j c j,k ψ j,k L 2 (D) ( q/p x q Bq(L s p (D)) 2 j(s+d(1/2 1/p))q cj,k) p. k j j=0 21/3

40 I.3 Besov Regularity In the sequel0 < p,q,s <. Assumption For allx = j=0 k j c j,k ψ j,k L 2 (D) ( q/p x q Bq(L s p (D)) 2 j(s+d(1/2 1/p))q cj,k) p. k j j=0 Theorem 3 C-D-D-K-L-R-R-S (2010) Letγ = 0. ThenX B s q(l p (D))P -a.s. iff in which casee α 1 2 ( ) X q Bq(L s p (D)) + β p > s d, <. 21/2

41 I.3 Besov Regularity In the sequel0 < p,q,s <. Assumption For allx = j=0 k j c j,k ψ j,k L 2 (D) ( q/p x q Bq(L s p (D)) 2 j(s+d(1/2 1/p))q cj,k) p. k j j=0 Theorem 3 C-D-D-K-L-R-R-S (2010) Letγ = 0. ThenX B s q(l p (D))P -a.s. iff in which casee α 1 2 ( ) X q Bq(L s p (D)) + β p > s d, <. Remark See Abramovich, Sapatinas, Silverman (1998) and Bochkina (2006) ford = 1 andp,q 1. Application: Bayesian non-parametric regression. 21/1

42 (a) α = 2.0,β = 0.0 (b) α = 1.8,β = 0.2 (c) α = 1.5,β = 0.5 (d) α = 1.2,β = /1

43 II. Adaptive Wavelet Algorithms for Stochastic Poisson Eqn s With X as previously, U = X U = 0 ond, on D defines a random functionu : Ω H 1 (D). 23/2

44 II. Adaptive Wavelet Algorithms for Stochastic Poisson Eqn s With X as previously, U = X U = 0 ond, on D defines a random functionu : Ω H 1 (D). Put 1/2. (E U e(û) = Û 2 H (D)) 1 We study best N -term approximation with infimum over allûn such that e(n) = infe(ûn) η(ûn) N a.s. 23/1

45 Theorem 4 C-D-D-K-L-R-R-S (2010) II.1 Approximation Rates Letd {2,3}. Put ρ = min ( 1 α+β 1, 2(d 1) ). 3d ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε. 24/3

46 Theorem 4 C-D-D-K-L-R-R-S (2010) II.1 Approximation Rates Letd {2,3}. Put ρ = min ( 1 α+β 1, 2(d 1) ). 3d ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε. Remark Uniform discretizations yieldn 1/(2d) on general Lipschitz domains. We haveρ > 1/(2d). 24/2

47 Theorem 4 C-D-D-K-L-R-R-S (2010) II.1 Approximation Rates Letd {2,3}. Put ρ = min ( 1 α+β 1, 2(d 1) ). 3d ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε. Remark Uniform discretizations yieldn 1/(2d) on general Lipschitz domains. We haveρ > 1/(2d). In the sequel: better results for more specific domains D. 24/1

48 Theorem 5 C-D-D-K-L-R-R-S (2010) LetD R 2 be a polygonal domain. Put ρ = α+β 2 ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε.. 25/2

49 Theorem 5 C-D-D-K-L-R-R-S (2010) LetD R 2 be a polygonal domain. Put ρ = α+β 2 ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε.. Theorem 6 C-D-D-K-L-R-R-S (2010) LetD R d be ac -domain. Put ρ = 1 ( α 1 1 β 2 ) +β + 1 d. ForN -term approximation ofu, with anyε > 0, e(n) N ρ+ε. 25/1

50 Remarks Adaptive wavelet schemes yield the rates from Theorems 4 6, provided that a suitable wavelet Riesz basis is available on D. Their the computational cost is proportional to N. 26/3

51 Remarks Adaptive wavelet schemes yield the rates from Theorems 4 6, provided that a suitable wavelet Riesz basis is available on D. Their the computational cost is proportional to N. For adaptive wavelet schemes for deterministic equations see Cohen, Dahmen, DeVore (2001), Gantumur, Harbrecht, Stevenson (2007), Stevenson, Werner (2008),... 26/2

52 Remarks Adaptive wavelet schemes yield the rates from Theorems 4 6, provided that a suitable wavelet Riesz basis is available on D. Their the computational cost is proportional to N. For adaptive wavelet schemes for deterministic equations see Cohen, Dahmen, DeVore (2001), Gantumur, Harbrecht, Stevenson (2007), Stevenson, Werner (2008),... For approximation of elliptic equations with random right-hand side/coefficients see R, Wasilkowski (1996), Xiu, Karniadakis (2002), Babuška, Tempone, Zouraris (2004), Nobile, Tempone, Webster (2008), Cohen, DeVore, Schwab (2010),... 26/1

53 II.2 Numerical Results Specifically D = [0,1], piecewise quadratic primal wavelet basis, see Primbs (2006), exact solution via master computation. 27/4

54 II.2 Numerical Results Specifically D = [0,1], piecewise quadratic primal wavelet basis, see Primbs (2006), exact solution via master computation ( 1/2, independent samples to estimate E U Û 2 H (D)) 1 95%-confidence intervals of length less that 3% of the estimate. 27/3

55 II.2 Numerical Results Specifically D = [0,1], piecewise quadratic primal wavelet basis, see Primbs (2006), exact solution via master computation ( 1/2, independent samples to estimate E U Û 2 H (D)) 1 95%-confidence intervals of length less that 3% of the estimate. adaptive wavelet scheme with 18 different accuracy levels, uniform scheme with 6 truncation levels. 27/2

56 II.2 Numerical Results Specifically D = [0,1], piecewise quadratic primal wavelet basis, see Primbs (2006), exact solution via master computation ( 1/2, independent samples to estimate E U Û 2 H (D)) 1 95%-confidence intervals of length less that 3% of the estimate. adaptive wavelet scheme with 18 different accuracy levels, uniform scheme with 6 truncation levels. TheH s -regularity of X kept constant. s < α+β /1

57 Smoothness of the right-hand sidex:α = 0.9,β = 0.2,γ = 0. 1 uniform adaptive log error log N 28/2

58 Smoothness of the right-hand sidex:α = 0.9,β = 0.2,γ = 0. 1 uniform adaptive log error log N Orders of convergence empirically 1.11 and 1.16, upper bounds 1.05 and 19/16 = /1

59 Smoothness of the right-hand sidex:α = 0.4,β = 0.7,γ = uniform adaptive 2 log error log N 29/2

60 Smoothness of the right-hand sidex:α = 0.4,β = 0.7,γ = uniform adaptive 2 log error log N Orders of convergence empirically 1.11 and 1.55, upper bounds1.05 and7/3 = /1

61 Summary Besov regularity and linear/non-linear approximation of random functions with sparse wavelet expansions α+β 1 2 vs. α+β 1 2(1 β). Adaptive wavelet algorithms for stochastic Poisson equations. 30/2

62 Summary Besov regularity and linear/non-linear approximation of random functions with sparse wavelet expansions α+β 1 2 vs. α+β 1 2(1 β). Adaptive wavelet algorithms for stochastic Poisson equations. Outlook Parabolic SPDE du(t) = ( U(t)+A(t,U(t)) ) dt+b(t,(u(t))dx(t). 30/1