Besov Regularity and Approximation of a Certain Class of Random Fields

Transcription

1 Besov Regularity and Approximation of a Certain Class of Random Fields Nicolas Döhring (TU Kaiserslautern) joint work with F. Lindner, R. Schilling (TU Dresden), K. Ritter (TU Kaiserslautern), T. Raasch (Uni Mainz), P. Cioica, S. Kinzel, S. Dahlke (Uni Marburg) supported by DFG priority programme /14

2 The Problem Consider a random function X : Ω L 2 (D) with a bounded Lipschitz domain D R d. 2/14

3 The Problem Consider a random function X : Ω L 2 (D) with a bounded Lipschitz domain D R d. Computational task: approximation of X. In particular linear vs. nonlinear approximation. 2/14

4 Motivation A centered Gaussian process X admits the Karhunen-Loève decomposition X(u) = i NZ i e i (u) with an orthonormal basis (e i ) i N of L 2 (D) and independent, centered Gaussian random variables (Z i ) i N. 3/14

5 Motivation A centered Gaussian process X admits the Karhunen-Loève decomposition X(u) = i NZ i e i (u) with an orthonormal basis (e i ) i N of L 2 (D) and independent, centered Gaussian random variables (Z i ) i N. Modify the representation, so that it is possible to introduce sparsity, work with wavelets. 3/14

6 The Model Definition A family (e i ) i N L 2 (D) ist called Riesz basis, if and only if f L 2 (D) 1 (c i ) l 2 (N) f = i c i e i and A,B > 0 f L 2 (D) A c l 2 f L2 (D) B c l 2. 4/14

7 The Model Definition A family (e i ) i N L 2 (D) ist called Riesz basis, if and only if f L 2 (D) 1 (c i ) l 2 (N) f = i c i e i and A,B > 0 f L 2 (D) A c l 2 f L2 (D) B c l 2. From now on let Ψ = (ψ j,k ) be an L 2 (D) (wavelet) Riesz basis with dyadic refinement level j j 0, spatial location k j, with j 2 jd. 4/14

8 The Model Let Y j,k,z j,k be a family of independent random variables with P(Y j,k = 1) = 1 P(Y j,k = 0) = 2 βjd, 0 β 1, Z j,k N(0,σj 2) with σ2 j = 2 αjd, α > 0, α+β > 1. 5/14

9 The Model Let Y j,k,z j,k be a family of independent random variables with P(Y j,k = 1) = 1 P(Y j,k = 0) = 2 βjd, 0 β 1, Z j,k N(0,σj 2) with σ2 j = 2 αjd, α > 0, α+β > 1. Define X(u) = Y j,k Z j,k ψ j,k (u). k j j=j 0 5/14

10 The Model Let Y j,k,z j,k be a family of independent random variables with P(Y j,k = 1) = 1 P(Y j,k = 0) = 2 βjd, 0 β 1, Z j,k N(0,σj 2) with σ2 j = 2 αjd, α > 0, α+β > 1. Define X(u) = Y j,k Z j,k ψ j,k (u). k j j=j 0 Thus α Gaussian decay parameter, β sparsity parameter. 5/14

11 Realisations of X α = 2.0, β = 0.0 (c.f. Brownian motion) coefficients sample path 6/14

12 Realisations of X α = 1.2, β = 0.8 coefficients sample path 6/14

13 Regularity of X Theorem (CDD... 10) Suppose that s > d ( 1 p 1) +. We have if and only if Furthermore E X q Bq s (Lp(D)) <. X B s q(l p (D)) a.s. ( α 1 s < d + β ). 2 p 7/14

14 Regularity of X Theorem (CDD... 10) Suppose that s > d ( 1 p 1) +. We have if and only if Furthermore E X q Bq s (Lp(D)) <. X B s q(l p (D)) a.s. ( α 1 s < d + β ). 2 p See Abramovich, Sapatinas, Silverman (1998) and Bochkina (2006) in the context of Bayesian nonparametric regression for the case d = 1, p,q 1. 7/14

15 Wavelet Characterisation Assumption The wavelet basis induces characterisation of Besov spaces B s q(l p (D)) of the form v B s q (L p(d)) j=j 0 2 j(s+d( p ))q q p v, ψ j,k L2 (D) p k j for 0 < p,q < and all s with d( 1 p 1) + < s < s 1 for some parameter s 1 > 0. 1 q, 8/14

16 Linear Approximation Definition Linear approximation error e lin N (X) = inf(e X X 2 L 2(D) )1 2, where X : Ω L 2 (D) measurable, such that dim(span(ˆx(ω))) N. X 9/14

17 Linear Approximation Definition Linear approximation error e lin N (X) = inf(e X X 2 L 2(D) )1 2, where X : Ω L 2 (D) measurable, such that dim(span(ˆx(ω))) N. X Theorem (CDD... 10) e lin N (X) N α+β /14

18 Nonlinear Approximation For f = j,k c j,k ψ j,k L 2 (D) let η(f) = #{(j,k) : c j,k 0}. 10/14

19 Nonlinear Approximation For f = j,k c j,k ψ j,k L 2 (D) let η(f) = #{(j,k) : c j,k 0}. Definition Best average N-term approximation error e non N (X) = inf(e X X 2 L 2 (D) X )1 2, where X : Ω L 2 (D) measurable, such that E(η( X)) N. 10/14

20 Nonlinear Approximation See..., DeVore (1998),... Cohen, d Ales (1997), Cohen, Daubechies, Guleryuz, Orchard (2002), Kon, Plaskota (2005), Creutzig, Müller-Gronbach, Ritter (2007), Dereich, Heidenreich (2010),... Definition Best average N-term approximation error e non N (X) = inf(e X X 2 L 2 (D) X )1 2, where X : Ω L 2 (D) measurable, such that E(η( X)) N. 10/14

21 Nonlinear Approximation Theorem (CDD... 10) e non N (X) { 2(1 β), if β [0,1), N α+β 1 2 αdn 2, if β = 1. 11/14

22 Nonlinear Approximation Theorem (CDD... 10) e non N (X) { 2(1 β), if β [0,1), N α+β 1 2 αdn 2, if β = 1. Recall: en lin (X) N α+β /14

23 Nonlinear Approximation Theorem (CDD... 10) e non N (X) { 2(1 β), if β [0,1), N α+β 1 2 αdn 2, if β = 1. Recall: en lin (X) N α+β 1 2 Remark For β (0,1) we have e non N 1 β e lin N. Upper bound in the theorem can be achieved by an algorithm at an average computational cost of order E(η( X)). 11/14

24 Application Consider the equation U = X in D, U = 0 on D. 12/14

25 Application Consider the equation U = X in D, U = 0 on D. Definition Approximation error e N,H 1 (D)(U) = inf(e U Û 2 H 1 (D) )1 2, where Û : Ω H1 (D) measurable, such that η(û) N a.s. 12/14

26 Application Theorem (CDD... 10) Let d {2,3} and ( ρ = min 1 α+β 1, + 2 2(d 1) 6 3d ). Then for every ε > 0 the error of the best N-term approximation satisfies e N,H 1 (D)(U) N ρ+ε. 13/14

27 Summary Recall: Gaussian decay parameter α, sparsity parameter β. Results: ( ) X Bq(L s α 1 p (D)) s < d 2 + β p. Linear approximation rate determined by α + β. 1 Nonlinear approximation rate determined by 1 β (α+β). For β 0 nonlinear approximation pays off. 14/14

28 Summary Recall: Gaussian decay parameter α, sparsity parameter β. Results: ( ) X Bq(L s α 1 p (D)) s < d 2 + β p. Linear approximation rate determined by α + β. 1 Nonlinear approximation rate determined by 1 β (α+β). For β 0 nonlinear approximation pays off. Application: Elliptic boundary value problems with random right-hand sides. Work in progress: lower bounds for en non(x), errors in Bq(L s p (D));... 14/14