Regularity of Stochastic Partial Differential Equations in Besov Spaces Related to Adaptive Schemes

Transcription

1 Regularity of Stochastic Partial Differential Equations in Besov Spaces Related to Adaptive Schemes Petru A. Cioica Philipps-Universität Marburg Workshop on New Discretization Methods for the Numerical Approximation of PDEs Oberwolfach January 2 6, 205 joint with: René L. Schilling (TU Dresden) Klaus Ritter, Felix Lindner, Nicolas Döhring (TU Kaiserslautern) Stephan Dahlke, Stefan Kinzel (Philipps-Universität Marburg) Kyeong-Hun Kim (Korea University) Kijung Lee (Ajou University) Thorsten Raasch (Johannes-Gutenberg-Universität Mainz) supported by: DFG-SPP 324 NRF Korea Philipps-Universität Marburg

2 Stochastic partial differential equation (SPDE) ( d ( d du = a ij u x i x )dt+ j +f σ ik u x i +g )dw k t k i,j= k= i= on Ω [0,T] O, u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. ( ) O R d bounded Lipschitz domain. {(w k t) t [0,T] : k N} independent standard Brownian motions... w.r.t. a normal Filtration (F t ) t [0,T]... on a complete probability space (Ω,F,P).

3 Stochastic partial differential equation (SPDE) ( d ( d du = a ij u x i x )dt+ j +f σ ik u x i +g )dw k t k i,j= k= i= on Ω [0,T] O, u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. ( ) O R d bounded Lipschitz domain. {(w k t) t [0,T] : k N} independent standard Brownian motions... w.r.t. a normal Filtration (F t ) t [0,T]... on a complete probability space (Ω,F,P). Solution u = u(ω,t,x) = u(randomness,time,space) u : Ω [0,T] O R

4 Stochastic partial differential equation (SPDE) ( d ( d du = a ij u x i x )dt+ j +f σ ik u x i +g )dw k t k i,j= k= i= on Ω [0,T] O, u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. ( ) O R d bounded Lipschitz domain. {(w k t) t [0,T] : k N} independent standard Brownian motions... w.r.t. a normal Filtration (F t ) t [0,T]... on a complete probability space (Ω,F,P). Solution u = u(ω,t,x) = u(randomness,time,space) u : Ω [0,T] O R u : Ω [0,T] D (O;R)

5 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d )

6 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ).

7 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ). Alternative: spatially adaptive wavelet methods

8 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ). Alternative: spatially adaptive wavelet methods [+] promise better convergence rates [ ] harder to implement and analyse

9 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ). Alternative: spatially adaptive wavelet methods [+] promise better convergence rates [ ] harder to implement and analyse Question: Does adaptivity really pay?

10 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ). Alternative: spatially adaptive wavelet methods [+] promise better convergence rates [ ] harder to implement and analyse Question: Does adaptivity really pay? benchmark: best N-term wavelet approximation v B α τ,τ (O), τ = α d + p v ṽ N Lp(O) = O(N α/d )

11 Motivation: Numerical treatment of SPDE ( ) So far: predominantly spatially uniform methods v W s p (O) v v N Lp(O) = O(N s/d ) Problem: Poor spatial Sobolev regularity of solution u to ( ). Alternative: spatially adaptive wavelet methods [+] promise better convergence rates [ ] harder to implement and analyse Question: Does adaptivity really pay? benchmark: best N-term wavelet approximation v B α τ,τ (O), τ = α d + p v ṽ N Lp(O) = O(N α/d ) Here: v = u(ω,t, ), where u solves SPDE ( )

12 Goal and motivation Goal: Motivation: Regularity analysis Does spatial of SPDE ( ) adaptivity in the adaptivity scale for SPDEs Bτ,τ(O), α τ = α d +. really pay? p

13 Starting point: Krylov s analytic approach to SPDEs ( d ( d du = a ij u x i x )dt+ j +f σ ik u x i +g )dw k t k i,j= k= i= on Ω [0,T] O, u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. ( )

14 Starting point: Krylov s analytic approach to SPDEs ( d ( d du = a ij u x i x )dt+ j +f σ ik u x i +g )dw k t k i,j= k= i= on Ω [0,T] O, u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. ( ) u = u(ω,t, ) D (O;R) solves ( ), if for all ϕ C 0 (O) P-a.s.: u(t, ),ϕ = u 0 ( ),ϕ t d + a ij (s, )u x i x j(s, )+f(s, ),ϕ ds + k= for all t [0,T]. 0 i,j= t 0 d σ ik (s, )u x i(s, )+g k (s, ),ϕ dws k, i=

15 Weighted Sobolev spaces H γ p,θ (O) (in the analytic approach) p > (integrability) γ R (smoothness) θ R (weight) weights: powers of the distance to the boundary for γ N 0 : v H γ p,θ (O) : v p H γ := p,θ (O) α γ ρ(x) := dist(x, O), x O. O ρ(x) α D α v(x) p ρ(x) θ d dx <

16 Weighted Sobolev spaces H γ p,θ (O) (in the analytic approach) p > (integrability) γ R (smoothness) θ R (weight) weights: powers of the distance to the boundary for γ N 0 : v H γ p,θ (O) : v p H γ := p,θ (O) α γ ρ(x) := dist(x, O), x O. O D α v(x) p dx <

17 Weighted Sobolev spaces H γ p,θ (O) (in the analytic approach) p > (integrability) γ R (smoothness) θ R (weight) weights: powers of the distance to the boundary for γ N 0 : v H γ p,θ (O) : v p H γ := p,θ (O) α γ ρ(x) := dist(x, O), x O. O ρ(x) α D α v(x) p ρ(x) θ d dx <

18 Weighted Sobolev spaces H γ p,θ (O) (in the analytic approach) p > (integrability) γ R (smoothness) θ R (weight) weights: powers of the distance to the boundary for γ N 0 : v H γ p,θ (O) : v p H γ := p,θ (O) α γ ρ(x) := dist(x, O), x O. O ρ(x) α D α v(x) p ρ(x) θ d dx < for γ (0, )\N: via complex interpolation for γ < 0: via duality

19 Weighted Sobolev spaces H γ p,θ (O) (in the analytic approach) p > (integrability) γ R (smoothness) θ R (weight) weights: powers of the distance to the boundary for γ N 0 : v H γ p,θ (O) : v p H γ := p,θ (O) α γ ρ(x) := dist(x, O), x O. O ρ(x) α D α v(x) p ρ(x) θ d dx < for γ (0, )\N: via complex interpolation for γ < 0: via duality analogous spaces for l 2 -valued functions: H γ p,θ (O;l 2)

20 ( d du = u = 0 u(0) = u 0 i,j= a ij u x i x j +f )dt+ on Ω [0,T] O, on Ω (0,T] O, on Ω O. ( d σ ik u x i +g )dw k t k k= i= ( ) Theorem (Existence and uniqueness K.-H. Kim 202) For γ R, p 2 and θ d+p 2: If f L p (Ω [0,T];H γ 2 p,θ+p (O)), g L p (Ω [0,T];H γ p,θ (O;l 2)), and u 0 L p (Ω;H γ 2/p p,θ p+2 (O)), then SPDE ( ) has a unique solution u L p (Ω [0,T];H γ p,θ p (O)).

21 ( d du = u = 0 u(0) = u 0 i,j= a ij u x i x j +f )dt+ on Ω [0,T] O, on Ω (0,T] O, on Ω O. ( d σ ik u x i +g )dw k t k k= i= ( ) Theorem (Existence and uniqueness K.-H. Kim 202) For γ R, p 2 and θ d+p 2: If f L p (Ω [0,T];H γ 2 p,θ+p (O)), g L p (Ω [0,T];H γ p,θ (O;l 2)), and u 0 L p (Ω;H γ 2/p p,θ p+2 (O)), then SPDE ( ) has a unique solution u L p (Ω [0,T];H γ p,θ p (O)).

22 Embeddings of weighted Sobolev spaces into Besov spaces Theorem (P.A.C., K.-H. Kim, K. Lee, F. Lindner 203) For p 2 and γ,ν > 0: H γ p,d νp (O) Bα τ,τ(o) for all α and τ with τ = α d + p & { 0 < α < min γ,ν d d }.

23 Embeddings of weighted Sobolev spaces into Besov spaces Theorem (P.A.C., K.-H. Kim, K. Lee, F. Lindner 203) For p 2 and γ,ν > 0: H γ p,d νp (O) Bα τ,τ(o) for all α and τ with τ = α d + p & { 0 < α < min γ,ν d d }. The regularity analysis in the Besov spaces from the adaptivity scale can be traced back to the analysis of the weighted Sobolev regularity.

24 Theorem (Existence and uniqueness K.-H. Kim 202) For γ R, p 2 and θ d+p 2: If f L p (Ω [0,T];H γ 2 p,θ+p (O)), g L p (Ω [0,T];H γ p,θ (O;l 2)), and u 0 L p (Ω;H γ 2/p p,θ p+2 (O)), then SPDE ( ) has a unique solution u L p (Ω [0,T];H γ p,θ p (O)) = L p(...;h γ p,d (+ d θ p )p(o)). Theorem (P.A.C., K.-H. Kim, K. Lee, F. Lindner 203) For p 2 and γ,ν > 0: H γ p,d νp (O) Bα τ,τ(o) for all α and τ with τ = α d + p & { 0 < α < min γ,ν d d }.

25 Theorem (P.A.C., K.-H. Kim, K. Lee, F. Lindner 203) For γ R, p 2 and θ d+p 2: If f L p (Ω [0,T];H γ 2 p,θ+p (O)), g L p (Ω [0,T];H γ p,θ (O;l 2)), and u 0 L p (Ω;H γ 2/p p,θ p+2 (O)), then SPDE ( ) has a unique solution u L p (Ω [0,T];H γ p,θ p (O)) = L p(...;h γ p,d (+ d θ p )p(o)). This solution fulfils u L p (Ω [0,T];B α τ,τ (O)) for all α and τ with τ = α d + & p { ( 0 < α < min γ, + d θ ) } d. p d

26 Example (γ = 2, p = 2, θ = d, d = 2) If f L 2 (Ω [0,T];H2,4 0 (O)), g L 2 (Ω [0,T];H2,2 (O;l 2)), and u 0 L 2 (Ω;H2,2 (O)), then SPDE ( ) has a unique solution u L 2 (Ω [0,T];H2,0(O)). 2 This solution fulfils u L 2 (Ω [0,T];Bτ,τ(O)) α for all α and τ with τ = α & 0 < α < 2.

27 Some References Besov regularity of SPDEs (adaptivity scale) P.A. Cioica Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains Dissertation, Philipps-Universität Marburg. Defended on February 7, 204. P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4. P.A. Cioica, S. Dahlke Spatial Besov regularity for semilinear SPDEs on bounded Lipschitz domains Int. J. Comput. Math. 89 (202), no. 8, P.A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains Stud. Math. 207 (20), no. 3, Sobolev regularity of SPDEs on polygons (upper bound) F. Lindner Singular behavior of the solution to the stochastic heat equation on a polygonal domain Stoch. PDE: Anal. Comp. 2 (204), no. 2, SPDEs in weighted Sobolev spaces ( analytic approach ) K.-H. Kim A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains J. Theoret. Probab. 27 (204), no., Thank you!

28 Regularity of Stochastic Partial Differential Equations in Besov Spaces Related to Adaptive Schemes Petru A. Cioica Philipps-Universität Marburg Workshop on New Discretization Methods for the Numerical Approximation of PDEs Oberwolfach January 2 6, 205 joint with: René L. Schilling (TU Dresden) Klaus Ritter, Felix Lindner, Nicolas Döhring (TU Kaiserslautern) Stephan Dahlke, Stefan Kinzel (Philipps-Universität Marburg) Kyeong-Hun Kim (Korea University) Kijung Lee (Ajou University) Thorsten Raasch (Johannes-Gutenberg-Universität Mainz) supported by: DFG-SPP 324 NRF Korea Philipps-Universität Marburg

29 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram (smoothness) r q (integrability)

30 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram (smoothness) r r B r q,q (O) q q (integrability)

31 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram (smoothness) r r B r q,q (O) B α τ,τ (O), τ = α q q (integrability)

32 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram (smoothness) r r 2 B r q,q (O) B 2 2/3,2/3 (O) B α τ,τ (O), τ = α q 3 2 q (integrability)

33 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram B2,2 s (O) = Ws 2 (O) (smoothness) r r 2 B r q,q (O) B 2 2/3,2/3 (O) B α τ,τ (O), τ = α q 3 2 q (integrability)

34 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram B2,2 s (O) = Ws 2 (O) (smoothness) r r 2 B r q,q (O) B 2 2/3,2/3 (O) B α τ,τ (O), τ = α W 2 (O) 2 q 3 2 q (integrability)

35 Example (γ = 2, p = 2, θ = d, d = 2): DeVore/Triebel diagram B2,2 s (O) = Ws 2 (O) (smoothness) r r 2 B r q,q (O) B 2 2/3,2/3 (O) [Lindner 204] 3/2 B α τ,τ (O), τ = α W 2 (O) 2 q 3 2 q (integrability)

36 Outlook

37 Outlook Analyse the time space regularity of the solution P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4.

38 Outlook Analyse the time space regularity of the solution P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4. Analyse the convergence rate of Rothe s method for SPDEs P.A. Cioica, S. Dahlke, N. Döhring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs Preprint, 205.

39 Outlook Analyse the time space regularity of the solution P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4. Analyse the convergence rate of Rothe s method for SPDEs P.A. Cioica, S. Dahlke, N. Döhring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs Preprint, 205. Develop spatially adaptive wavelet methods for SPDEs work in progress (in particular S. Kinzel)

40 Outlook Analyse the time space regularity of the solution P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4. Analyse the convergence rate of Rothe s method for SPDEs P.A. Cioica, S. Dahlke, N. Döhring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs Preprint, 205. Develop spatially adaptive wavelet methods for SPDEs work in progress (in particular S. Kinzel) Improve regularity results on polygonal (R 2 ) and polyhedral (R 3 ) domains work in progress (with F. Lindner and K. Lee)

41 Outlook Analyse the time space regularity of the solution P.A. Cioica, K.-H. Kim, K. Lee, F. Lindner On the L q(l p)-regularity and Besov smoothness of SPDEs on bounded Lipschitz domains Electron. J. Probab 8 (203), no. 82, 4. Analyse the convergence rate of Rothe s method for SPDEs P.A. Cioica, S. Dahlke, N. Döhring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R.L. Schilling On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs Preprint, 205. Develop spatially adaptive wavelet methods for SPDEs work in progress (in particular S. Kinzel) Improve regularity results on polygonal (R 2 ) and polyhedral (R 3 ) domains work in progress (with F. Lindner and K. Lee) Obtain a more direct approach by developing a stochastic integration theory in quasi-banach spaces work in progress (with M.C. Veraar and S.G. Cox)

42 Besov Spaces (Oleg V. Besov) B s p,q (O) Let p,q,s (0, ) and r := [s]+. For u L p (O): u B s p,q(o) : u q Bp,q s (O) := (sup r h u L p(o) ) q sq dt t 0 h t t }{{} =: ω r (u,t) p <. rth difference with step h R d r = : hu(x) := u(x+h) u(x) r N : r+ h u(x) := h ( r h u(x)) if x+jh O for all j =,...,r.

43 Besov Spaces (Oleg V. Besov) B s p,q (O) Let p,q,s (0, ) and r := [s]+. For u L p (O): u B s p,q(o) : u q Bp,q s (O) := (sup r h u L p(o) ) q sq dt t 0 h t t }{{} =: ω r (u,t) p <. rth difference with step h R d r = : hu(x) := u(x+h) u(x) r N : r+ h u(x) := h ( r h u(x)) if x+jh O for all j =,...,r. = Bτ α(o), τ = α d +, α > 0, p 2 fixed p

44 Semigroup approach du = udt+ g k dwt k on Ω [0,T] O, k= u = 0 on Ω (0,T] O, u(0) = u 0 on Ω O. du(t) = AU(t)dt+G(t)dW H (t) on Ω [0,T], U(0) = u 0 on Ω. Mild solution: U(t) = S(t)u 0 + Problem () t Typically: U L p (Ω [0,T];D(A)) D(A) = W2 2(O) W 2 (O) if O is smooth 0 S(t s)g(s)dw H (s)