adaptive wavelet methods for spdes: theoretical analysis and … · adaptive wavelet methods for...
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![Page 1: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/1.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Adaptive Wavelet Methods for SPDEs:Theoretical Analysis and Practical
Realization
Stephan Dahlke
FB 12 Mathematics and Computer SciencesPhilipps–Universitat Marburg
Workshop on
Numerical Analysis of Multiscale Problems & StochasticModelling
December 12–16, 2011
joint work with P. Cioica, S. Kinzel, F. Lindner, N.
Doring, T. Raasch, K. Ritter, R. L. Schilling
![Page 2: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/2.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Outline
Motivation
Theoretical AnalysisDoes Adaptivity Pay?The Model EquationSPDEs in Weighted Sobolev SpacesBesov Regularity
Practical RealizationDiscretization SchemeThe Noise ModelStochastic Elliptic Equations
![Page 3: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/3.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Outline
Motivation
Theoretical AnalysisDoes Adaptivity Pay?The Model EquationSPDEs in Weighted Sobolev SpacesBesov Regularity
Practical RealizationDiscretization SchemeThe Noise ModelStochastic Elliptic Equations
![Page 4: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/4.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Motivation:
I Numerical treatment of SPDEs:
du(t) = (A(u(t)) + F (t, u(t)))dt+ Σ(t, u(t))dWt,
in O ⊆ Rd , bounded Lipschitz.
I Computational finance, epidemiology, populationgenetics...
I as usual:
I much is known concerning existence and uniqueness...I but how does the solution look like?I numerical approximation scheme needed!
![Page 5: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/5.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Motivation:
I Numerical treatment of SPDEs:
du(t) = (A(u(t)) + F (t, u(t)))dt+ Σ(t, u(t))dWt,
in O ⊆ Rd , bounded Lipschitz.
I Computational finance, epidemiology, populationgenetics...
I as usual:
I much is known concerning existence and uniqueness...I but how does the solution look like?I numerical approximation scheme needed!
![Page 6: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/6.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Motivation:
I Numerical treatment of SPDEs:
du(t) = (A(u(t)) + F (t, u(t)))dt+ Σ(t, u(t))dWt,
in O ⊆ Rd , bounded Lipschitz.
I Computational finance, epidemiology, populationgenetics...
I as usual:I much is known concerning existence and uniqueness...I but how does the solution look like?I numerical approximation scheme needed!
![Page 7: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/7.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 8: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/8.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 9: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/9.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 10: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/10.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 11: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/11.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 12: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/12.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Approaches:
I First natural idea: classical nonadaptive schemes.
I based on uniform space/time refinements
I approximation spaces/degrees of freedom a priori fixed
I ‘easy’ to implement/analyze
I but: convergence might be slow!
I Alternative: use adaptive schemes!
I nonuniform space/time refinements
I updating strategy
I degrees of freedom adjusted to the unknown solution
I a posteriori error estimator, refinement strategy....
I heavy to implement/analyze
I but convergence might be faster!
![Page 13: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/13.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Wavelets:
I Multiresolution Analysis Vjj≥0
V0 ⊂ V1 ⊂ V2 ⊂ . . .∞⋃j=0
Vj = L2(O)
I
Vj+1 = Vj ⊕Wj+1 V0 = W0 L2(O) = ⊕∞j=0Wj
Wj = spanψj,k, k ∈ Jj
I
λ = (j, k), |λ| = j, J =∞⋃j=0
(j × Jj)
![Page 14: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/14.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Wavelets:
I Multiresolution Analysis Vjj≥0
V0 ⊂ V1 ⊂ V2 ⊂ . . .∞⋃j=0
Vj = L2(O)
I
Vj+1 = Vj ⊕Wj+1 V0 = W0 L2(O) = ⊕∞j=0Wj
Wj = spanψj,k, k ∈ Jj
I
λ = (j, k), |λ| = j, J =∞⋃j=0
(j × Jj)
![Page 15: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/15.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Wavelets:
I Multiresolution Analysis Vjj≥0
V0 ⊂ V1 ⊂ V2 ⊂ . . .∞⋃j=0
Vj = L2(O)
I
Vj+1 = Vj ⊕Wj+1 V0 = W0 L2(O) = ⊕∞j=0Wj
Wj = spanψj,k, k ∈ Jj
I
λ = (j, k), |λ| = j, J =∞⋃j=0
(j × Jj)
![Page 16: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/16.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Basic Properties:
I diam (suppψλ) ∼ 2−|λ|, λ ∈ J
I ∫Oxγψλ(x)dx = 0, |γ| ≤ N
I
‖f‖Bsq(Lp(O) ∼ ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q
∑λ∈J ,|λ|=j
|〈f, ψλ〉|pq/p
1/q
,
where Ψ = ψλ : λ ∈ J satisfies 〈ψλ, ψν〉 = δλ,ν ,
![Page 17: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/17.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Basic Properties:
I diam (suppψλ) ∼ 2−|λ|, λ ∈ J
I ∫Oxγψλ(x)dx = 0, |γ| ≤ N
I
‖f‖Bsq(Lp(O) ∼ ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q
∑λ∈J ,|λ|=j
|〈f, ψλ〉|pq/p
1/q
,
where Ψ = ψλ : λ ∈ J satisfies 〈ψλ, ψν〉 = δλ,ν ,
![Page 18: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/18.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Basic Properties:
I diam (suppψλ) ∼ 2−|λ|, λ ∈ J
I ∫Oxγψλ(x)dx = 0, |γ| ≤ N
I
‖f‖Bsq(Lp(O) ∼ ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q
∑λ∈J ,|λ|=j
|〈f, ψλ〉|pq/p
1/q
,
where Ψ = ψλ : λ ∈ J satisfies 〈ψλ, ψν〉 = δλ,ν ,
![Page 19: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/19.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Outline
Motivation
Theoretical AnalysisDoes Adaptivity Pay?The Model EquationSPDEs in Weighted Sobolev SpacesBesov Regularity
Practical RealizationDiscretization SchemeThe Noise ModelStochastic Elliptic Equations
![Page 20: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/20.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
General Question: Does Adaptivity Really Pay?I nonadaptive schemes
Ej(u) = infg∈Vj‖u− g‖L2(O) <∼ 2−sj |u|W s
2 (O))
Ej(u) = O(n−s/dj )⇐= u ∈W s2 (O)
(linear approximation)
I adaptive schemes
“ideal” algorithm: best n–term approximation(nonlinear approximation)
σn(u)L2(O) := ‖u− gn‖L2(O)
gn =∑
(j,k)∈Λn
dj,kψj,k, Λn = n biggest wavelet coefficients
σn(u)L2(O) = O(n−s/d)⇐ u ∈ Bsτ (Lτ (O))
1τ
=(s
d+
12
)I Question: u ∈ Bs
τ (Lτ (O)), 0 < s < s∗?
![Page 21: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/21.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
General Question: Does Adaptivity Really Pay?I nonadaptive schemes
Ej(u) = infg∈Vj‖u− g‖L2(O) <∼ 2−sj |u|W s
2 (O))
Ej(u) = O(n−s/dj )⇐= u ∈W s2 (O)
(linear approximation)I adaptive schemes
“ideal” algorithm: best n–term approximation(nonlinear approximation)
σn(u)L2(O) := ‖u− gn‖L2(O)
gn =∑
(j,k)∈Λn
dj,kψj,k, Λn = n biggest wavelet coefficients
σn(u)L2(O) = O(n−s/d)⇐ u ∈ Bsτ (Lτ (O))
1τ
=(s
d+
12
)
I Question: u ∈ Bsτ (Lτ (O)), 0 < s < s∗?
![Page 22: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/22.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
General Question: Does Adaptivity Really Pay?I nonadaptive schemes
Ej(u) = infg∈Vj‖u− g‖L2(O) <∼ 2−sj |u|W s
2 (O))
Ej(u) = O(n−s/dj )⇐= u ∈W s2 (O)
(linear approximation)I adaptive schemes
“ideal” algorithm: best n–term approximation(nonlinear approximation)
σn(u)L2(O) := ‖u− gn‖L2(O)
gn =∑
(j,k)∈Λn
dj,kψj,k, Λn = n biggest wavelet coefficients
σn(u)L2(O) = O(n−s/d)⇐ u ∈ Bsτ (Lτ (O))
1τ
=(s
d+
12
)I Question: u ∈ Bs
τ (Lτ (O)), 0 < s < s∗?
![Page 23: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/23.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Model Equation:I Fix T ∈ (0,∞) and O ⊆ Rd bounded Lipschitz domain,
(Ω,F ,P) → probability space.
I Stochastic evolution equation
du(t) =d∑
µ,ν=1
aµνuxµxνdt+∞∑k=1
gk(t)dwkt ,
I (aµν)1≤ν,µ≤d ∈ Rd×d symmetric positive definite, theBrownian motions wkt are independent.
I ϕ ∈ C∞0 (O) =⇒
〈u(t, ω), ϕ〉=〈u0(ω), ϕ〉+d∑
ν,µ=1
∫ t
0〈aµνuxµxν (s, ω), ϕ〉ds
+∞∑k=1
∫ t
0〈gk(s), ϕ〉dwks(t, ω)︸ ︷︷ ︸= Intwk
(〈gk,ϕ〉
)(t,·)︸ ︷︷ ︸
convergence w.r.t. ‖·‖M2,cT
P-a.s.
![Page 24: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/24.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Model Equation:I Fix T ∈ (0,∞) and O ⊆ Rd bounded Lipschitz domain,
(Ω,F ,P) → probability space.I Stochastic evolution equation
du(t) =d∑
µ,ν=1
aµνuxµxνdt+∞∑k=1
gk(t)dwkt ,
I (aµν)1≤ν,µ≤d ∈ Rd×d symmetric positive definite, theBrownian motions wkt are independent.
I ϕ ∈ C∞0 (O) =⇒
〈u(t, ω), ϕ〉=〈u0(ω), ϕ〉+d∑
ν,µ=1
∫ t
0〈aµνuxµxν (s, ω), ϕ〉ds
+∞∑k=1
∫ t
0〈gk(s), ϕ〉dwks(t, ω)︸ ︷︷ ︸= Intwk
(〈gk,ϕ〉
)(t,·)︸ ︷︷ ︸
convergence w.r.t. ‖·‖M2,cT
P-a.s.
![Page 25: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/25.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Model Equation:I Fix T ∈ (0,∞) and O ⊆ Rd bounded Lipschitz domain,
(Ω,F ,P) → probability space.I Stochastic evolution equation
du(t) =d∑
µ,ν=1
aµνuxµxνdt+∞∑k=1
gk(t)dwkt ,
I (aµν)1≤ν,µ≤d ∈ Rd×d symmetric positive definite, theBrownian motions wkt are independent.
I ϕ ∈ C∞0 (O) =⇒
〈u(t, ω), ϕ〉=〈u0(ω), ϕ〉+d∑
ν,µ=1
∫ t
0〈aµνuxµxν (s, ω), ϕ〉 ds
+∞∑k=1
∫ t
0〈gk(s), ϕ〉 dwks(t, ω)︸ ︷︷ ︸= Intwk
(〈gk,ϕ〉
)(t,·)︸ ︷︷ ︸
convergence w.r.t. ‖·‖M2,cT
P-a.s.
![Page 26: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/26.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Spaces:
I ρ(x) := dist(x, ∂O) for x ∈ O
I γ ∈ N0; θ ∈ R
u ∈ Hγ2,θ(O) :⇔
‖u‖2Hγ2,θ(O) :=
∑α∈Nd0|α|≤γ
∫O
∣∣ρ(x)|α|Dαu(x)∣∣2ρ(x)θ−ddx
is finite.
![Page 27: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/27.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Spaces for Sequences:
I γ ∈ N0, θ ∈ R
g =(gk)k∈N ∈ H
γ2,θ(O; `2)
:⇔
‖g‖2Hγ2,θ(O;`2) :=
∑α∈Nd0|α|≤γ
∫O
(ρ|α|∣∣∣(Dαgk
)k∈N
∣∣∣`2
)2ρθ−d dx
is finite.
I γ ∈ R: Hγ2,θ(O) and Hγ
2,θ(O; `2)I by complex interpolation
I see [Lototsky(2000)]
![Page 28: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/28.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Spaces for Sequences:
I γ ∈ N0, θ ∈ R
g =(gk)k∈N ∈ H
γ2,θ(O; `2)
:⇔
‖g‖2Hγ2,θ(O;`2) :=
∑α∈Nd0|α|≤γ
∫O
(ρ|α|∣∣∣(Dαgk
)k∈N
∣∣∣`2
)2ρθ−d dx
is finite.
I γ ∈ R: Hγ2,θ(O) and Hγ
2,θ(O; `2)I by complex interpolation
I see [Lototsky(2000)]
![Page 29: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/29.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Space for StochasticProcesses:
I γ ∈ R, θ ∈ R
Hγ2,θ(O, T ) : = L2
([0, T ]× Ω,dt⊗ P;Hγ
2,θ(O))
‖u‖2Hγ2,θ(O,T ) =
∫Ω
∫ T
0‖u(t, ω, ·)‖2Hγ
2,θ(O) dtP(dω)
I γ ∈ R, θ ∈ R for sequences
Hγ2,θ(O, T ; `2) := L2
([0, T ]× Ω,dt⊗ P;Hγ
2,θ(O; `2))
![Page 30: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/30.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Space for StochasticProcesses:
I γ ∈ R, θ ∈ R
Hγ2,θ(O, T ) : = L2
([0, T ]× Ω,dt⊗ P;Hγ
2,θ(O))
‖u‖2Hγ2,θ(O,T ) =
∫Ω
∫ T
0‖u(t, ω, ·)‖2Hγ
2,θ(O) dtP(dω)
I γ ∈ R, θ ∈ R for sequences
Hγ2,θ(O, T ; `2) := L2
([0, T ]× Ω,dt⊗ P;Hγ
2,θ(O; `2))
![Page 31: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/31.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Existence/Uniqueness of Solutions in Hγ2,θ−2 :
du(t) =d∑
µ,ν=1
aµνuxµxνdt+∞∑k=1
gk(t)dwkt (•)
Theorem (Kim 2008)
∃κ = κ(d,O) ∈(0, 1),
such that
∀ θ ∈(d− κ, d+ κ
),
(•) has a unique solution in the class Hγ2,θ−2
(O, T
), provided(
gk)k∈N ∈ H
γ−12,θ
(O, T ; `2
)& u0 ∈ L2
(Ω;Hγ−1
2,θ (O)).
![Page 32: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/32.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Besov Regularity:
Theorem (2010)
If γ ∈ N and
u ∈ L2
([0, T ]× Ω;W s
2 (O))
for some
s ∈(
0, γ ∧ 1 +d− θ
2
),
then
u ∈ Lτ(
[0, T ]× Ω;Bατ,τ (O)
),
1τ
=α
d+
12,
for all
α ∈(
0, γ ∧ s d
d− 1
).
![Page 33: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/33.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example: d = 2, θ = d, γ = 2:
I Suppose that(gk)k∈N ∈ H1
2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).
I Then: Kim’s theorem =⇒ ∃! solution Hγ2,θ−2
(O, T
)I ⇒ u ∈ L2(. . . ;W 1
2 (O))
I Our theorem ⇒
u ∈ Lτ (. . . ;Bατ,τ (O)),
1τ
=α
2+
12
for all α < 2
I Adaptivity completely justified!
![Page 34: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/34.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example: d = 2, θ = d, γ = 2:
I Suppose that(gk)k∈N ∈ H1
2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).
I Then: Kim’s theorem =⇒ ∃! solution Hγ2,θ−2
(O, T
)
I ⇒ u ∈ L2(. . . ;W 12 (O))
I Our theorem ⇒
u ∈ Lτ (. . . ;Bατ,τ (O)),
1τ
=α
2+
12
for all α < 2
I Adaptivity completely justified!
![Page 35: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/35.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example: d = 2, θ = d, γ = 2:
I Suppose that(gk)k∈N ∈ H1
2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).
I Then: Kim’s theorem =⇒ ∃! solution Hγ2,θ−2
(O, T
)I ⇒ u ∈ L2(. . . ;W 1
2 (O))
I Our theorem ⇒
u ∈ Lτ (. . . ;Bατ,τ (O)),
1τ
=α
2+
12
for all α < 2
I Adaptivity completely justified!
![Page 36: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/36.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example: d = 2, θ = d, γ = 2:
I Suppose that(gk)k∈N ∈ H1
2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).
I Then: Kim’s theorem =⇒ ∃! solution Hγ2,θ−2
(O, T
)I ⇒ u ∈ L2(. . . ;W 1
2 (O))
I Our theorem ⇒
u ∈ Lτ (. . . ;Bατ,τ (O)),
1τ
=α
2+
12
for all α < 2
I Adaptivity completely justified!
![Page 37: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/37.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example: d = 2, θ = d, γ = 2:
I Suppose that(gk)k∈N ∈ H1
2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).
I Then: Kim’s theorem =⇒ ∃! solution Hγ2,θ−2
(O, T
)I ⇒ u ∈ L2(. . . ;W 1
2 (O))
I Our theorem ⇒
u ∈ Lτ (. . . ;Bατ,τ (O)),
1τ
=α
2+
12
for all α < 2
I Adaptivity completely justified!
![Page 38: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/38.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Idea of the Proof:I Wavelet characterization of Besov spaces
‖u‖Bsp,q(Rd) ∼( ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q( ∑λ∈J|λ|=j
|〈u, ψλ〉|p)q/p)1/q
I Weighted Sobolev space estimate (Kim 2008)
‖u‖Hγ2,θ−2(O) + ‖d∑
µ.ν=1
aµνuxµxν‖Hγ−22,θ+2(O)
≤ C(‖g‖
Hγ−12,θ (O)
+ ‖u0‖L2(Hγ−12,θ (O))
)
I P.A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch,K. Ritter, R.L. Schilling: Spatial Besov Regularity for SPDEs onLipschitz Domains, Preprint Nr. 66, DFG Priority Program 1324”Extraction of Quantifiable Information from Complex Systems”,Nov. 2010, appear in: Studia Math.
![Page 39: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/39.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Idea of the Proof:I Wavelet characterization of Besov spaces
‖u‖Bsp,q(Rd) ∼( ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q( ∑λ∈J|λ|=j
|〈u, ψλ〉|p)q/p)1/q
I Weighted Sobolev space estimate (Kim 2008)
‖u‖Hγ2,θ−2(O) + ‖d∑
µ.ν=1
aµνuxµxν‖Hγ−22,θ+2(O)
≤ C(‖g‖
Hγ−12,θ (O)
+ ‖u0‖L2(Hγ−12,θ (O))
)I P.A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch,
K. Ritter, R.L. Schilling: Spatial Besov Regularity for SPDEs onLipschitz Domains, Preprint Nr. 66, DFG Priority Program 1324”Extraction of Quantifiable Information from Complex Systems”,Nov. 2010, appear in: Studia Math.
![Page 40: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/40.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Generalizations:
I More general noise models, mutliplicative noise
I Semilinear stochastic evolution equations [Cioica/D.]
P.A. Cioica, S. Dahlke: Spatial Besov Regularity for Semilinear
SPDEs on Lipschitz Domains, Preprint Nr. 99, DFG Priority
Program 1324 ”Extraction of Quantifiable Information from
Complex Systems”, July 2011, to appear in: Int. J. Comput. Math.
![Page 41: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/41.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Generalizations:
I More general noise models, mutliplicative noise
I Semilinear stochastic evolution equations [Cioica/D.]
P.A. Cioica, S. Dahlke: Spatial Besov Regularity for Semilinear
SPDEs on Lipschitz Domains, Preprint Nr. 99, DFG Priority
Program 1324 ”Extraction of Quantifiable Information from
Complex Systems”, July 2011, to appear in: Int. J. Comput. Math.
![Page 42: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/42.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Outline
Motivation
Theoretical AnalysisDoes Adaptivity Pay?The Model EquationSPDEs in Weighted Sobolev SpacesBesov Regularity
Practical RealizationDiscretization SchemeThe Noise ModelStochastic Elliptic Equations
![Page 43: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/43.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Possible Approaches:
I Vertical method of lines (first in space, then in time)[Gyongy/Krylov/Millet/Morien],[Walsh], [Yan].... Hardto combine with adaptivity...
I Full space-time adaptive wavelet algorithms[Schwab/Stevenson]...
I Horizontal method of lines, Rothe method (first in time,then in space) [Debussche/Printems]
I Abstract Cauchy problemI ODE in suitable functions spaces, ODE-solver with
adaptive step-size control
![Page 44: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/44.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Possible Approaches:
I Vertical method of lines (first in space, then in time)[Gyongy/Krylov/Millet/Morien],[Walsh], [Yan].... Hardto combine with adaptivity...
I Full space-time adaptive wavelet algorithms[Schwab/Stevenson]...
I Horizontal method of lines, Rothe method (first in time,then in space) [Debussche/Printems]
I Abstract Cauchy problemI ODE in suitable functions spaces, ODE-solver with
adaptive step-size control
![Page 45: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/45.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Possible Approaches:
I Vertical method of lines (first in space, then in time)[Gyongy/Krylov/Millet/Morien],[Walsh], [Yan].... Hardto combine with adaptivity...
I Full space-time adaptive wavelet algorithms[Schwab/Stevenson]...
I Horizontal method of lines, Rothe method (first in time,then in space) [Debussche/Printems]
I Abstract Cauchy problemI ODE in suitable functions spaces, ODE-solver with
adaptive step-size control
![Page 46: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/46.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Rothe Method:
I Stiff problem y implicit discretization in time:(I − (tn+1 − tn)A
)Utn+1
= Utn + (tn+1−tn)F (tn, Utn) + Σ(tn, Utn)(Wtn+1−Wtn
)
I leads to elliptic subproblems, ; model problem:
−∆V = X(ω) in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....realize the convercence orderof best n-term approximation!
![Page 47: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/47.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Rothe Method:
I Stiff problem y implicit discretization in time:(I − (tn+1 − tn)A
)Utn+1
= Utn + (tn+1−tn)F (tn, Utn) + Σ(tn, Utn)(Wtn+1−Wtn
)I leads to elliptic subproblems, ; model problem:
−∆V = X(ω) in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....realize the convercence orderof best n-term approximation!
![Page 48: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/48.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Rothe Method:
I Stiff problem y implicit discretization in time:(I − (tn+1 − tn)A
)Utn+1
= Utn + (tn+1−tn)F (tn, Utn) + Σ(tn, Utn)(Wtn+1−Wtn
)I leads to elliptic subproblems, ; model problem:
−∆V = X(ω) in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....realize the convercence orderof best n-term approximation!
![Page 49: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/49.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I Usually: noise modeled by means of the Eigenfunctionsof A.
I We need: noise model based on wavelets, control ofBesov regularity.
I Choose parameters α > 0 and 0 ≤ β ≤ 1 s.t. α+ β > 1I Let Yλ, (wλt )t∈[0,T ], λ ∈ J be independent,
Yλ ∼ B(1, 2−βjd). Set
gλ(ω, t, ·) := σjYλψλ(·), σj := (j−(j0−2))cd2 2−
α(j−j0−1))d2
I ; in each time step
−4V = X(ω) in O, V = 0 on ∂O
X =∑λ∈J
YλZλψλ, Zλ ∼ N(0, 2−α|λ|d)
![Page 50: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/50.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I Usually: noise modeled by means of the Eigenfunctionsof A.
I We need: noise model based on wavelets, control ofBesov regularity.
I Choose parameters α > 0 and 0 ≤ β ≤ 1 s.t. α+ β > 1I Let Yλ, (wλt )t∈[0,T ], λ ∈ J be independent,
Yλ ∼ B(1, 2−βjd). Set
gλ(ω, t, ·) := σjYλψλ(·), σj := (j−(j0−2))cd2 2−
α(j−j0−1))d2
I ; in each time step
−4V = X(ω) in O, V = 0 on ∂O
X =∑λ∈J
YλZλψλ, Zλ ∼ N(0, 2−α|λ|d)
![Page 51: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/51.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I Usually: noise modeled by means of the Eigenfunctionsof A.
I We need: noise model based on wavelets, control ofBesov regularity.
I Choose parameters α > 0 and 0 ≤ β ≤ 1 s.t. α+ β > 1I Let Yλ, (wλt )t∈[0,T ], λ ∈ J be independent,
Yλ ∼ B(1, 2−βjd). Set
gλ(ω, t, ·) := σjYλψλ(·), σj := (j−(j0−2))cd2 2−
α(j−j0−1))d2
I ; in each time step
−4V = X(ω) in O, V = 0 on ∂O
X =∑λ∈J
YλZλψλ, Zλ ∼ N(0, 2−α|λ|d)
![Page 52: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/52.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I
X =∑λ∈J
YλZλ · ψλ (∗)
I Note that X is Gaussian iff β = 0.
I Moreover, (∗) is the Karhunen-Loeve expansion iff(ψλ)λ∈J is an ONB
![Page 53: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/53.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I
X =∑λ∈J
YλZλ · ψλ (∗)
I Note that X is Gaussian iff β = 0.
I Moreover, (∗) is the Karhunen-Loeve expansion iff(ψλ)λ∈J is an ONB
![Page 54: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/54.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The Noise Model:
I
X =∑λ∈J
YλZλ · ψλ (∗)
I Note that X is Gaussian iff β = 0.
I Moreover, (∗) is the Karhunen-Loeve expansion iff(ψλ)λ∈J is an ONB
![Page 55: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/55.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Besov Regularity:
‖f‖Bsq(Lp(O)) ∼ ∞∑|λ|=j0
2|λ|(s+d( 12− 1p
))q
∑λ∈J ,|λ|=j
|〈f, ψλ〉|pq/p
1/q
,
where Ψ = ψλ : λ ∈ J satisfies 〈ψλ, ψν〉 = δλ,ν ,
Theorem
X ∈ Bsq(Lp(O)) P -a.s. iff
α− 12
+β
p>s
d,
in which case E(‖X‖qBsq(Lp(D))
)<∞.
[Abramovich/Sapatinas/Silverman], [Bochkina] for d = 1and p, q ≥ 1.
![Page 56: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/56.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Besov Regularity:
Corollary
X ∈W s2 (O) P -a.s. iff
s < d
(α+ β − 1
2
)
Corollary
Let 1/τ = s/d+ 1/2. X ∈ Bsτ (Lτ (O)) P -a.s. iff
s < d
(α+ β − 12(1− β)
)
I β is a sparsity parameter
I α+ β fixed, β → 1 Sobolev smoothness fixed, arbitraryhigh Besov regularity!
![Page 57: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/57.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Besov Regularity:
Corollary
X ∈W s2 (O) P -a.s. iff
s < d
(α+ β − 1
2
)
Corollary
Let 1/τ = s/d+ 1/2. X ∈ Bsτ (Lτ (O)) P -a.s. iff
s < d
(α+ β − 12(1− β)
)
I β is a sparsity parameter
I α+ β fixed, β → 1 Sobolev smoothness fixed, arbitraryhigh Besov regularity!
![Page 58: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/58.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Realizations:
(a) α = 2.0, β = 0.0
k
leve
l j
2
3
4
5
6
7
8
9
10
11
−6
−5
−4
−3
−2
−1
0
(b) α = 2.0, β = 0.0
(c) α = 1.8, β = 0.2
k
leve
l j
2
3
4
5
6
7
8
9
10
11
−6
−5
−4
−3
−2
−1
0
(d) α = 1.8, β = 0.2
![Page 59: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/59.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Realizations:
(e) α = 1.5, β = 0.5
k
leve
l j
2
3
4
5
6
7
8
9
10
11
−6
−5
−4
−3
−2
−1
0
(f) α = 1.5, β = 0.5
(g) α = 1.2, β = 0.8
k
leve
l j
2
3
4
5
6
7
8
9
10
11
−6
−5
−4
−3
−2
−1
0
(h) α = 1.2, β = 0.8
![Page 60: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/60.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Stochastic Elliptic Equations:
I Rothe method leads to elliptic subproblem:
−∆V = X in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....
I Optimal in the energy norm,
η(g) := #λ ∈ J : cλ 6= 0, g =∑λ∈J
cλψλ
e(V ) =(E‖V − V ‖2H1(O)
)1/2
en,H1(V ) = inf e(Vn), η(Vn) ≤ n a.s.
![Page 61: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/61.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Stochastic Elliptic Equations:
I Rothe method leads to elliptic subproblem:
−∆V = X in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....
I Optimal in the energy norm,
η(g) := #λ ∈ J : cλ 6= 0, g =∑λ∈J
cλψλ
e(V ) =(E‖V − V ‖2H1(O)
)1/2
en,H1(V ) = inf e(Vn), η(Vn) ≤ n a.s.
![Page 62: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/62.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Stochastic Elliptic Equations:
I Rothe method leads to elliptic subproblem:
−∆V = X in O, V = 0 on ∂O
I Will be treated by (stochastic versions of) optimallyconvergent adaptive wavelet (frame) algorithms[Cohen/Dahmen/DeVore], [Stevenson/Schwab],[D./Fornasier/Raasch]....
I Optimal in the energy norm,
η(g) := #λ ∈ J : cλ 6= 0, g =∑λ∈J
cλψλ
e(V ) =(E‖V − V ‖2H1(O)
)1/2
en,H1(V ) = inf e(Vn), η(Vn) ≤ n a.s.
![Page 63: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/63.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Approximation Rates:
Theorem
Let d ∈ 2, 3. Put
ρ = min(
12(d− 1)
,α+ β − 1
6+
23d
).
For n-term approximation of V , with any ε > 0,
en,H1(V ) n−ρ+ε.
I Uniform discretizations yield n−1/(2d) on generalLipschitz domains. We have ρ > 1/(2d).
I Better results for more specific domains, e.g. polygonalO.
I Convergence order realized by adaptive waveletalgorithms.
![Page 64: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/64.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Approximation Rates:
Theorem
Let d ∈ 2, 3. Put
ρ = min(
12(d− 1)
,α+ β − 1
6+
23d
).
For n-term approximation of V , with any ε > 0,
en,H1(V ) n−ρ+ε.
I Uniform discretizations yield n−1/(2d) on generalLipschitz domains. We have ρ > 1/(2d).
I Better results for more specific domains, e.g. polygonalO.
I Convergence order realized by adaptive waveletalgorithms.
![Page 65: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/65.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Approximation Rates:
Theorem
Let d ∈ 2, 3. Put
ρ = min(
12(d− 1)
,α+ β − 1
6+
23d
).
For n-term approximation of V , with any ε > 0,
en,H1(V ) n−ρ+ε.
I Uniform discretizations yield n−1/(2d) on generalLipschitz domains. We have ρ > 1/(2d).
I Better results for more specific domains, e.g. polygonalO.
I Convergence order realized by adaptive waveletalgorithms.
![Page 66: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/66.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Numerical Results (1D):
Specifically
I D = [0, 1],
I Problem completely regular, Sobolev/Besov smoothnessonly depends on smoothness of the right-hand side!
I ‘Exact’ solution via master computation.
I adaptive wavelet scheme ←→ uniform scheme
I The W s2 -regularity
s <α+ β − 1
2
of X kept constant, Besov smoothness varies
![Page 67: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/67.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Comparison:
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2−4
−3.5
−3
−2.5
−2
−1.5
−1
log N
log
err
or
uniform
adaptive
Convergence Rates: α = 0.9, β = 0.2
Orders of convergence:upper bounds 1.05 and 19/16 = 1.1875.
![Page 68: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/68.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Comparison:
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
log N
log
err
or
uniform
adaptive
Convergence Rates: α = 0.4, β = 0.7
![Page 69: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/69.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Comparison:
0.5 1 1.5 2 2.5 3 3.5−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
log N
log
err
or
uniform
adaptive
Convergence Rates: α = −0.87, β = 0.97
![Page 70: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/70.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Example in 2D:
(a) exact solution (b) exact right-hand side
(c) α = 1.0, β = 0.1 (d) α = 1.0, β = 0.9
![Page 71: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/71.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 72: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/72.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 73: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/73.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?
I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 74: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/74.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 75: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/75.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 76: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/76.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 77: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/77.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 78: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/78.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 79: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/79.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Summary:
I Adaptive numerical treatment of SPDEs
I Theoretical analysis: Besov regularity
I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?I weighted Sobolev estimates + wavelet expansions ;
new regularity results, s∗ sufficiently large
I Practical realization
I Rothe method, implicit discretization scheme
I new noise model, prescribed Besov regularity
I elliptic subproblems, solved by adaptive waveletalgorithms
I numerical experiments
![Page 80: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/80.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
P. A. Cioica, S. Dahlke (2011): Spacial Besov Regularity for Semilinear Elliptic Equations.
Preprint Nr. 99, DFG Priority Program 1324 ”Extraction of Quantifiable Information fromComplex Systems”, July. 2011, to appear in: Int. J. Comput. Math.
P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling (2010):
Spatial Besov Regularity for SPDEs on Lipschitz Domains. Preprint Nr. 66, DFG PriorityProgram 1324 ”Extraction of Quantifiable Information from Complex Systems”, Nov. 2010, toappear in: Stud. Math.
P. A. Cioica, S. Dahlke, N. Doring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling
(2010): Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations. PreprintNr. 77, DFG Priority Program 1324 ”Extraction of Quantifiable Information from ComplexSystems”, Jan. 2011, to appear in: BIT.
A. Cohen, W. Dahmen, R. DeVore (2001): Adaptive Wavelet Methods for Elliptic Operator
Equations: Convergence Rates. Math. Comp. 70, 21–75.
S. Dahlke, R. DeVore (1997): Besov regularity for elliptic boundary value problems. Comm.
Partial Differential Equations 22, 1–16.
K.-H. Kim (2008): An Lp-Theory of SPDEs on Lipschitz Domains. Potential Anal. 29, 303–326.
N. V. Krylov (1999): An Analytic Approach to SPDEs. in: B.L. Rozovskii, R. Carmora (eds.),
Stochastic Partial Differential Equations. Six Perspectives, AMS, 185–242.
S. V. Lototsky (2000): Sobolev Spaces with Weights in Domains and Boundary Value Problems
for Degenerate Elliptic Equations, Methods Appl. Anal. 7(1), 195–204.
C. Prevot, M. Rockner (2007): A Concise Course on Stochastic Partial Differential Equations.
Springer.
R. Stevenson (2003): Adaptive solution of operator equations using wavelet frames. SIAM
J. Numer. Anal 41, 1074–1100.
Thank you for the attention!
![Page 81: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/81.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
P. A. Cioica, S. Dahlke (2011): Spacial Besov Regularity for Semilinear Elliptic Equations.
Preprint Nr. 99, DFG Priority Program 1324 ”Extraction of Quantifiable Information fromComplex Systems”, July. 2011, to appear in: Int. J. Comput. Math.
P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling (2010):
Spatial Besov Regularity for SPDEs on Lipschitz Domains. Preprint Nr. 66, DFG PriorityProgram 1324 ”Extraction of Quantifiable Information from Complex Systems”, Nov. 2010, toappear in: Stud. Math.
P. A. Cioica, S. Dahlke, N. Doring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling
(2010): Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations. PreprintNr. 77, DFG Priority Program 1324 ”Extraction of Quantifiable Information from ComplexSystems”, Jan. 2011, to appear in: BIT.
A. Cohen, W. Dahmen, R. DeVore (2001): Adaptive Wavelet Methods for Elliptic Operator
Equations: Convergence Rates. Math. Comp. 70, 21–75.
S. Dahlke, R. DeVore (1997): Besov regularity for elliptic boundary value problems. Comm.
Partial Differential Equations 22, 1–16.
K.-H. Kim (2008): An Lp-Theory of SPDEs on Lipschitz Domains. Potential Anal. 29, 303–326.
N. V. Krylov (1999): An Analytic Approach to SPDEs. in: B.L. Rozovskii, R. Carmora (eds.),
Stochastic Partial Differential Equations. Six Perspectives, AMS, 185–242.
S. V. Lototsky (2000): Sobolev Spaces with Weights in Domains and Boundary Value Problems
for Degenerate Elliptic Equations, Methods Appl. Anal. 7(1), 195–204.
C. Prevot, M. Rockner (2007): A Concise Course on Stochastic Partial Differential Equations.
Springer.
R. Stevenson (2003): Adaptive solution of operator equations using wavelet frames. SIAM
J. Numer. Anal 41, 1074–1100.
Thank you for the attention!
![Page 82: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/82.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
The DeVore-Triebel Diagram:
-
6
```````
s
1τ
12 1 2
1τ = s
2 + 12qW
3/22 (O)
![Page 83: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/83.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
I Nonlinear (best n-term) approximation of deterministicfunctions ..., [DeVore/Jawerth/Popov (1992)] [DeVore(1998)],...
I Nonlinear approximation of stochastic processes:
I wavelet methods for piecewise stationary processes:[Cohen/d’Ales (1997)], [Cohen/Daubechies/Guleryuz(2002)]
I free Knot splines for Brownian motion, SDEs:[Kon/Plaskota (2005)][Creutzig/Muller-Gronbach/Ritter (2007)], [Slassi(2010)]
I free Knot splines for Levy driven SDEs: [Dereich (2010),Dereich/Heidenreich (2010)]
I Nonlinear approximation for elliptic PDEs with randomcoefficients:
I [Cohen, DeVore, Hansen, Kuo, Nicols, Schwab,Sloan,Scheichl...]
![Page 84: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/84.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Extensions
I More general linear equations of the type:
du =d∑
i,j=1
(aijuxixj + biuxi + cu+ f
)dt
+∞∑k=1
(σikuxi + ηku+ gk
)dwkt ,
u(0, · ) = u0,
with random functions aij , bi, c, σik, ηk, f and gk
depending on t and x
![Page 85: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/85.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Spaces
Set ρ(x) := dist(x, ∂O) for x ∈ O.
I (ζn)n∈Z ⊆ C∞0 (O), such that:
I∑n∈Z ζn(x) = 1, x ∈ O,
I supp ζn ⊆ On := x ∈ O : 2−n−1 < ρ(x) < 2−n+1,I |Dmζn| ≤ N(m) 2mn, m ∈ N0, n ∈ Z.
I For γ ∈ R and θ ∈ R define:
Hγ2, θ(O) :=
u ∈ D′(O) : ‖u‖2Hγ
2,θ(O) <∞
I with:
‖u‖2Hγ2,θ(O) :=
∑n∈Z
2nθ‖ζ−n(2n·)u(2n·)‖2Hγ2 (Rd)
I and ‖f‖Hγ2 (Rd) = ‖(1−∆)γ/2f‖L2(Rd).
![Page 86: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model](https://reader033.vdocument.in/reader033/viewer/2022041810/5e574d20ab71913891438141/html5/thumbnails/86.jpg)
Adaptive WaveletMethods for
SPDEs
Stephan Dahlke
Motivation
TheoreticalAnalysis
Does Adaptivity Pay?
The Model Equation
SPDEs in WeightedSobolev Spaces
Besov Regularity
PracticalRealization
Discretization Scheme
The Noise Model
Stochastic EllipticEquations
Weighted Sobolev Spaces for Sequences
I For γ ∈ R and θ ∈ R define:
Hγ2,θ(O; `2) :=
g = (gk)k∈N ∈
[D′(O)
]N :
‖g‖Hγ2,θ(O;`2) <∞
I where
‖g‖2Hγ2,θ(O;`2) :=
∑n∈Z
2nθ‖ζ−n(2n·)g(2n·)‖2Hγ2 (Rd;`2)
I and f = (fk)k∈N ∈ Hγ2 (Rd; `2) :⇔
I fk ∈ Hγ2 for all k ∈ N, and
I ‖f‖Hγ2 (Rd;`2) :=∥∥ ∣∣((1−∆)γ/2fk
)k∈N
∣∣`2
∥∥L2(Rd)
<∞.