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
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
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
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!
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!
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!
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!
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!
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!
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!
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!
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!
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)
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)
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)
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 ,
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 ,
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 ,
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
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∗?
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∗?
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∗?
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.
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.
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.
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.
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)]
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)]
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))
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))
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)).
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
).
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!
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!
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!
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!
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!
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.
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.
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.
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.
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
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
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
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
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!
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!
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!
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)
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)
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)
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
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
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
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.
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!
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!
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
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
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.
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.
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.
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.
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.
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.
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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!
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!
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)
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...]
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
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).
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)
<∞.
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