Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and … · Adaptive Wavelet Methods for SPDEs Stephan Dahlke Motivation Theoretical Analysis Does Adaptivity Pay? The Model

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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Outline

Motivation

Theoretical AnalysisDoes Adaptivity Pay?The Model EquationSPDEs in Weighted Sobolev SpacesBesov Regularity

Practical RealizationDiscretization SchemeThe Noise ModelStochastic Elliptic Equations


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Outline

Motivation




SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Motivation:





I as usual:

I much is known concerning existence and uniqueness...I but how does the solution look like?I numerical approximation scheme needed!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Motivation:





I as usual:I much is known concerning existence and uniqueness...I but how does the solution look like?I numerical approximation scheme needed!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model










I updating strategy






SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model










I updating strategy






SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model










I updating strategy






SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model










I updating strategy






SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model










I updating strategy






SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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)


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Wavelets:


V0 ⊂ V1 ⊂ V2 ⊂ . . .∞⋃j=0

Vj = L2(O)

I

Vj+1 = Vj ⊕Wj+1 V0 = W0 L2(O) = ⊕∞j=0Wj


I

λ = (j, k), |λ| = j, J =∞⋃j=0

(j × Jj)


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Wavelets:


V0 ⊂ V1 ⊂ V2 ⊂ . . .∞⋃j=0

Vj = L2(O)

I

Vj+1 = Vj ⊕Wj+1 V0 = W0 L2(O) = ⊕∞j=0Wj


I

λ = (j, k), |λ| = j, J =∞⋃j=0

(j × Jj)


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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 ,


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Basic Properties:



I

‖f‖Bsq(Lp(O) ∼ ∞∑|λ|=j0

2|λ|(s+d( 12− 1p

))q

∑λ∈J ,|λ|=j

|〈f, ψλ〉|pq/p

1/q

,



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Basic Properties:



I

‖f‖Bsq(Lp(O) ∼ ∞∑|λ|=j0

2|λ|(s+d( 12− 1p

))q

∑λ∈J ,|λ|=j

|〈f, ψλ〉|pq/p

1/q

,



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Outline

Motivation




SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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∗?


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model




2 (O))

Ej(u) = O(n−s/dj )⇐= u ∈W s2 (O)

(linear approximation)I adaptive schemes


σn(u)L2(O) := ‖u− gn‖L2(O)

gn =∑

(j,k)∈Λn



1τ

=(s

d+

12

)

I Question: u ∈ Bsτ (Lτ (O)), 0 < s < s∗?


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model




2 (O))

Ej(u) = O(n−s/dj )⇐= u ∈W s2 (O)

(linear approximation)I adaptive schemes


σn(u)L2(O) := ‖u− gn‖L2(O)

gn =∑

(j,k)∈Λn



1τ

=(s

d+

12

)I Question: u ∈ Bs

τ (Lτ (O)), 0 < s < s∗?


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model



(Ω,F ,P) → probability space.I Stochastic evolution equation

du(t) =d∑

µ,ν=1


gk(t)dwkt ,


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,·)︸︷︷︸


P-a.s.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model



(Ω,F ,P) → probability space.I Stochastic evolution equation

du(t) =d∑

µ,ν=1


gk(t)dwkt ,


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,·)︸︷︷︸


P-a.s.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Weighted Sobolev Spaces for Sequences:

I γ ∈ N0, θ ∈ R

g =(gk)k∈N ∈ H

γ2,θ(O; `2)

:⇔

‖g‖2Hγ2,θ(O;`2) :=


∫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)]


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Weighted Sobolev Spaces for Sequences:

I γ ∈ N0, θ ∈ R

g =(gk)k∈N ∈ H

γ2,θ(O; `2)

:⇔

‖g‖2Hγ2,θ(O;`2) :=


∫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)]


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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))


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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))


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Existence/Uniqueness of Solutions in Hγ2,θ−2 :

du(t) =d∑

µ,ν=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)).


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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

).


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Example: d = 2, θ = d, γ = 2:


2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).


(O, T

)

I ⇒ u ∈ L2(. . . ;W 12 (O))

I Our theorem ⇒

u ∈ Lτ (. . . ;Bατ,τ (O)),

1τ

=α

2+

12

for all α < 2



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Example: d = 2, θ = d, γ = 2:


2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).


(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



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Example: d = 2, θ = d, γ = 2:


2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).


(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



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Example: d = 2, θ = d, γ = 2:


2,2(O, T ; `2) and u0 ∈ L2(Ω, H12,2(O)).


(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



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Outline

Motivation




SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model









SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model









SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Rothe Method:


)Utn+1


)I leads to elliptic subproblems, ; model problem:

−∆V = X(ω) in O, V = 0 on ∂O



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Rothe Method:


)Utn+1


)I leads to elliptic subproblems, ; model problem:

−∆V = X(ω) in O, V = 0 on ∂O



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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)


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


The Noise Model:






α(j−j0−1))d2


−4V = X(ω) in O, V = 0 on ∂O

X =∑λ∈J



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


The Noise Model:






α(j−j0−1))d2


−4V = X(ω) in O, V = 0 on ∂O

X =∑λ∈J



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


The Noise Model:

I

X =∑λ∈J





SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


The Noise Model:

I

X =∑λ∈J





SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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

,


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model




−∆V = X in O, V = 0 on ∂O



η(g) := #λ ∈ J : cλ 6= 0, g =∑λ∈J

cλψλ

e(V ) =(E‖V − V ‖2H1(O)

)1/2



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model




−∆V = X in O, V = 0 on ∂O



η(g) := #λ ∈ J : cλ 6= 0, g =∑λ∈J

cλψλ

e(V ) =(E‖V − V ‖2H1(O)

)1/2



SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model



Theorem

Let d ∈ 2, 3. Put

ρ = min(

12(d− 1)

,α+ β − 1

6+

23d

).







SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model



Theorem

Let d ∈ 2, 3. Put

ρ = min(

12(d− 1)

,α+ β − 1

6+

23d

).







SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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.


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:



I u ∈ Bsτ (Lτ (Ω)), 1/τ = s/d+ 1/2, 0 < s < s∗?

I weighted Sobolev estimates + wavelet expansions ;








SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


Summary:











SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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!


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


The DeVore-Triebel Diagram:

-

6

```````

s

1τ

12 1 2

1τ = s

2 + 12qW

3/22 (O)


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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...]


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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).


SPDEs

Stephan Dahlke

Motivation

TheoreticalAnalysis


The Model Equation


Besov Regularity



The Noise Model


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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