Adaptive waveletGalerkin methods: Algorithm & Applications Rob - - PowerPoint PPT Presentation

Adaptive wavelet–Galerkin methods: Algorithm & Applications

Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam joint work with Christoph Schwab and Rafaela Guberovic (Z¨ urich), Monique Dauge (Rennes), Stephan Dahlke and Ulrich Friedrich (Marburg), Nabi Chegini and Tammo Jan Dijkema (A’dam)

Contents

• Adaptive wavelet-Galerkin method
• Application to least-squares formulations
• Applications:

– Elliptic eqs with iso- and anisotropic wavelets – Parabolic problems – Navier-Stokes – First Order Systems Least Squares (FOSLS)

1/40

(Nonlinear) operator equations

F : X → X ′, solve F(u) = 0. We assume that a sol. u exists, and that (c1) F is continuously Fr´ echet differentiable in a neighborhood of u, (c2) DF(u)(v)(w) is inner product, with “energy”-norm · u · X. ❀ Solution u locally unique.

2/40

(Nonlinear) operator equations

F : X → X ′, solve F(u) = 0. We assume that a sol. u exists, and that (c1) F is continuously Fr´ echet differentiable in a neighborhood of u, (c2) DF(u)(v)(w) is inner product, with “energy”-norm · u · X. ❀ Solution u locally unique. Ex 1. X = H1

0(Ω), F(u)(w) = ∇u, ∇w + u3, w − f, w,

so DF(u)(v)(w) = ∇v, ∇w + 3u2v, w.

2/40

(Nonlinear) operator equations

F : X → X ′, solve F(u) = 0. We assume that a sol. u exists, and that (c1) F is continuously Fr´ echet differentiable in a neighborhood of u, (c2) DF(u)(v)(w) is inner product, with “energy”-norm · u · X. ❀ Solution u locally unique. Ex 1. X = H1

0(Ω), F(u)(w) = ∇u, ∇w + u3, w − f, w,

so DF(u)(v)(w) = ∇v, ∇w + 3u2v, w. Galerkin: SΛ ⊂ X closed, find uΛ ∈ SΛ s.t. F(uΛ)(vΛ) = 0 (vΛ ∈ SΛ). Lem 1. If d(u, SΛ) suff. small, then Galerkin has unique sol near u, and u − uΛX d(u, SΛ).

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Adaptive Galerkin method

Let Ψ = {ψλ : λ ∈ ∇} with closX spanΨ = X. For Λ ⊂ ∇, SΛ := closX span{ψλ : λ ∈ Λ}.

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Adaptive Galerkin method

Let Ψ = {ψλ : λ ∈ ∇} with closX spanΨ = X. For Λ ⊂ ∇, SΛ := closX span{ψλ : λ ∈ Λ}. Goal: Whenever for some s > 0, u ∈ As := {v ∈ X : vAs := sup

N∈I N

N s inf

{Λ⊂∇:#Λ=N} d(v, SΛ) < ∞},

i.e., s is rate of best N-term approximation, to find Λ0 ⊂ Λ1 ⊂ · · · , and wΛi ∈ SΛi, with u − wΛiX (#Λi)−s.

3/40

Adaptive Galerkin method

Let Ψ = {ψλ : λ ∈ ∇} with closX spanΨ = X. For Λ ⊂ ∇, SΛ := closX span{ψλ : λ ∈ Λ}. Goal: Whenever for some s > 0, u ∈ As := {v ∈ X : vAs := sup

N∈I N

N s inf

{Λ⊂∇:#Λ=N} d(v, SΛ) < ∞},

i.e., s is rate of best N-term approximation, to find Λ0 ⊂ Λ1 ⊂ · · · , and wΛi ∈ SΛi, with u − wΛiX (#Λi)−s. Let QΛ : X → X be some projector onto SΛ, unif. bound. in Λ ⊂ ∇.

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Adaptive Galerkin method

Alg 1 (agm). % Let 0 < µ0 < 1, δ, γ > 0 be suff. small constants. % Let Λ0 ⊂ ∇, wΛ0 ∈ SΛ0 with u − wΛ0X suff. small. for i = 0, 1, . . . do Compute ri ∈ X ′ such that ri − F(wΛi)X ′ ≤ δriX ′. Determine a near-smallest Λi+1 ⊃ Λi with Q′

Λi+1riX ′ ≥ µ0riX ′

Compute wΛi+1 ∈ SΛi+1 with wΛi+1 − uΛi+1X ≤ γriX ′. enfor Thm 2 (St. ’11). For some α < 1, u − wΛi+1u ≤ αu − wΛiu. If, for whatever s > 0, u ∈ As, then u − wΛiX (#Λi)−s.

4/40

Rephrasing in coordinates

Let Ψ be Riesz basis (wavelets) for X, i.e. analysis and so synthesis

• perators

F ∈ L(X ′, ℓ2(∇)) : g → [g(ψλ)]λ∈∇, F′ ∈ L(ℓ2(∇), X) : v → v⊤Ψ :=

• λ∈∇

vλψλ. are boundedly invertible. Not.: F′v = v. One has v ∈ As ⇐ ⇒ v − vN N −s.

5/40

Rephrasing in coordinates

Let Ψ be Riesz basis (wavelets) for X, i.e. analysis and so synthesis

• perators

F ∈ L(X ′, ℓ2(∇)) : g → [g(ψλ)]λ∈∇, F′ ∈ L(ℓ2(∇), X) : v → v⊤Ψ :=

• λ∈∇

vλψλ. are boundedly invertible. Not.: F′v = v. One has v ∈ As ⇐ ⇒ v − vN N −s. With F := FFF′, F(u) = 0 ⇐ ⇒ F(u) = 0. IΛ : ℓ2(Λ) → ℓ2(∇) extension with zeros, I′

Λ : ℓ2(∇) → ℓ2(Λ) restriction.

With FΛ := I′

ΛFIΛ, F(uΛ)(vΛ) = 0 (vΛ ∈ SΛ) ⇐

⇒ FΛ(uΛ) = 0.

5/40

Rephrasing in coordinates

Let Ψ be Riesz basis (wavelets) for X, i.e. analysis and so synthesis

• perators

F ∈ L(X ′, ℓ2(∇)) : g → [g(ψλ)]λ∈∇, F′ ∈ L(ℓ2(∇), X) : v → v⊤Ψ :=

• λ∈∇

vλψλ. are boundedly invertible. Not.: F′v = v. One has v ∈ As ⇐ ⇒ v − vN N −s. With F := FFF′, F(u) = 0 ⇐ ⇒ F(u) = 0. IΛ : ℓ2(Λ) → ℓ2(∇) extension with zeros, I′

Λ : ℓ2(∇) → ℓ2(Λ) restriction.

With FΛ := I′

ΛFIΛ, F(uΛ)(vΛ) = 0 (vΛ ∈ SΛ) ⇐

⇒ FΛ(uΛ) = 0. Take QΛ := F′IΛI′

Λ(F′)−1 (biorth. proj.), so that Q′ Λ := F−1IΛI′ ΛF,

and equip X with (F′)−1 · ℓ2(∇).

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Adaptive (wavelet) Galerkin method

Alg 3 (awgm). % Let 0 < µ0 < 1, δ, γ > 0 be suff. small constants. % Let Λ0 ⊂ ∇, with d(u, SΛ0) suff. small., and let wΛ0 ∈ ℓ2(Λ0) % with FΛ0(wΛ0) suff. small. for i = 0, 1, . . . do Compute ri ∈ ℓ2(∇) such that ri − F(wΛi) ≤ δri. Determine a near-smallest Λi+1 ⊃ Λi with ri|Λi+1 ≥ µ0ri Compute wΛi+1 ∈ ℓ2(Λi+1) with FΛi+1(wΛi+1) ≤ γri. enfor

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Historical notes

For affine (linear) problems, i.e. F(u) = Lu − f, current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries.

7/40

Historical notes

For affine (linear) problems, i.e. F(u) = Lu − f, current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations.

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Historical notes

For affine (linear) problems, i.e. F(u) = Lu − f, current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations. In 2007, Gantumur, Harbrecht & St. showed optimality of CDD1 without

• coarsening. Quantitively more efficient than CDD2, for which coarsening is
• required. Spin-off: optimally convergent adaptive fem.

7/40

Historical notes

For affine (linear) problems, i.e. F(u) = Lu − f, current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations. In 2007, Gantumur, Harbrecht & St. showed optimality of CDD1 without

• coarsening. Quantitively more efficient than CDD2, for which coarsening is
• required. Spin-off: optimally convergent adaptive fem.

In 2003, Y. Xu and Q. Zhou extended CDD1 to nonlinear equations and removed coarsening. Optimality proof hinges on equivalence of sequence norms and Besov norms which can be avoided.

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General well-posed nonlinear eqs: Least-squares approach

G : X → Y′, solve G(u) = 0. We assume that a sol. u exists, and that (c′

1) G is two times continuously Fr´

echet differentiable near u, (c′

2) DG(u)

∈ L(X, Y′) is homeomorphism

• nto

its range, i.e., DG(u)(v)Y′ vX (v ∈ X).

8/40

General well-posed nonlinear eqs: Least-squares approach

G : X → Y′, solve G(u) = 0. We assume that a sol. u exists, and that (c′

1) G is two times continuously Fr´

echet differentiable near u, (c′

2) DG(u)

∈ L(X, Y′) is homeomorphism

• nto

its range, i.e., DG(u)(v)Y′ vX (v ∈ X). With Q(v) := 1

2G(v)2 Y′, u = argminv∈X Q(v), and so

F(u)(v) := DQ(u)(v) = DG(u)(v), G(u)Y′ = 0 (v ∈ X). F : X → X ′ and (c1) F is continuously Fr´ echet differentiable in a neighborhood of u, (c2) DF(u)(v)(w) = DG(u)(v), DG(u)(w)Y′ is inner product, with “energy”-norm · u · X.

8/40

Least-squares approach

Riesz basis ΨX for X. To circumvent eval. of ·, ·Y′, Riesz basis ΨY for

• Y. Corr. analyses ops. FX, FY.

Equip X, Y with (F′

X)−1 · ℓ2(∇X ), (F′ Y)−1 · ℓ2(∇Y).

With G := FYGF′

X, DG(v) = FYDG(v)F′ X and

F(v) (= FXFF′

X(v) ) = DG(v)⊤G(v).

(For G = Lv − f, L⊤(Lv − f), normal eqs.)

9/40

Least-squares approach

Riesz basis ΨX for X. To circumvent eval. of ·, ·Y′, Riesz basis ΨY for

• Y. Corr. analyses ops. FX, FY.

Equip X, Y with (F′

X)−1 · ℓ2(∇X ), (F′ Y)−1 · ℓ2(∇Y).

With G := FYGF′

X, DG(v) = FYDG(v)F′ X and

F(v) (= FXFF′

X(v) ) = DG(v)⊤G(v).

(For G = Lv − f, L⊤(Lv − f), normal eqs.) Special case: For Y′ = L2(Ω) (mild variational formulations of PDEs, in particular first order systems), not nec. to select a basis for Y. Using standard inner product: F(v) =

• DG(v)ψλ, G(v)L2(Ω)
• λ∈∇.

9/40

• Comput. compl. of AWGM

Determined by that of approx res. eval.

• Opt. comput compl., i.e., for u ∈ As, not only u − wΛX (#Λ)−s

but also u − wΛX (#ops)−s if r − F(wΛ)X ′ ≤ ε in O(ε−1/s(1 + wΛ1/s

As ) + #Λ) ops.

Term ε−1/s ass. to rhs approx (ignore). Needed to approx F h(wΛ) within tol ε in O(ε−1/swΛ1/s

As + #Λ) ops.

s unknown. So nec. for ∀s ≤ smax, being best possible rate (e.g., d−m

n ).

10/40

• Comput. compl. of AWGM

CDD1: Algorithm for linear F h (apply-routine): Approx columns of Fh with accuracy proportional to size of corresponding entry. For a large class of PDO’s and sing. integr. ops, and suitable wavelet bases (vanishing moments, piecewise smoothness), sufficient near-sparsity of Fh (s∗-compressibility with s∗ > smax) was shown later.

11/40

• Comput. compl. of AWGM

CDD1: Algorithm for linear F h (apply-routine): Approx columns of Fh with accuracy proportional to size of corresponding entry. For a large class of PDO’s and sing. integr. ops, and suitable wavelet bases (vanishing moments, piecewise smoothness), sufficient near-sparsity of Fh (s∗-compressibility with s∗ > smax) was shown later. CDD4: results for classes of nonlinear operators. Needed to restrict to Λ ⊂ ∇ that are trees, meaning that λ ∈ Λ = ⇒ µ ∈ Λ when |µ| = |λ| − 1 and supp ψλ ∩ supp ψµ = ∅. Not clear whether valid ∀s ≤ smax.

Applications (first linear ones):

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Poisson-type problems with isotropic wavelets

V0 ⊂ V1 ⊂ · · · ⊂ X multi-resolution analysis. Vℓ+1 = Vℓ ⊕ Wℓ+1. Equip Wℓ with basis. Under conds, union over ℓ is Riesz basis for X.

• Comput. less efficient than AFEM, which is very sim. with role residual in

wav coordinates being played by a posteriori error estimator.

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Poisson-type problems with anisotropic wavelets

Let Ω =

i Ωi. Let Ψi be a Riesz basis for L2(Ωi), then ⊗iΨi Riesz basis

for ⊗iL2(Ωi) ∼ L2(

i Ωi). Also for Hs( i Ωi) for s = 0.

Supports basis functions anisotropic. “Linear” approximation rates: Ω = ✷ := (0, 1)n, wavelets of order d. Isotropic: inf

v∈span{ψλ:|λ|≤J} u − vHm(✷) “N”−d−m

n uHd(✷). 13/40

Poisson-type problems with anisotropic wavelets

Let Ω =

i Ωi. Let Ψi be a Riesz basis for L2(Ωi), then ⊗iΨi Riesz basis

for ⊗iL2(Ωi) ∼ L2(

i Ωi). Also for Hs( i Ωi) for s = 0.

Supports basis functions anisotropic. “Linear” approximation rates: Ω = ✷ := (0, 1)n, wavelets of order d. Isotropic: inf

v∈span{ψλ:|λ|≤J} u − vHm(✷) “N”−d−m

n uHd(✷).

Anisotropic: for σ ∈ (1,

d d−m)

inf

v∈span{ψλ1⊗···⊗ψλn:σ P

i |λi|+(1−σ) maxi |λi|≤J}

u − vHm(✷) “N”−(d−m)uHd(0,1)⊗···Hd(0,1). (optimized sparse grids, [Griebel & Knapek ’00]).

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Poisson-type problems with anisotropic wavelets

Even for smooth f, sol of Poisson generally does not satisfy u ∈ Hd(✷) or u ∈ Hd(0, 1) ⊗ · · · Hd(0, 1). With optimized sparse grids instead of rate d − 1, generally only 1

2 + 1 n.

14/40

Poisson-type problems with anisotropic wavelets

Even for smooth f, sol of Poisson generally does not satisfy u ∈ Hd(✷) or u ∈ Hd(0, 1) ⊗ · · · Hd(0, 1). With optimized sparse grids instead of rate d − 1, generally only 1

2 + 1 n.

[Dauge & St. ’09]: Suff. for rate d − m in Hm(✷) of best N-term approx. is u ∈ Hd

θ−min(1,θ)(0, 1) ⊗ Hd θ(0, 1) ⊗ · · · ⊗ Hd θ(0, 1) ∩

. . . ∩ Hd

θ(0, 1) ⊗ · · · ⊗ Hd θ(0, 1) ⊗ Hd θ−min(1,θ)(0, 1)

for some θ ∈ [0, d), with weighted Sobolev norm reading as vHd

ω(0,1) :=

d

i=0

1

0 |xω(1 − x)ωv(i)(x)|2dx

1

2 , 14/40

Poisson-type problems with anisotropic wavelets

Even for smooth f, sol of Poisson generally does not satisfy u ∈ Hd(✷) or u ∈ Hd(0, 1) ⊗ · · · Hd(0, 1). With optimized sparse grids instead of rate d − 1, generally only 1

2 + 1 n.

[Dauge & St. ’09]: Suff. for rate d − m in Hm(✷) of best N-term approx. is u ∈ Hd

θ−min(1,θ)(0, 1) ⊗ Hd θ(0, 1) ⊗ · · · ⊗ Hd θ(0, 1) ∩

. . . ∩ Hd

θ(0, 1) ⊗ · · · ⊗ Hd θ(0, 1) ⊗ Hd θ−min(1,θ)(0, 1)

for some θ ∈ [0, d), with weighted Sobolev norm reading as vHd

ω(0,1) :=

d

i=0

1

0 |xω(1 − x)ωv(i)(x)|2dx

1

2 ,

and for sufficiently smooth rhs, the solution of Poisson’s problem on ✷, i.e., m = 1, satisfies this anisotropic regularity condition for arbitrary n and arbitrary d, for θ ∈ (d − 1

n, d).

14/40

Poisson-type problems with an isotropic wavelets

With suitable wavs, bi-infinite stiffness matrix is s∗ > d − 1 compressible (also with suff. smooth variable coeffs), so AWGM gives rate d − 1 in opt.

• comput. compl.
• Num. results: Poisson on ✷, f = 1, hom. Dir on all faces that contain
• rigin. 1 ≤ n ≤ 10; L2-orth. wavs. with d = 2.

15/40

Poisson-type problems with an isotropic wavelets

With suitable wavs, bi-infinite stiffness matrix is s∗ > d − 1 compressible (also with suff. smooth variable coeffs), so AWGM gives rate d − 1 in opt.

• comput. compl.
• Num. results: Poisson on ✷, f = 1, hom. Dir on all faces that contain
• rigin. 1 ≤ n ≤ 10; L2-orth. wavs. with d = 2.

1 101 102 103 104 105 103 102 101 1

1

˜ f− ˜ Ag uN f

N

15/40

Poisson-type problems with anisotropic wavelets

Extensions

To avoid apply-routine: Special wavelets, d = 5, s.t. any PDO of order 2 with c.c. gives rise to a truly sparse stiffness matrix.

16/40

Poisson-type problems with anisotropic wavelets

Extensions

To avoid apply-routine: Special wavelets, d = 5, s.t. any PDO of order 2 with c.c. gives rise to a truly sparse stiffness matrix. General domains: domain decomposition, piecewise tensor product approx., scale-dependent extension operators.

16/40

Poisson-type problems with anisotropic wavelets

Extensions

To avoid apply-routine: Special wavelets, d = 5, s.t. any PDO of order 2 with c.c. gives rise to a truly sparse stiffness matrix. General domains: domain decomposition, piecewise tensor product approx., scale-dependent extension operators. Poisson, f = 1, hom. Dirichlet, L-shaped domain (n = 2) subdivided into 3 squares. Energy error vs. support length. Optimal slope −4.

10 10

1

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−6

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−5

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−1

10

16/40

Poisson-type problems with anisotropic wavelets

Centers supports

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

17/40

Poisson-type problems with anisotropic wavelets

Poisson, f = 1, hom. Dirichlet, “thick” L-shaped domain (n = 3) subdivided into 6 cubes. Energy error vs. support length. Optimal slope −4.

10 10

2

10

4

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18/40

Evolution problems: Overview

• With time marching methods, an optimal distribution of grid points over

space and time is hard to realize.

19/40

Evolution problems: Overview

• With time marching methods, an optimal distribution of grid points over

space and time is hard to realize.

• We apply an adaptive method to a simultaneously space-time variational

formulation.

19/40

Evolution problems: Overview

• With time marching methods, an optimal distribution of grid points over

space and time is hard to realize.

• We apply an adaptive method to a simultaneously space-time variational

formulation.

• While keeping discrete solutions on all time levels is prohibitive for time

marching methods, thanks to the use of tensorized multi-level bases

• ur method produces approximations simultaneously in space and time

without penalty in complexity because of the additional time dimension.

19/40

Parabolic problems

Let V ֒ → H ֒ → V ′, I := (0, T). Consider parabolic problem

∂u ∂t(t, ·) + A(t)u(t, ·) = g(t, ·)

in V ′, u(0, ·) = u0 in H, where a(t; η, ζ) := (A(t)(η))(ζ) satisfies for a.e. t ∈ I, |a(t; η, ζ)| ≤ MaηV ζV (η, ζ ∈ V ) (boundedness), ℜa(t; η, η) + λη2

H ≥ αη2 V

(η ∈ V ) (G˚ arding inequality). E.g., A(t) differential or integrodifferential operator of order 2m ≥ 0, H = L2(Ω), V = Hm(Ω) (Hm

0 (Ω)).

20/40

Parabolic problems

Weak formulations

Multiplication with smooth v with v(T, ·) = 0, int. by parts over space and time ❀ Find u ∈ X := L2(I; V ) s.t. b(u, v) = f(v) (v ∈ Y := L2(I; V ) ∩ H1

0,{T }(I; V ′))

where b(u, v) :=

• I

−u(t, ·), ∂v

∂t(t, ·)H + a(t; u(t, ·), v(t·))dt,

f(v) :=

• I

g(t, ·), v(t, ·)Hdt + u0, v(0, ·)H.

21/40

Parabolic problems

Weak formulations

Multiplication with smooth v with v(T, ·) = 0, int. by parts over space and time ❀ Find u ∈ X := L2(I; V ) s.t. b(u, v) = f(v) (v ∈ Y := L2(I; V ) ∩ H1

0,{T }(I; V ′))

where b(u, v) :=

• I

−u(t, ·), ∂v

∂t(t, ·)H + a(t; u(t, ·), v(t·))dt,

f(v) :=

• I

g(t, ·), v(t, ·)Hdt + u0, v(0, ·)H. Thm 4 (Dautray & Lions ’92, Wloka ’82, Schwab & St.’08). B : X → Y′ defined by (Bu)(v) = b(u, v) is boundedly invertible.

21/40

Parabolic problems

Weak formulations

Multiplication with smooth v with v(T, ·) = 0, int. by parts over space and time ❀ Find u ∈ X := L2(I; V ) s.t. b(u, v) = f(v) (v ∈ Y := L2(I; V ) ∩ H1

0,{T }(I; V ′))

where b(u, v) :=

• I

−u(t, ·), ∂v

∂t(t, ·)H + a(t; u(t, ·), v(t·))dt,

f(v) :=

• I

g(t, ·), v(t, ·)Hdt + u0, v(0, ·)H. Thm 4 (Dautray & Lions ’92, Wloka ’82, Schwab & St.’08). B : X → Y′ defined by (Bu)(v) = b(u, v) is boundedly invertible. Thm

• 5. Let W

֒ → V , A(·)′ ∈ C(¯ I, L(W, H)), and A(t)′ + λI : W → H is boundedly invertible. Then with X := L2(I; H) and Y := L2(I; W) ∩ H1

0,{T }(I; H), B ∈ L(X, Y′) is boundedly invertible.

21/40

Tensor product bases

Let ΘX, ΘY, and ΣX, ΣY be collections of temporal or spatial functions such that, normalized in the corresponding norms, ΘX is a Riesz basis for L2(I), ΘY ” L2(I) and for H1

0,{T }(I),

ΣX ” H, ΣY ” W ” H. Then, with X = L2(I; H) and Y = L2(I; W) ∩ H1

0,{T }(I; H), normalized

in the corresponding norms, ΘX ⊗ ΣX is a Riesz basis for X, ΘY ⊗ ΣY ” L2(I; W), H1

0,{T }(I; H), and so for Y,

22/40

Best possible rates

H = L2(Ω), Ω ⊂ I Rn, so X = L2(I; L2(Ω)) smax =

• min(dt, dx

n )

isotropic spatial wavelets min(dt, dx) anisotropic spatial wavelets up to log-factors when dt = dx

n or dt = dx.

23/40

Best possible rates

H = L2(Ω), Ω ⊂ I Rn, so X = L2(I; L2(Ω)) smax =

• min(dt, dx

n )

isotropic spatial wavelets min(dt, dx) anisotropic spatial wavelets up to log-factors when dt = dx

n or dt = dx.

With linear approx, these rates require boundedness of certain mixed derivatives in L2. Relaxed regularity conditions with best N-term approx.

23/40

Best possible rates

H = L2(Ω), Ω ⊂ I Rn, so X = L2(I; L2(Ω)) smax =

• min(dt, dx

n )

isotropic spatial wavelets min(dt, dx) anisotropic spatial wavelets up to log-factors when dt = dx

n or dt = dx.

With linear approx, these rates require boundedness of certain mixed derivatives in L2. Relaxed regularity conditions with best N-term approx. Realization of rates of best N-term approximation: Adaptive wavelet Galerkin scheme to B⊤(Bu − f) = 0.

23/40

Numerics heat equation

Heat equation

∂ ∂tu − ∆xu = g

• n (0, T) × ✷,

u = 0

• n (0, T) × ∂✷

u(0, ·) = u0. Temporal, and anisotropic spatial wavelets with dt = dx = 5.

24/40

Numerics heat equation

10 10

1

10

2

10

3

10

4

10

5

10

−15

10

−10

10

−5

10 10

5

Figure 1: Heat eqn. in n = 1 spatial dimension, right-hand side g = 1 and initial condition u0 = 0. Buε − f/f vs. N = #supp uε for the AWGM (solid), full-grid (dashed) and sparse-grid method (dashed-dotted). The dotted line is a multiple of N −5(log N)51

2. 25/40

Numerics heat equation

10 10

1

10

2

10

3

10

4

10

5

10

−10

10

−5

10 10

5

10

10

Figure 2: AWGM applied to heat eqn. in n = 1 spatial dimension, right-hand side g = 1 and initial condition u0 = 1. Buε − f/f vs. N = #supp uε. The dotted line is a multiple of N −5(log N)51

2. 26/40

Numerics heat equation

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0 0.2 0.4 0.6 0.8 1.0 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

Figure 3: Heat eqn. in n = 1 spatial dimension and right-hand side g = 1. Centers of the supports of the wavelets selected by the AWGM. Left u0 = 0 and #uε = 13420. Right u0 = 1 and #uε = 13917. A zoom in near t = 0 is given at the bottom row.

27/40

Numerics heat equation

10 10

1

10

2

10

3

10

4

10

5

10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

Figure 4: Heat eqn. in n = 2 spatial dimensions, right-hand side g = 1 and initial condition u0 = 0. Buε − f/f vs. N = #supp uε for the AWGM (solid), full-grid (dashed) and sparse-grid method (dashed-dotted). The dotted line is a multiple of N −5(log N)11.

28/40

(Navier–) Stokes eqs

An approach: reduction to (nonlinear) parabolic problem for divergence-free velocities. Disadvantage: Wavelet bases for spaces of divergence-free velocities available essentially for rectangular domains only. [Guberovic, Schwab, St.’11]: Space-time variational saddle-point formulations of (N)SE:

29/40

(Navier–) Stokes eqs

An approach: reduction to (nonlinear) parabolic problem for divergence-free velocities. Disadvantage: Wavelet bases for spaces of divergence-free velocities available essentially for rectangular domains only. [Guberovic, Schwab, St.’11]: Space-time variational saddle-point formulations of (N)SE: Ω ⊂ I R2 or I

• R3. C2 or convex.

      

∂u ∂t − ν∆xu + u · ∇x u + ∇x p = f

• n [0, T] × Ω,

divx u = g

• n [0, T] × Ω,

u = 0

• n [0, T] × ∂Ω,

u(0, ·) = 0

• n Ω.

Finding (u, p) ∈ X = U × P (G(u, p))(v, q) := a(u, v) + b(p, v) + c(u, q) + n(u, u, v) = f(v) + g(q) ((v, q) ∈ Y = V × Q),

29/40

(Navier–) Stokes eqs

                                         a(u, v) = T

• Ω

∂u ∂t · v dxdt +

T

• Ω

ν∇xu : ∇xv dxdt, b(p, v) = T

• Ω

v · ∇p dxdt, c(u, q) = − T

• Ω

u · ∇q dxdt, f(v) = T

• Ω

f · v dxdt, g(q) = T

• Ω

g q dxdt. n(y, z, v) = T

• Ω

y · ∇x z · v dxdt.

30/40

(Navier–) Stokes eqs

                                         a(u, v) = T

• Ω

∂u ∂t · v dxdt +

T

• Ω

ν∇xu : ∇xv dxdt, b(p, v) = T

• Ω

v · ∇p dxdt, c(u, q) = − T

• Ω

u · ∇q dxdt, f(v) = T

• Ω

f · v dxdt, g(q) = T

• Ω

g q dxdt. n(y, z, v) = T

• Ω

y · ∇x z · v dxdt. U := L2((0, T); (H1

0(Ω) ∩ H2(Ω))n) ∩ H1 0,{0}((0, T); L2(Ω)n),

P := L2((0, T); H1(Ω)/I R), V := L2((0, T); L2(Ω)n), Q :=

• L2((0, T); H1(Ω)/I

R) ∩ H1

0,{0}

• (0, T); (H1(Ω)/I

R)′′ .

30/40

(Navier–) Stokes eqs

Then G : X → Y′, and, in any case for sufficiently small f and g, there exists a unique sol (u, p) in a neighborhood of 0. Furthermore, DG(u, p) : (˜ u, ˜ p) →

• (v, q) →

a(˜ u, v) + n(u, ˜ u, v) + n(˜ u, u, v) + b(˜ p, v) + c(˜ u, q)

• ∈ L(X, Y′)

is boundedly invertible. Corresponding strong operator       

∂˜ u ∂t − ν∆x˜

u + u · ∇x˜ u + ˜ u · ∇xu + ∇x ˜ p = · · ·

• n [0, T] × Ω,

divx ˜ u = · · ·

• n [0, T] × Ω,

˜ u = 0

• n [0, T] × ∂Ω,

˜ u(0, ·) = 0

• n Ω.

31/40

FOSLS

Ex 2. Polyhedron Ω ⊂ I R3, −∆p + p = f

• n Ω,

∂p ∂n = h

• n ∂Ω.

= ⇒    u − ∇p = 0

• n Ω,

p − div u = f

• n Ω,

u · n = h

• n ∂Ω.

When h = 0, with H0(div; Ω) := {v ∈ H(div; Ω) : v · n = 0 on ∂Ω} (u, p) → (u − ∇p, p − div u) ∈ L(H0(div; Ω) × H1(Ω), L2(Ω)3 × L2(Ω)) is boundedly invertible.

32/40

FOSLS

Ex 2. Polyhedron Ω ⊂ I R3, −∆p + p = f

• n Ω,

∂p ∂n = h

• n ∂Ω.

= ⇒    u − ∇p = 0

• n Ω,

p − div u = f

• n Ω,

u · n = h

• n ∂Ω.

When h = 0, with H0(div; Ω) := {v ∈ H(div; Ω) : v · n = 0 on ∂Ω} (u, p) → (u − ∇p, p − div u) ∈ L(H0(div; Ω) × H1(Ω), L2(Ω)3 × L2(Ω)) is boundedly invertible. Add redundant eq.:        u − ∇p = 0

• n Ω,

p − div u = f

• n Ω,

curl u = 0

• n Ω,

u · n = h

• n ∂Ω.

32/40

Again for h = 0, with H1

0(Ω) := {v ∈ H1(Ω)3 : v · n = 0 on ∂Ω}, and

assuming Ω convex, (u, p) → (u−∇p, p−div u, curl u) ∈ L(H1

0(Ω)×H1(Ω), L2(Ω)3×L2(Ω)×L2(Ω)3)

is a homeomorphism onto its range. In [Cai, Manteuffel, McCormick ’97] generalization to wide class of 2nd order ellip. bvp.

33/40

Again for h = 0, with H1

0(Ω) := {v ∈ H1(Ω)3 : v · n = 0 on ∂Ω}, and

assuming Ω convex, (u, p) → (u−∇p, p−div u, curl u) ∈ L(H1

0(Ω)×H1(Ω), L2(Ω)3×L2(Ω)×L2(Ω)3)

is a homeomorphism onto its range. In [Cai, Manteuffel, McCormick ’97] generalization to wide class of 2nd order ellip. bvp. With Γ1, . . . , ΓK denoting the faces of ∂Ω with normal vectors n1 . . . , nK, (u, p) → (u − ∇p, p − div u, curl u,

K

• r=1

(u · nr)|Γr) ∈ L(H1(Ω)3 × H1(Ω), L2(Ω)3 × L2(Ω) × L2(Ω)3 ×

K

• r=1

H

1 2(Γr))

is a homeomorphism onto its range.

33/40

Application of AWGM to FOSLS

Setting: X = H1(Ω)M, Y = Y1 × · · · × YL, G = (G1, . . . , GL)⊤ : X → Y. G(u) = 0, DG(u) ∈ L(X, Y′) homeomorphism onto range.

34/40

Application of AWGM to FOSLS

Setting: X = H1(Ω)M, Y = Y1 × · · · × YL, G = (G1, . . . , GL)⊤ : X → Y. G(u) = 0, DG(u) ∈ L(X, Y′) homeomorphism onto range. Search u as argminv∈X Q(v), with Q(v) = 1

2G(v)2 Y, and so as solution

• f

0 = F(u)(v) = DG(u)(v), G(u)Y′ =

• i

DGi(u)(v), Gi(u)Y′

i

(v ∈ X). Given wΛ, to approx. F(wΛ) within some suff. small rel. tol. Thanks to G(u) = 0, equiv. to evaluating each v → DGi(u)(v), Gi(u)Y′

i within suff.

small rel. tol.

34/40

Application of AWGM to FOSLS

Setting: X = H1(Ω)M, Y = Y1 × · · · × YL, G = (G1, . . . , GL)⊤ : X → Y. G(u) = 0, DG(u) ∈ L(X, Y′) homeomorphism onto range. Search u as argminv∈X Q(v), with Q(v) = 1

2G(v)2 Y, and so as solution

• f

0 = F(u)(v) = DG(u)(v), G(u)Y′ =

• i

DGi(u)(v), Gi(u)Y′

i

(v ∈ X). Given wΛ, to approx. F(wΛ) within some suff. small rel. tol. Thanks to G(u) = 0, equiv. to evaluating each v → DGi(u)(v), Gi(u)Y′

i within suff.

small rel. tol.

• Yi = L2(Ω) and Gh

i is PDO, or

• Yi = H

1 2(Γr) and Gh

i : w → w · a|Γ for some constant a.

34/40

Application of AWGM

• Equip. H1(Ω) with standard, isotropic wavelet basis.

By selecting suitable matching wavelet basis for H

1 2(Γr), trace operators

give no problems. To approximate [DGi(w⊤

ΛΨ)(ψλ), Gh i (w⊤ ΛΨ) − fL2(Ω)]λ∈∇:

35/40

Application of AWGM

• Equip. H1(Ω) with standard, isotropic wavelet basis.

By selecting suitable matching wavelet basis for H

1 2(Γr), trace operators

give no problems. To approximate [DGi(w⊤

ΛΨ)(ψλ), Gh i (w⊤ ΛΨ) − fL2(Ω)]λ∈∇:

Restricting to Λ that are trees, express w⊤

ΛΨ in a locally finite single-scale

• repr. Is piecewise polynomial w.r.t. some partition. Assume f is piecewise

polynomial w.r.t. this part (...). Then Gh

i (w⊤ ΛΨ) − f exact.

35/40

Application of AWGM

• Equip. H1(Ω) with standard, isotropic wavelet basis.

By selecting suitable matching wavelet basis for H

1 2(Γr), trace operators

give no problems. To approximate [DGi(w⊤

ΛΨ)(ψλ), Gh i (w⊤ ΛΨ) − fL2(Ω)]λ∈∇:

Restricting to Λ that are trees, express w⊤

ΛΨ in a locally finite single-scale

• repr. Is piecewise polynomial w.r.t. some partition. Assume f is piecewise

polynomial w.r.t. this part (...). Then Gh

i (w⊤ ΛΨ) − f exact.

Gh

i (w⊤ ΛΨ)

inherits some piecewise smoothness from its argument. Therefore, for some constant k (k = 1 in experiment), λ with |λ| > |µ| + k for all µ ∈ Λ with supp ψλ∩supp ψµ = ∅ can be ignored. Remaining entries first computed in “single-scale” repr., then “backtransformation”. Algorithm close to what is used in FOSLS AFEM setting.

35/40

Numerical example

Scalar ODE

• u′ + αu3

= f

• n (0, 1),

u(0) = u0, where α =

• x → 1 + 3x
• n (0, 1

3),

x → 5 − 3x

• n (1

3, 1).

G = (G1, G2) : H1(0, 1) → L2(0, 1)×I R : w → (w′ +αw3 −f, w(0)−u0),

36/40

Numerical example

Scalar ODE

• u′ + αu3

= f

• n (0, 1),

u(0) = u0, where α =

• x → 1 + 3x
• n (0, 1

3),

x → 5 − 3x

• n (1

3, 1).

G = (G1, G2) : H1(0, 1) → L2(0, 1)×I R : w → (w′ +αw3 −f, w(0)−u0), DG(w) : H1(0, 1) → L2(0, 1)×I R : v → (v′+α3w2v, v(0)) boundedly invertible. F(w)(v) = DG(w)v, G(w)L2(0,1)×I

R

= v′ + α3w2v, w′ + αw3 − fL2(0,1) + v(0)(w(0) − u0).

36/40

Numerical example

Scalar ODE

• u′ + αu3

= f

• n (0, 1),

u(0) = u0, where α =

• x → 1 + 3x
• n (0, 1

3),

x → 5 − 3x

• n (1

3, 1).

G = (G1, G2) : H1(0, 1) → L2(0, 1)×I R : w → (w′ +αw3 −f, w(0)−u0), DG(w) : H1(0, 1) → L2(0, 1)×I R : v → (v′+α3w2v, v(0)) boundedly invertible. F(w)(v) = DG(w)v, G(w)L2(0,1)×I

R

= v′ + α3w2v, w′ + αw3 − fL2(0,1) + v(0)(w(0) − u0). Biorthogonal wavelet basis (2, 2). u0 = 1, f = 1. Galerkin problems solved with damped Rich., with at most 8 iterations.

36/40

Numerical example

10 10

1

10

2

10

3

10

4

10

5

10

6

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

Figure 5: #Λ vs. F(wΛ)|¯

Λℓ2(¯ Λ). The hypothenuse has slope −1. With

2.5 · 105 unknowns, max level is 52

37/40

Numerical example

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6: Distribution of the wavelet coefficients over levels and locations for a support size #Λ = 169.

38/40

Numerical example

0.0 0.2 0.4 0.6 0.8 1.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Figure 7: Approximate solutions being linear combinations of 5, 10 and 15

• wavelets. The exact solution is indicated with the solid line.

39/40

Summary

• Adaptive wavelet Galerkin method solves a wide class of operator eqs

with optimal rates.

• Promising applications with tensor product approximations as they

naturally arise with space-time variational formulations.

• Efficient residual evaluations for linear constant coefficient ops, and with

FOSLS also for nonlinear ops.

40/40