Computation of operators in wavelet coordinates Tsogtgerel Gantumur - - PowerPoint PPT Presentation

Computation of operators in wavelet coordinates

Tsogtgerel Gantumur and Rob Stevenson Department of Mathematics Utrecht University The Netherlands

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.1/19

Contents

Settings: linear operator equation, wavelet basis Optimal complexity of CDD2 algorithm Differential operators Compressibility Computability Boundary integral operators Compressibility Computability Conclusion

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.2/19

Linear operator equations

Let Ω be an n-dimensional domain or smooth manifold

Ht ⊂ Ht(Ω) be a subspace, and H−t be its dual space

Consider the problem of finding u from

Lu = g

where L : Ht → H−t is a self-adjoint elliptic operator of

• rder 2t

and g ∈ H−t is a linear functional

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.3/19

Differential operators

Partial differential operators of order 2t

v, Lu =

• |α|,|β|≤t

∂αv, aαβ∂βu,

Example: The reaction-diffusion equation (t = 1)

v, Lu =

• Ω

∇v · ∇u + κ2vu,

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.4/19

Singular integral operators

Boundary integral operators

[Lu](x) =

• Ω

K(x, y)u(y)dΩy

with the kernel K(x, y) singular at x = y Example: The single layer operator for the Laplace BVP in 3-d domain (t = −1

2)

K(x, y) = 1 4π|x − y|

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.5/19

Wavelet bases

Multiresolution: S0 ⊂ S1 ⊂ . . . ⊂ Ht

dim Sj = O(2jn)

(dyadic refinements)

Sj contains all piecewise pols of degree d − 1 Sj is globally Cr-smooth γ := r + 3

2

Ψ = {ψλ : λ ∈ Λ} is a Riesz basis for Ht span{ψλ : |λ| ≤ j} = Sj diam(supp ψλ) 2−|λ| Ψ has ˜ d vanishing moments

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.6/19

Nonlinear approximation

u = (uλ)λ ∈ ℓ2 s.t. u =

λ uλψλ

uN best approximation of u with #supp uN ≤ N

If u ∈ Bt+ns

τ,τ

with 1

τ = 1 2 + s for some s < d−t n

εN =

λ[uN]λψλ − uHt uN − u N−s

u is given as the solution to Lu = g.

[Dahlke, DeVore]: u ∈ Bt+ns

τ,τ

under mild requirements

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.7/19

Equivalent problem in ℓ2

Wavelet basis Ψ = {ψλ : λ ∈ Λ} Stiffness L = Lψλ, ψµλ,µ and load g = g, ψλλ Linear equation in ℓ2(Λ)

Lu = g L : ℓ2(Λ) → ℓ2(Λ) SPD and g ∈ ℓ2(Λ) u =

λ uλψλ is the solution of Lu = g

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.8/19

Richardson iterations in ℓ2

[Cohen, Dahmen, DeVore ’02]

u(0) = 0 u(i+1) = u(i) + α[g − Lu(i)] i = 0, 1, . . . g and Lu(i) are infinitely supported

Approximate them by finitely supported sequences Algorithm SOLVE[N, L, g] → u[N] (N operations)

#supp u[N] N and ε[N] = u[N] − u → 0

as N → ∞

ε[N] speed of convergence?

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.9/19

Complexity of SOLVE

Matrix L is called q∗-computable, when for each N one can construct an infinite matrix L∗

N s.t.

for any q < q∗, L∗

N − L N−q

having in each column O(N) non-zero entries whose computation takes O(N) operations [CDD’02]: Suppose that

uN − u N−s [s < d−t

n ]

L is q∗-computable with q∗ > s [q∗ ≥ d−t

n is suff. ]

Some condition on g then u[N] = SOLVE[N, L, g] satisfies

u[N] − u N−s

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.10/19

Compressibility of diff. ops.

[Stevenson ’04]: Suppose L is a diff. op.

L, L′ : Ht+σ → H−t+σ are bounded with σ ≥ d − t Ψ are piecewise polynomial wavelets that

are smooth: γ ≥ d − d−t

n

have vanishing moments: ˜

d ≥ d − 2t

then we can construct LN by dropping entries from L, s.t. with some q∗ ≥ d−t

n

(> s)

for any q < q∗, LN − L N−q

LN has N non-zeros in each column

Need to spend O(N) ops. for each column of LN

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.11/19

Quadrature for diff. ops.

Lλ,µ =

αβ

• Θ aαβ∂αψλ∂βψµ

Θ = supp ψλ ∩ supp ψµ aαβ are piecewise smooth Ψ are piecewise pols of order e (degree e − 1)

Internal, positive, interpolatory quadratures Composite quadrature of rank W: Subdivide Θ into W subdomains Apply quadrature of order p to each subdomain Fixed order p, variable rank W

❀

work = O(W)

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.12/19

Computability of diff. ops.

[T.G., Stevenson ’04] Fix quadrature order p > q∗n + e − 1 − t Fix θ and ̺ satisfying q∗n

p ≤ θ ≤ ̺ < 1 − e−1−t p

L∗

N — computed approximation of LN

Work WN,λ,µ max{1, Nθ2−||λ|−|µ||n̺} for [L∗

N]λ,µ

Then

LN − L∗

N N−q∗

(⇒ ∀q < q∗ : L − L∗

N N−q)

Work for each column of L∗

N is O(N)

Therefore L is q∗-computable with q∗ ≥ d−t

n

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.13/19

Compressibility of b.i.o.

[Stevenson ’04]: Suppose L is a b.i.o.

Ω is sufficiently smooth K(x, y) satisfies the Calderon-Zygmund estimate Ψ are sufficiently smooth and have sufficiently many

vanishing moments Then we can construct LN by dropping entries from L, s.t. with some q∗ ≥ d−t

n

(> s)

for any q < q∗, LN − L N−q

LN has N(log2 N)−2n−1 non-zeros in each column

Need to spend O(N) ops. for each column of LN

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.14/19

Quadrature for b.i.o.

Lλ,µ =

• Θ
• Θ′ K(x, y)ψλψµ

Θ = supp ψλ, Θ′ = supp ψµ

Far field: dist(Θ, Θ′) 2− min{|λ|,|µ|} Uniform mesh, variable order p

❀

work = O(p2n) Near field: dist(Θ, Θ′) 2− min{|λ|,|µ|} Adaptive (non-uniform) mesh, variable order p

❀

work = O(p2n(1 + ||λ| − |µ||))

[Schneider ’95], [von Petersdorff, Schwab ’97], [Lage, Schwab ’99], [Harbrecht ’01] Singular integrals: dist(Θ, Θ′) = 0 Duffy’s transformation [Duffy ’82], [Sauter ’96], [vPS’97]

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Computability of b.i.o.

[T.G. ’04]

L∗

N — computed approximation of LN

With sufficiently large, fixed θ and ̺ Choose the quadrature order

p = θ log2 N + ̺||λ| − |µ|| + const.

for [L∗

N]λ,µ

Then

LN − L∗

N N−q∗

(⇒ ∀q < q∗ : L − L∗

N N−q)

Work for each column of L∗

N is O(N)

Therefore L is q∗-computable with q∗ ≥ d−t

n

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.16/19

Conclusion

Given d and t, for wavelets that are sufficiently smooth have sufficiently many vanishing moments

• n sufficiently smooth manifolds

any internal, positive, interpolatory quadrature formula yields computa- tional schemes for the infinite stiffness matrix L, such that the CDD2 al- gorithm converges at the same rate as that of best N-term approximation.

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.17/19

References

• A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the

elliptic case. Found. Comput. Math., 2(3):203–245, 2002. R.P . Stevenson. On the compressibility of operators in wavelet coordinates. SIAM J.

• Math. Anal., 35(5):1110–1132, 2004.
• T. Gantumur and R.P

. Stevenson. Computation of differential operators in wavelet

• coordinates. Technical Report 1306, Utrecht University, August 2004. Submitted.
• T. Gantumur. Computation of singular integral operators in wavelet coordinates.

Technical Report, Utrecht University, 2004. To appear. This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”.

Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.18/19

(extra slide)

Suppose ∀q < q∗ : LN − L N −q and # of non-zeros in each column of LN is N. With αN = (log2 N)−c, define ˜ LN := L[NαN] Non-zeros per column for ˜ LN: NαN = N(log2 N)−c For arbitrary q < q∗, choose q′ ∈ (q, q∗) ˜ LN − L = L[NαN] − L (NαN)−q′ = N −q′(log2 N)cq′ N −q

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