Computation of operators in wavelet coordinates Tsogtgerel Gantumur and Rob Stevenson Department of Mathematics Utrecht University Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.1/22
Overview Linear operator equation Lu = g with L : H → H ′ Riesz basis Ψ = { ψ λ } of H , e.g. u = � λ u λ ψ λ Infinite dimensional matrix-vector system Lu = g , with u = ( u λ ) λ and L : ℓ 2 → ℓ 2 Convergent iterations such as u ( i +1) = u ( i ) + α [ g − Lu ( i ) ] We can approximate Lu ( i ) by a finitely supported vector How cheap can we compute this approximation? The answer will depend on L and Ψ Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.2/22
Linear operator equations Let Ω be an n -dimensional domain or smooth manifold H t ⊂ H t (Ω) be a subspace, and H − t be its dual space Consider the problem of finding u from Lu = g where L : H t → H − t is a self-adjoint elliptic operator of order 2 t and g ∈ H − t is a linear functional Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.3/22
Differential operators Partial differential operators of order 2 t � � ∂ α v, a αβ ∂ β u � , � v, Lu � = | α | , | β |≤ t Example: The reaction-diffusion equation ( t = 1 ) � ∇ v · ∇ u + κ 2 vu, � v, Lu � = Ω Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.4/22
Singular integral operators Boundary integral operators � [ Lu ]( x ) = K ( x, y ) u ( y ) d Ω y Ω 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 ) 1 K ( x, y ) = 4 π | x − y | Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.5/22
Multiresolution analysis S 0 ⊂ ˜ ˜ S 1 ⊂ . . . ⊂ H − t S 0 ⊂ S 1 ⊂ . . . ⊂ H t and dim S j , dim ˜ S j = O (2 jn ) (dyadic refinements) S j contains all piecewise pols of degree d − 1 S j contains all piecewise pols of degree ˜ ˜ d − 1 S j is globally C r -smooth Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.6/22
Wavelet bases Ψ = { ψ λ : λ ∈ Λ } is a Riesz basis for H t – each v ∈ H t has a unique expansion � c � v � ≤ � v � H t ≤ C � v � v = v λ ψ λ s.t. λ ∈ Λ For every index λ ∈ Λ , there is a number | λ | ∈ I N 0 called the level of ψ λ span { ψ λ : | λ | ≤ j } = S j � ψ λ , v � = 0 for any v ∈ ˜ S | λ |− 1 diam(supp ψ λ ) = O (2 −| λ | ) Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.7/22
Typical wavelets ψ λ ψ µ x ψ λ is a piecewise polynomial of degree d − 1 � x k ψ λ ( x ) dx = 0 for k < ˜ ( ˜ d vanishing moments) d Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.8/22
Galerkin methods Wavelet basis Ψ j := { ψ λ : | λ | ≤ j } of S j Stiffness L ( j ) = � Lψ λ , ψ µ � | λ | , | µ |≤ j load g ( j ) = � g, ψ λ � | λ |≤ j R N j Linear equation in I ( N j := dim S j ) L ( j ) u ( j ) = g ( j ) R N j SPD and g ( j ) ∈ I R N j → I R N j L ( j ) : I u ( j ) = � λ [ u ( j ) ] λ ψ λ approximates the solution of Lu = g Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.9/22
Galerkin approximation If u ∈ H s for some s ∈ [ t, d ] ε ( j ) := � u ( j ) − u � H t ≤ O (2 − j ( s − t ) ) N j = dim S j = O (2 jn ) ε ( j ) ≤ O ( N − s − t ) n j Solve L ( j ) u ( j ) = g ( j ) with CG ❀ complexity O ( N j ) Similar estimates for FEM Better convergence? Adaptive methods? Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.10/22
Nonlinear approximation Given u = ( u λ ) λ ∈ ℓ 2 Approximate u using N coeffs | u λ | λ Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.11/22
Nonlinear approximation Given u = ( u λ ) λ ∈ ℓ 2 Approximate u using N coeffs | u λ | λ (arranged) Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.11/22
Nonlinear approximation Given u = ( u λ ) λ ∈ ℓ 2 Approximate u using N coeffs | u λ | � �� � N biggest u N best approximation of u with #supp u N ≤ N Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.11/22
Nonlinear approximation Given u = ( u λ ) λ ∈ ℓ 2 Approximate u using N coeffs | [ u N − u ] λ | N zeroes u N best approximation of u with #supp u N ≤ N Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.11/22
Nonlinear vs. linear approximation τ,τ with 1 τ = 1 2 + s − t If u ∈ B s n for some s < d ε N = � u N − u � ≤ O ( N − s − t n ) If u ∈ H s for some s ≤ d , uniform refinement ε ( j ) = � u ( j ) − u � ≤ O ( N − s − t ) n j B s τ,τ is bigger than H s Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.12/22
Besov vs. Sobolev regularity s d τ = 1 1 2 + s − t n B s τ,τ t 0 1 1 2 τ τ,τ with 1 τ = 1 2 + d − t [Dahlke, DeVore]: u ∈ B d "often" n Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.13/22
Equivalent problem in ℓ 2 [Cohen, Dahmen, DeVore ’02] Wavelet basis Ψ = { ψ λ : λ ∈ Λ } Stiffness L = � Lψ λ , ψ µ � λ,µ and load g = � g, ψ λ � λ Linear equation in ℓ 2 (Λ) Lu = g L : ℓ 2 (Λ) → ℓ 2 (Λ) SPD and g ∈ ℓ 2 (Λ) u = � λ u λ ψ λ is the solution of Lu = g Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.14/22
Richardson iterations in ℓ 2 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 ] ≤ O ( N ) and ε [ N ] = � u [ N ] − u � → 0 as N → ∞ ε [ N ] speed of convergence? Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.15/22
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 � ≤ O ( N − q ) having in each column O ( N ) non-zero entries whose computation takes O ( N ) operations [CDD’02]: Suppose that [ s < d − t � u N − u � ≤ O ( N − s ) n ] L is q ∗ -computable with q ∗ > s then for suitable g , u [ N ] = SOLVE [ N, L , g ] satisfies � u [ N ] − u � ≤ O ( N − s ) Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.16/22
Computability [ L λ,µ ] λ ∈ Λ – µ -th column | L λ,µ | λ Approximate by N entries? Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.17/22
Computability [ L λ,µ ] λ ∈ Λ – µ -th column arranged by modulus | L λ,µ | λ (arranged) N biggest entries? Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.17/22
Computability [ L λ,µ ] λ ∈ Λ – µ -th column | L λ,µ | � �� � N Compute the N biggest entries Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.17/22
Computability The µ -th column of the difference | [ L N − L ] λ,µ | � �� � N Need to locate the biggest entries a priori Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.17/22
Compressibility L is called q ∗ -compressible, when L is q ∗ -computable assuming each entry of L is available at unit cost [CDD’01], [Stevenson ’04]: Suppose { ψ λ } are piecewise polynomial wavelets that are sufficiently smooth and have sufficiently many vanishing moments L is either differential or singular integral operator then L is q ∗ -compressible for some q ∗ ≥ d − t ( > s ) n Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.18/22
Computability Distribute computational works over the entries W N Require: shaded area = O ( N ) Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - Sixth Minisimposium TIANA. Amsterdam. Oct 2004. – p.19/22
Recommend
More recommend
