Optimal adaptive wavelet methods for linear operator equations T. - PowerPoint PPT Presentation

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Optimal adaptive wavelet methods for linear operator equations T. Gantumur R. Stevenson Numerical Colloquium 17 June Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Overview Linear operator equation Au = g with A : H → H ′ Riesz basis Ψ = { ψ λ } of H , e.g. u = � λ u λ ψ λ Infinite dimensional matrix-vector system Au = g , with u = ( u λ ) λ and A : ℓ 2 → ℓ 2 Convergent iterations such as u ( i + 1 ) = u ( i ) + α [ g − Au ( i ) ] We can approximate Au ( i ) by a finitely supported vector How cheap can we compute this approximation? The answer will depend on A and Ψ Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Outline Continuous problem, discretization, and convergent iterations 1 Linear operator equations Discretization Convergent iterations in discrete space Complexity analysis 2 Uniform methods - convergence, complexity Nonlinear approximation Optimal complexity Computability An adaptive Galerkin method 3 Optimal complexity with coarsening Optimal complexity without coarsening Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Linear Operator Equations Let H be a separable Hilbert space, H ′ be its dual A : H → H ′ is boundedly invertible g ∈ H ′ is a linear functional Problem u ∈ H is such that Au = g For v ∈ H and h ∈ H ′ , � h , v � = h ( v ) the duality pairing Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Sobolev Spaces Let Ω be an n -dimensional domain or smooth manifold H = H t ⊂ H t (Ω) is a closed subspace H ′ = H − t the dual space Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Linear Differential Operators Partial differential operators of order 2 t � � a αβ ∂ β u , ∂ α v � , � Au , v � = | α | , | β |≤ t Example: The reaction-diffusion equation ( t = 1) � ∇ u · ∇ v + κ 2 uv , � Au , v � = Ω Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Boundary Integral Operators Boundary integral operators � � � Au , v � = v ( x ) K ( x , y ) u ( y ) d Ω y d Ω x Ω 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 | Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Convergent Iterations in Continuous Space Gradient Iterations u ( i + 1 ) = u ( i ) + B i ( g − Au ( i ) ) , B i : H ′ → H u − u ( i + 1 ) = u − u ( i ) − B i A ( u − u ( i ) ) = ( I − B i A )( u − u ( i ) ) � u − u ( i + 1 ) � H ≤ � I − B i A � H→H � u − u ( i ) � H Convergence ρ i := � I − B i A � H→H < 1 Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Normal Equations Observation Let R : H ′ → H be self-adjoint: � Rf , h � = � f , Rh � for f , h ∈ H ′ and H ′ -elliptic: with some α > 0 � Rf , f � ≥ α � f � 2 H for f ∈ H ′ . Then A ′ RA : H → H ′ is self-adjoint and H -elliptic. Normal Equation A ′ RAu = A ′ Rg Au = g = ⇒ Assumption A is self-adjoint and H -elliptic. Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Riesz bases Ψ = { ψ λ : λ ∈ ∇} is a Riesz basis for H – each v ∈ H has a unique expansion | d λ ( v ) | 2 ≤ C � v � 2 � � c � v � 2 v = d λ ( v ) ψ λ H ≤ s.t. H λ ∈∇ λ ∈∇ d λ ∈ H ′ and d λ ( ψ µ ) = δ λµ { d λ : λ ∈ ∇} is a Riesz basis for H ′ Ψ = { ˜ ˜ ψ λ } := { d λ } is the dual basis: � ˜ ψ λ , ψ µ � = δ λµ For v ∈ H , we have v = { v λ } := { d λ ( v ) } ∈ ℓ 2 ( ∇ ) Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Wavelet bases Ψ Riesz basis for H = H t Nested index sets ∇ 0 ⊂ ∇ 1 ⊂ . . . ⊂ ∇ j ⊂ . . . ⊂ ∇ , S j = span { ψ λ : λ ∈ ∇ j } ⊂ H and ˜ S j = span { ˜ ψ λ : λ ∈ ∇ j } ⊂ H ′ Locality, Polynomial exactness and Vanishing moments diam ( supp ψ λ ) = O ( 2 − j ) if λ ∈ ∇ j \ ∇ j − 1 All polynomials of degree d − 1, P d − 1 ⊂ S 0 d − 1 ⊂ ˜ d − 1 , ·� L 2 ⊂ ˜ S 0 more precisely, � P ˜ S 0 P ˜ {S j } has a good approximation property If λ ∈ ∇ \ ∇ 0 , we have � P ˜ d − 1 , ψ λ � L 2 = 0 � cancellation property Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Equivalent Discrete Problem [Cohen, Dahmen, DeVore ’02] Wavelet basis Ψ = { ψ λ : λ ∈ ∇} Stiffness A = � A ψ λ , ψ µ � λ,µ and load g = � g , ψ λ � λ Linear equation in ℓ 2 ( ∇ ) Au = g , A : ℓ 2 ( ∇ ) → ℓ 2 ( ∇ ) SPD and g ∈ ℓ 2 ( ∇ ) u = � λ u λ ψ λ is the solution of Au = g � u − v � ℓ 2 ( ∇ ) � � u − v � H with v = � λ v λ ψ λ A good approx. of u induces a good approx. of u Ψ defines a topological isomorphism between H and ℓ 2 ( ∇ ) Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Convergent Iterations in Discrete Space Richardson’s iterations u ( 0 ) = 0 u ( i + 1 ) = u ( i ) + α [ g − Au ( i ) ] i = 0 , 1 , . . . u − u ( i + 1 ) = u − u ( i ) − α A ( u − u ( i ) ) = ( I − α A )( u − u ( i ) ) � u − u ( i + 1 ) � ℓ 2 ≤ � I − α A � ℓ 2 → ℓ 2 � u − u ( i ) � ℓ 2 Convergence ρ := � I − α A � ℓ 2 → ℓ 2 < 1 g and Au ( i ) are infinitely supported Approximate them by finitely supported sequences Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Approximate Iterations Approximate right-hand side RHS [ g , ε ] → g ε satisfies � g − g ε � ℓ 2 ≤ ε Approximate application of the matrix APPLY [ A , v , ε ] → w ε satisfies � Av − w ε � ℓ 2 ≤ ε Approximate Richardson’s iterations u ( 0 ) = 0 ˜ u ( i + 1 ) = ˜ u ( i ) + α � u ( i ) , ε i ] � ˜ RHS [ g , ε i ] − APPLY [ A , ˜ i = 0 , 1 , . . . Choose ε i such that � u ( i ) − ˜ u ( i ) � � � u − u ( i ) � Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Convergence u ( 0 ) , ε fin ] → ˜ u ( i ) RICHARDSON [˜ for i = 0 , 1 , . . . r ( i ) := RHS [ g , ε i ] − APPLY [ A , ˜ u ( i ) , ε i ] ε i := C ρ i ; ˜ r ( i ) � ℓ 2 + 2 ε i ≤ ε fin then terminate; if � ˜ u ( i + 1 ) := ˜ u ( i ) + α r ( i ) ˜ endfor Lemma u ( 0 ) , ε ] → ˜ RICHARDSON [˜ u terminates with � g − A ˜ u � ℓ 2 ≤ ε u ( 0 ) , ε ] depending on ε ? Computational cost of RICHARDSON [˜ Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Uniform Refinement Galerkin Methods Wavelet basis Ψ j := { ψ λ : λ ∈ ∇ j } of S j Stiffness A j = � A ψ λ , ψ µ � λ,µ ∈∇ j Load g j = � g , ψ λ � λ ∈∇ j Linear equation in ℓ 2 ( ∇ j ) A j u j = g j , A j : ℓ 2 ( ∇ j ) → ℓ 2 ( ∇ j ) SPD and g j ∈ ℓ 2 ( ∇ j ) u j = � λ [ u j ] λ ψ λ ∈ S j approximates the solution of Au = g With the orthogonal projector P j : ℓ 2 ( ∇ ) → ℓ 2 ( ∇ j ) , the above equation is equivalent to P j Au j = P j g Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Convergence and Complexity If u ∈ H t + ns for some s ∈ ( 0 , d − t n ] v ∈S j � u − v � H t ≤ O ( 2 − jns ) ε j := � u − u j � H t ≤ C inf N j = dim S j = O ( 2 jn ) ε j ≤ O ( N − s j ) Solve A j u j = g j with Cascadic CG � complexity O ( N j ) Similar estimates for FEM Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Best N -term Approximation Given u = ( u λ ) λ ∈ ℓ 2 , approximate u using N nonzero coeffs � ℵ N := ℓ 2 (Λ) Λ ⊂∇ :#Λ= N ℵ N is a nonlinear manifold Let u N be such that � u − u N � ℓ 2 ≤ � u − v � ℓ 2 for v ∈ ℵ N u N is a best approximation of u with # supp u N ≤ N u N can be constructed by picking N largest in modulus coeffs from u Gantumur, Stevenson

Convergent iterations Complexity analysis An adaptive Galerkin method Summary Nonlinear vs. linear approximation Nonlinear approximation If u ∈ B t + ns ( L τ ) with 1 τ = 1 2 + s for some s ∈ ( 0 , d − t n ) τ ε N = � u N − u � ≤ O ( N − s ) Linear approximation If u ∈ H t + ns for some s ∈ ( 0 , d − t n ] , uniform refinement ε j = � u j − u � ≤ O ( N − s j ) H t + ns is a proper subset of B t + ns ( L τ ) τ [Dahlke, DeVore]: u ∈ B t + ns ( L τ ) much milder than u ∈ H t + ns τ Gantumur, Stevenson