for strongly elliptic operator equations. Tsogtgerel Gantumur. (Utrecht University). Workshop on Fast Numerical Solution of PDE's. 20-22 December 2005 Utrecht ...
An optimal adaptive wavelet method
Continuous and discrete problems Linear operator equations
for strongly elliptic operator equations
Discretization
A convergent adaptive Galerkin method
Tsogtgerel Gantumur (Utrecht University)
Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
Workshop on Fast Numerical Solution of PDE’s 20-22 December 2005 Utrecht
1
Motivation and overview
[Cohen, Dahmen, DeVore ’01, ’02] I
Using a wavalet basis Ψ, transform Au = g to a matrix-vector system Au = g
I
Solve it by an iterative method
I
They apply to A symmetric positive definite
Continuous and discrete problems Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation
I
If A is perturbed by a compact operator?
I
Normal equation: AT Au = AT g
I
I
Convergence
Complexity analysis Nonlinear approximation Optimal complexity
Condition number squared, application of AT A expensive We modified [Gantumur, Harbrecht, Stevenson ’05]
2
Sobolev spaces
Continuous and discrete problems Linear operator equations Discretization
I
Let Ω be an n-dimensional domain
I
H s := (L2 (Ω), H01 (Ω))s,2
I
Hs
:=
I
Hs
:=
H s (Ω)∩H01 (Ω) (H −s )0 for
for 0 ≤ s ≤ 1 for s > 1
s 0)
H −1
Find u ∈ H 1 s.t. I
u ∈ H1
Regularity: g ∈ H −1+σ
Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Lu = g ⇒
(g ∈ H −1 )
kuk1+σ . kgk−1+σ
Complexity analysis Nonlinear approximation Optimal complexity
Example: Helmholtz equation Z hLu, vi = ∇u · ∇v − κ2 uv Ω
4
Riesz bases
Continuous and discrete problems Linear operator equations
Ψ = {ψλ : λ ∈ ∇} is a Riesz basis for H 1 – each v ∈ H 1 has a unique expansion X X |hψ˜λ , vi|2 ≤ Ckvk21 v= hψ˜λ , viψλ s.t. ckvk21 ≤ λ∈∇
λ∈∇
Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
I I
˜ = {ψ˜λ } ⊂ H −1 is the dual basis: hψ˜λ , ψµ i = δλµ Ψ For v ∈ H 1 , we have v = {vλ } := {hψ˜λ , vi} ∈ `2 (∇)
5
Wavelet basis I
Ψ = {ψλ } Riesz basis for H 1
I
Nested index sets ∇0 ⊂ ∇1 ⊂ . . . ⊂ ∇j ⊂ . . . ⊂ ∇,
Continuous and discrete problems Linear operator equations Discretization
H1
I
Sj = span{ψλ : λ ∈ ∇j } ⊂
I
diam(supp ψλ ) = O(2−j ) if λ ∈ ∇j \ ∇j−1
I
All polynomials of degree d − 1, Pd−1 ⊂ S0
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
inf kv − vj k1 ≤ C · 2−j(s−1)/n kvks
vj ∈Sj
I
v ∈ H s (1 ≤ s ≤ d)
If λ ∈ ∇ \ ∇0 , we have hP˜d−1 , ψλ iL2 = 0
6
Equivalent discrete problem
[CDD01, CDD02] I I
Wavelet basis Ψ = {ψλ : λ ∈ ∇} Stiffness L = hLψλ , ψµ iλ,µ and load g = hg, ψλ iλ
Linear equation in `2 (∇) Lu = g, L : `2 (∇) → `2 (∇) invertible and g ∈ `2 (∇) P I u = λ uλ ψλ is the solution of Lu = g P I ku − vk `2 h ku − vk1 with v = λ vλ ψλ I
Continuous and discrete problems Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
A good approx. of u induces a good approx. of u
7
Galerkin solutions I
A := hAψλ , ψµ iλ,µ SPD,
recall L = A + B
1 2
I
||| · ||| := hA·, ·i is a norm on `2
I
Λ⊂∇
I
Continuous and discrete problems Linear operator equations Discretization
LΛ := PΛ L|`2 (Λ) : `2 (Λ) → `2 (Λ), and gΛ := PΛ g ∈ `2 (Λ)
A convergent adaptive Galerkin method Galerkin approximation Convergence
Lemma
Complexity analysis
∃j0 : If Λ ⊃ ∇j with j ≥ j0 , a unique solution uΛ ∈ `2 (Λ) to LΛ uΛ = gΛ exists, and
Nonlinear approximation Optimal complexity
|||u − uΛ ||| ≤ [1 + O(2−jσ/n )] inf |||u − v||| v∈`2 (Λ)
Ref: [Schatz ’74] 8
Quasi-orthogonality
Continuous and discrete problems I
Linear operator equations
j ≥ j0
Discretization
I
∇ j ⊂ Λ 0 ⊂ Λ1
I
LΛi ui = gΛi , i = 0, 1
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis
|||u − u0 |||2 − |||u − u1 |||2 − |||u1 − u0 |||2 −jσ/n
≤ O(2
Nonlinear approximation Optimal complexity
2
) |||u − u0 ||| + |||u − u1 |||
2
Ref: [Mekchay, Nochetto ’04]
9
A sketch of a proof
Continuous and discrete problems
|||u − u0 |||2 = |||u − u1 |||2 + |||u1 − u0 |||2 + 2hA(u − u1 ), u1 − u0 i
Linear operator equations Discretization
A convergent adaptive Galerkin method
hL(u − u1 ), u1 − u0 i = 0
Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
hA(u − u1 ), u1 − u0 i = −hB(u − u1 ), u1 − u0 i = −hB(u − u1 ), u1 − u0 i ≤ kBk1−σ→−1 ku − u1 k1−σ ku1 − u0 k1 ku − u1 k1−σ ≤ O(2−jσ/n )ku − u1 k1
(Aubin-Nitsche)
10
Error reduction
|||u − u1 |||2 ≤ [1 + O(2−jσ/n )] |||u − u0 |||2 − |||u1 − u0 |||2
Continuous and discrete problems Linear operator equations Discretization
A convergent adaptive Galerkin method
Lemma
Galerkin approximation Convergence
Let µ ∈ (0, 1), and Λ1 be s.t.
Complexity analysis
kPΛ1 (g − Lu0 )k ≥ µkg − Lu0 k
Nonlinear approximation Optimal complexity
Then we have 1
|||u − u1 ||| ≤ [1 − κ(A)−1 µ2 + O(2−jσ/n )] 2 |||u − u0 ||| Ref: [CDD01] 11
Exact algorithm
Continuous and discrete problems Linear operator equations
SOLVE[ε] → uk k := 0; Λ0 := ∇j do Solve LΛk uk = gΛk rk := g − Luk determine a set Λk+1 ⊃ Λk , with minimal cardinality, such that kPΛk+1 rk k ≥ µkrk k k := k + 1 while krk k > ε
Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
12
Approximate Iterations
Approximate right-hand side
Continuous and discrete problems Linear operator equations Discretization
RHS[ε] → gε with kg − gε k`2 ≤ ε
Approximate application of the matrix
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis
APPLY[v, ε] → wε with kLv − wε k`2 ≤ ε
Nonlinear approximation Optimal complexity
Approximate residual RES[v, ε] := RHS[ε/2] − APPLY[v, ε/2]
13
Best N-term approximation
Continuous and discrete problems
Given u = (uλ )λ ∈ `2 , coeffs
approximate u using N nonzero
ℵN :=
Linear operator equations Discretization
A convergent adaptive Galerkin method
[
Galerkin approximation
`2 (Λ)
Λ⊂∇:#Λ=N
Convergence
Complexity analysis Nonlinear approximation
I
ℵN is a nonlinear manifold
I
Let uN be a best approximation of u with #supp uN ≤ N
I
uN can be constructed by picking N largest in modulus coeffs from u
Optimal complexity
14
Nonlinear vs. linear approximation Nonlinear approximation If u ∈ B1+ns (Lτ ) with τ
1 τ
=
1 2
+ s for some s ∈ (0, d−1 n )
Continuous and discrete problems Linear operator equations Discretization
εN = kuN − uk ≤ O(N −s )
A convergent adaptive Galerkin method Galerkin approximation Convergence
Linear approximation If u ∈ H 1+ns for some s ∈ (0, d−1 n ], uniform refinement
Complexity analysis Nonlinear approximation Optimal complexity
εj = kuj − uk ≤
O(Nj−s )
I
H 1+ns is a proper subset of B1+ns (Lτ ) τ
I
[Dahlke, DeVore]: u ∈ B1+ns (Lτ )\H 1+ns τ
"often"
15
Approximation spaces
I
Approximation space As := {v ∈ `2 : kv − vN k`2 ≤ O(N −s )}
I
Quasi-norm |v|
I
As
:= supN∈N
u ∈ B1+ns (Lτ ) with τ ⇒ u ∈ As
1 τ
=
1 2
N s kv
Continuous and discrete problems Linear operator equations Discretization
− v N k `2
+ s for some s ∈ (0, d−1 n )
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis
Assumption
Nonlinear approximation Optimal complexity
u∈
As
for some s ∈ (0,
d−1 n )
Best approximation 1/s
ku − vk ≤ ε satisfies #supp v ≤ ε−1/s |u|As
16
Requirements on the subroutines
Complexity of RHS
Continuous and discrete problems
RHS[ε] → gε terminates with kg − gε k`2 ≤ ε I I
1/s #supp gε . ε−1/s |u|As 1/s flops, memory . ε−1/s |u|As
Linear operator equations Discretization
A convergent adaptive Galerkin method
+1
Galerkin approximation Convergence
Complexity analysis
Complexity of APPLY
Nonlinear approximation
For #supp v < ∞ APPLY[v, ε] → wε terminates with kLv − wε k`2 ≤ ε
Optimal complexity
1/s
I
#supp wε . ε−1/s |v|As
I
flops, memory . ε−1/s |v|As + #supp v + 1
1/s
17
The subroutine APPLY
Continuous and discrete problems Linear operator equations
I
{ψλ } are piecewise polynomial wavelets that are sufficiently smooth and have sufficiently many vanishing moments
Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
I
L is either differential or singular integral operator
Then we can construct APPLY satisfying the requirements. Ref: [CDD01], [Stevenson ’04], [Gantumur, Stevenson ’05,’06], [Dahmen, Harbrecht, Schneider ’05]
Complexity analysis Nonlinear approximation Optimal complexity
18
Optimal expansion
Continuous and discrete problems
1
I
µ ∈ (0, κ(A)− 2 ).
I
∇j ⊂ Λ0 with a sufficiently large j
I
LΛ0 u0 = gΛ0
Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation
Then the smallest set Λ1 ⊃ Λ0 with
Convergence
kPΛ1 (g − Lu0 )k ≥ µkg − Lu0 k
Complexity analysis Nonlinear approximation Optimal complexity
satisfies 1/s
#(Λ1 \ Λ0 ) . ku − u0 k−1/s |u|As Ref: [GHS05]
19
Adaptive Galerkin method
Continuous and discrete problems
SOLVE[ε] → wk k := 0; Λ0 := ∇j do Compute an appr.solution wk of LΛk uk = gΛk Compute an appr.residual rk for wk Determine a set Λk+1 ⊃ Λk , with modulo constant factor minimal cardinality, such that kPΛk+1 rk k ≥ µkrk k k := k + 1 while krk k > ε
Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
20
Conclusion
Continuous and discrete problems
SOLVE[ε] → w terminates with kg − Lwk`2 ≤ ε. 1/s ε−1/s |u|As
I
#supp w .
I
flops, memory . the same expression
Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Ref: [CDD01, GHS05] Open: I
Singularly perturbed problems
I
Adaptive initial index set
Complexity analysis Nonlinear approximation Optimal complexity
21
References [CDD01] A. Cohen, W. Dahmen, R. DeVore. Adaptive wavelet methods for elliptic operator equations — Convergence rates. Math. Comp., 70:27–75, 2001. [GHS05] Ts. Gantumur, H. Harbrecht, R.P. Stevenson. An optimal adaptive wavelet method without coarsening of the iterands. Technical Report 1325, Utrecht University, March 2005. To appear in Math. Comp.. [Gan05] Ts. Gantumur. An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems. To appear.
Continuous and discrete problems Linear operator equations Discretization
A convergent adaptive Galerkin method Galerkin approximation Convergence
Complexity analysis Nonlinear approximation Optimal complexity
This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”.
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