adaptive eigenvalue computation for elliptic operators
TRANSCRIPT
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Adaptive Eigenvalue Computation forElliptic Operators
Joint work with Wolfgang Dahmen (RWTH Aachen),Thorsten Rohwedder (Uni Kiel), Reinhold Schneider (TU Berlin)
Andreas ZeiserInstitut fur Mathematik
Technische Universitat Berlin
Workshop ”Adaptive numerical methods for PDEs” Wien, January 23, 2008
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A Model Eigenvalue Problem
Poisson eigenvalue problem on Ω ⊂ Rd with Dirichlet bc.:
−∆u = λu, u|∂Ω = 0.
Weak formulation on H10(Ω):
∫
Ω
∇u(x) · ∇v(x) dx︸ ︷︷ ︸
:=a(u,v)
= λ
∫
Ω
u(x)v(x) dx︸ ︷︷ ︸L2(Ω) inner product
, for all v ∈ H10 .
Properties of bilinear form a : H10(Ω)×H1
0(Ω)→ R:
• a(u, v) = a(v, u) symmetric
• a(u, v) . ‖u‖H10(Ω)‖v‖H1
0(Ω) bounded
• a(u, u) & ‖u‖2H1
0(Ω)
strongly positive
for all u, v ∈ H10(Ω).
a(·, ·) induces energy norm on H10(Ω) equivalent to ‖ · ‖H1
0(Ω).
Andreas Zeiser 2 Wien, January 23, 2008
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Abstract Eigenvalue Problems
Gelfand-triple of Hilbert- (H, (·, ·), | · |) and Banach-space (V, ‖ · ‖)
Vd→ H ∼= H∗
d→ V ∗,
with dual pairing 〈·, ·〉 on V ∗ × V .
In model problem: V = H10(Ω), H = L2(Ω)
Operator formulation:
Au = λEu in V ∗
with operators A : V → V ∗ and E : H → H∗ such that
〈Au, v〉 = a(u, v), 〈Eu, v〉 = (u, v),
for given bounded, symmetric, strongly positive bilinear forma : V × V → R.
In model problem: A : H10(Ω)→ H−1(Ω).
Andreas Zeiser 3 Wien, January 23, 2008
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Goal
Find lowest eigenpair (λ1, u1) of abstract eigenvalue problem
Au = λEu in V ∗.
λ2λ10 other spectrumsimple
Desired properties of algorithm:
• convergent with desired accuracy in finite number of steps,
• adaptive with quasi-optimal number of degrees of freedom,
• computational complexity linear in number of degrees of freedom.
Andreas Zeiser 4 Wien, January 23, 2008
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Boundary value problems
Solution of boundary value problem
Au = f in V ∗
Optimal adaptive algorithm (Cohen/Dahmen/deVore 2001):
• Richardson-iteration with damping parameter γn:
un+1 = un − γnB−1(Aun − f ),
• stable wavelet-discretization for equivalent formulation in ℓ2,
• approximation of operators (compressibility),
• nonlinear best N -term approximation.
Andreas Zeiser 5 Wien, January 23, 2008
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Outline of the talk
(I) Iteration scheme and perturbations
(II) Numerical realization
(III) Optimality
(IV) (Preliminary) numerical results
Andreas Zeiser 6 Wien, January 23, 2008
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Part I: Iteration scheme
Andreas Zeiser 7 Wien, January 23, 2008
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Preconditioned inverse iteration
Steepest descent of Rayleigh-quotient
µ(v) =〈Av, v〉
〈Ev, v〉, ∇µ(v) ∼ Av − µ(v)Ev.
Recursion:
v′ = v − αB−1(Av − µ(v)Ev), (PINVIT)
Preconditioner B : V → V ∗ with
δ0〈Bv, v〉 ≤ 〈Av, v〉 ≤ δ1〈Bv, v〉
Perfect preconditioner (B = A):
v′ = αµ(v)A−1Ev.
Andreas Zeiser 8 Wien, January 23, 2008
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Convergence of PINVIT (I)
Thm 1 (D’yakonov/Orekhov, 1980). (matrix case)If µ(v) < λ2 then the error in the Rayleigh-quotient decreases like
µ(v′)− λ1 ≤ q · (µ(v)− λ1), q = q(µ(v)) < 1,
from step to step, where the monotonically decreasing function q(µ)depends asymptotically only on δ1/δ0 and λ2/λ1.
Improvements: Samokish (1958), Bramble/Knyacev/Pasciak (1996),Neymeyr (2001), Knyacev/Neymeyr (200x)
Andreas Zeiser 9 Wien, January 23, 2008
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Convergence of PINVIT (II)
Thm 2 (Rohwedder/Schneider/Z., 2007). (operator case)If µ(v) < λ2 then the error in the Rayleigh-quotient decreases like
µ(v′)− λ1 ≤ q · (µ(v)− λ1), q = q(µ(v)) < 1,
from step to step, where the monotonically decreasing function q(µ)depends asymptotically only on δ1/δ0 and λ2/λ1.
Sketch of proof.Main estimate. Temple-Kato like inequality:
‖Av − µ(v)Ev‖2A−1 ≥ µ(v)(µ(v)− λ1)(λ2 − µ(v))
λ1λ2|v|2,
through spectral resolution. Rest of the proof: algebraic reasoning,analogously to D’yakonov/Orekhov (1980).
Andreas Zeiser 10 Wien, January 23, 2008
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Perturbed PINVIT
For every ǫ > 0 perturbed preconditioned inverse iteration
v′ǫ = v − αB−1(Aǫ(v)− µǫ(v)Eǫ(v)), (PPINVIT).
with nonlinear approximations
‖Aǫ(v)− Av‖∗ ≤ ǫ‖v‖, |Eǫ(v)− Ev|∗ ≤ ǫ|v|
and perturbed Rayleigh quotient
µǫ(v) =〈Aǫ(v), v〉
〈Eǫ(v), v〉.
Andreas Zeiser 11 Wien, January 23, 2008
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Convergence of PPINVIT (I)Thm 3 (Rohwedder/Schneider/Z., 2007). (perturbed scheme)If µ(v) < λ2 then the error in the Rayleigh-quotient decreases like
µ(v′ǫ)− λ1 ≤ q · (µ(v)− λ1)+Cǫ, q = q(µ(v)) < 1,
from step to step, where the monotonically decreasing function q(µ)depends asymptotically only on δ1/δ0 and λ2/λ1. C can be boundedindependently of v and ǫ.
Sketch of proof.
Perturbation argument, i.e. bound perturbations by multiple of ǫ, e.g.
|µǫ(v)− µ(v)| . ǫ, ‖v′ − v′ǫ‖ . ǫ.
Resulting Rayleigh-quotient satisfies
|µ(v′)− µ(v′ǫ)| . ǫ,
proof by similar arguments as for matrices, see D’yakonov (1996).
Andreas Zeiser 12 Wien, January 23, 2008
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Subspace convergence
v
v
1
φspanu
Estimate for angle sinφ . (µ(v)− λ1)1/2 (with respect to 〈A·, ·〉).
For appropriately scaled vectors:
accuracy ǫ in (V, ‖ · ‖) ∼ application of A,E with tolerance ǫ2
Andreas Zeiser 13 Wien, January 23, 2008
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Convergence of PPINVIT (II)
Thm 4 (Dahmen/Rohwedder/Schneider/Z., 2007). (improvement)If ∠(v, u1) is sufficiently small then the angle decreases like
∠(v′ǫ, u1) ≤ q · ∠(v, u1) + Cǫ, q < 1
from step to step, and C can be bounded independently of v and ǫ.
Sketch of proof.
Concentration on vector and its projection onto eigenspace.Estimation of reduction perpendicular to eigenspace throughellipticity.
For appropriately scaled vectors:
accuracy ǫ in (V, ‖ · ‖) ∼ application of A,E with tolerance ǫ
Andreas Zeiser 14 Wien, January 23, 2008
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A convergent algorithm
Starting vector v0 and target accuracy τ
ADAPTIVE(v0, τ )
Require: ∠(v0, u1) sufficiently smallτ 0 ← ∠(v0, u1)i← 0while τ i > τ dovi+1 ← PPINVIT(vi, ατ i, N) N steps with tolerance ατ iτ i+1 ← 1
2τi
i← i + 1end whilereturn vi .
N steps of PPINVIT to halve the error, independent of vi and τ i.
Andreas Zeiser 15 Wien, January 23, 2008
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Part II: Numerical realization
Andreas Zeiser 16 Wien, January 23, 2008
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Stable wavelet bases
For basis (ψi)i∈I with index set I
u =∑
i∈I
uiψi ∈ V ⇔ u =
...ui...
∈ ℓ2(I).
Stable bases (Riesz bases) for H and V :
‖u‖ℓ2(I) ∼ ‖u‖, ‖D−1
u‖ℓ2(I) ∼ |u|, D diagonal.
Standard example:
V = H t(Ω), H = L2(Ω), D = (δij2t|i|)i,j∈I,
where |i| level of ψi.
Andreas Zeiser 17 Wien, January 23, 2008
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Equivalent ℓ2 problem
With stable basis (ψi)i∈I
Au = λEu⇔ Au = λEu,
where
A = (〈Aψi, ψj〉)i,j∈I, E = (〈Eψi, ψj〉)i,j∈I.
Built-in preconditioning:
〈Au,u〉ℓ2 ∼ ‖u‖2ℓ2
equivalent problem to Au = λEu through norm-equivalence
Andreas Zeiser 18 Wien, January 23, 2008
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Quasi-sparsityLevel 0
Level 3
Level 2
Level 1
Original matrix B.
Level 1
Level 2
Level 3
Level 0
Compressed matrix Bj.
Controlled approximation: ‖B−Bj‖ℓ2 ≈ 2−js, with ≈ 2j entries perrow/column (s∗-computable).
• wide class of discretized operators quasi-sparse
• simple approximation of Au through
Bǫ(u) = Bju, j appropriate.
• more advanced nonlinear scheme in Cohen/Dahmen/deVore(2001).
Andreas Zeiser 19 Wien, January 23, 2008
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Exploding support
Approximate operator application:
#v <∞⇒ #Aǫ(v),#Eǫ(v) <∞
with #v number of non-zero entries.
A computable algorithm:
• Initial vector v0 with #v0 <∞
• Iterate
vn+1 = v
n − α (Aǫ(vn)− µǫ(v
n)Eǫ(vn))
Support of vn stays finite but explodes:
#vn ∼ Cn, C > 1.
Non-optimal behavior in number of degrees of freedom.
Andreas Zeiser 20 Wien, January 23, 2008
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Part III: Optimality
Andreas Zeiser 21 Wien, January 23, 2008
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Adaptivity
Let u ∈ V
u =∑
i∈I
uiψi ⇒ u =
...ui...
.
For finite index set I0 with ui = 0, i 6∈ I0:
u =∑
i∈I0
uiψi finite sum⇔ u =
...0∗...
finite number of entries.
Non-zero entry ui↔ basis function ψi needed in expansion.
Andreas Zeiser 22 Wien, January 23, 2008
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Best-N-term approximation
Best-approximation in ℓ2(I) with N non-zero elements:
σN(u) = inf‖u− v‖ℓ2,#v ≤ N.
Approximation space
As = u ∈ ℓ2(I) : ‖u‖ℓ2 + |u|As <∞, |u|As = supN≥1
N sσN(u).
Approximation of u ∈ As
accuracy ǫ↔ number of degrees of freedom ǫ−1/s|u|As
Andreas Zeiser 23 Wien, January 23, 2008
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Coarsening and approximation
Thm 5 (Cohen/Dahmen/deVore, 2001).Coarsening with appropriate tolerance of approximation v with
‖v − u‖ ≤ ǫ, u ∈ As
leads to quasi-optimal approximation w:
‖w‖As . ‖u‖As, #w . ‖u‖1/sAs ǫ
−1/s, ‖u−w‖ . ǫ.
Time to time coarsening prevents exponential growth of theiterates.
Andreas Zeiser 24 Wien, January 23, 2008
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Optimal adaptive algorithm
Starting vector v0, target accuracy τ .
MINIEIG(v0, τ )
Require: ∠(v0, u1) sufficiently smallτ 0 ← ∠(v0, u1)i← 0while τ i > τ dovi+1 ← PPINVIT(vi, α1τ
i, N) α1 fixedvi+1 ← APPROX(vi+1, α2τ
i) α2 fixed, Coarsening stepτ i+1 ← 1
2τi
i← i + 1end whilereturn vi
Simplified version from Dahmen/Rohwedder/Schneider/Z. (2007).Includes efficient and reliable error estimator.
Andreas Zeiser 25 Wien, January 23, 2008
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Optimal convergence
Thm 6 (Dahmen/Rohwedder/Schneider/Z., 2007).For compressible operators A and E, the algorithm gives for eachtarget accuracy τ a vector uτ :
∠(uτ ,u1) . τ, |µτ(uτ)− λ1| . τ.
If the eigenvector u1 ∈ As then
#uτ . τ−1/s
given sufficiently compressible operators. The floating pointoperations remain bounded linearly.
Sketch of proof.Convergence of PPINVIT, estimate perturbation due to coarsening.Bookkeeping of ‖ · ‖As, support sizes and flops.
Andreas Zeiser 26 Wien, January 23, 2008
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Part IV: (Preliminary) numerical results
Andreas Zeiser 27 Wien, January 23, 2008
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A model problem
Poisson eigenvalue problem in R2:
−∆u = λu on Ω,
u|∂Ω = 0.Ω
−1
0
1−1
π3/2
• Reduced Sobolev regularity due to corner. Classical theory:
u1 ∈ Hs, s < 1 +
π
α=
5
3rates: ‖u1 − uh‖ . N−1/3, |λ1 − µ(uh)| . N−2/3.
• Higher Besov regularity (Dahlke 1999)
u1 ∈ B2s+1τ,τ ,
1
τ= s +
1
2, 0 < s <
13
12.
rates: ‖u1 − uh‖ . N−1/2, |λ1 − µ(uh)| . N−1
due to linear wavelets. Likely not optimal (bootstrapping).
Andreas Zeiser 28 Wien, January 23, 2008
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Numerical result
x
y
z
Joint work with Jurgen Vorloepper (RWTH Aachen).• Linear wavelets with tree-structured index sets• Exact application of operators for given index set• Galerkin-solution with frozen index sets
Andreas Zeiser 29 Wien, January 23, 2008
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Convergence rates
10−5
10−4
10−3
10−2
10−1
100
102 103 104 105 106 107
Err
or R
ayle
igh−
quot
ient
degrees of freedom
Adaptive wavelet
Uniform refined FEM
Andreas Zeiser 30 Wien, January 23, 2008
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Index sets - Initial iterate
0.05
0
−0.05 0.05 0−0.05
161 degrees of freedom
Andreas Zeiser 31 Wien, January 23, 2008
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Index sets - 1st iterate
0.05
0
−0.05 0.05 0−0.05
1859 degrees of freedom
Andreas Zeiser 32 Wien, January 23, 2008
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Index sets - 2nd iterate
0.05
0
−0.05 0.05 0−0.05
11393 degrees of freedom
Andreas Zeiser 33 Wien, January 23, 2008
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Index sets - 3rd iterate
0.05
0
−0.05 0.05 0−0.05
34124 degrees of freedom
Andreas Zeiser 34 Wien, January 23, 2008
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Index sets - 4th iterate
0.05
0
−0.05 0.05 0−0.05
127355 degrees of freedom
Andreas Zeiser 35 Wien, January 23, 2008
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Summary
• Preconditioned inverse iteration as abstract iteration scheme
• Robustness of PINVIT against perturbations
• Numerical realization through stable wavelet bases
• Optimality through coarsening
• First (preliminary) numerical realization
Andreas Zeiser 36 Wien, January 23, 2008
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References
[1] T. Rohwedder, R. Schneider, A. Zeiser, Perturbed preconditionedinverse iteration for operator eigenvalue problems withapplications to adaptive wavelet discretizations, submitted 2007.
[2] W. Dahmen, T. Rohwedder, R. Schneider, A. Zeiser, AdaptiveEigenvalue Computation – Complexity Estimates, submitted 2007.
available at http://www.math.tu-berlin.de/∼zeiser.
Thank you for your attention.
Andreas Zeiser 37 Wien, January 23, 2008