Post on 07-Aug-2020
Some perturbation theorems for nonlineareigenvalue problems
David Bindel
Department of Computer ScienceCornell University
8 January 2013
(Cornell University) Dissipative Spectral Theory 1 / 46
Some favorite examples
Why nonlinear eigenvalue problems?
(Cornell University) Dissipative Spectral Theory 2 / 46
The problem
The general setting
Nonlinear eigenvalue problem:
T (λ)v = 0, v 6= 0.
whereT : Ω→ Cn×n analytic on simply connected Ω ⊂ Cdet(T ) 6≡ 0 (i.e. T is regular)
Write the set of nonlinear eigenvalues as Λ(T ).
Source: transform methods on almost anything with damping!For many examples, see:
NLEVP collectionSurvey by Mehrmann and Voss
(Cornell University) Dissipative Spectral Theory 3 / 46
The problem
Quadratic problems
Example: Damped free vibrations of a mechanical system
Mu′′ +Bu′ +Ku = 0.
Laplace transform:(s2M + sB +K)U = 0.
Approach directly or convert to first order:
Bv +Ku = sMv
v = su
(Cornell University) Dissipative Spectral Theory 4 / 46
The problem
Polynomial problems
More general is polynomial eigenvalue problem:
T (λ)v = 0, T (z) ≡ zdI + zd−1Ad + . . .+ zA1 +A0
Common approach: define uj = λjv, and solve−Ad−1 −Ad−2 . . . −A1 −A0
I 0I 0
. . . . . .I 0
ud−1
ud−2...u1
v
= λ
ud−1
ud−2...u1
v
This is one of many possible linearizations.Can do something similar with rational problems.
(Cornell University) Dissipative Spectral Theory 5 / 46
The problem
A special rational problem
Consider the eigenvalue equation[A− λI BC D − λI
] [vv
]= 0.
If λ 6∈ Λ(D), partial Gaussian elimination yields T (λ)v = 0, where
T (z) = A− zI −B(D − zI)−1C.
This is a spectral Schur complement problem.
(c.f. Feschbach, Lifschitz, Grushin).
(Cornell University) Dissipative Spectral Theory 6 / 46
The problem
Solving general NEPs
T (λ)x = 0, x 6= 0, T : Ω→ Cn×n analytic
Computational approaches:Local polynomial / rational approximation of TMethods based on contour integration
Either way, we want:A starting point (expansion point, contour)Error estimates for the results
(Cornell University) Dissipative Spectral Theory 7 / 46
The problem
Perturbation and localization
Many uses for perturbation theory in linear case:Backward error analysis (first-order theory, pseudospectra)Crude bounds for choosing algorithm parameters (Gerschgorin)Crude bounds for stability testing (Gerschgorin)Reasoning about dynamics (pseudospectra)
Want the same theory for nonlinear problems!
(Cornell University) Dissipative Spectral Theory 8 / 46
NEP perturbation theorems
First-order perturbation theory
Small, analytic E, consider
T = T + E
Given a simple eigentriple (λ, u, w∗) of T :
T (λ)u = 0, w∗T (λ) = 0.
First-order perturbation theory gives:
δλ = −w∗E(λ)u
w∗T ′(λ)u
Great! What about large perturbations, multiple eigenvalues, ...?
(Cornell University) Dissipative Spectral Theory 9 / 46
NEP perturbation theorems
Beyond first order
SupposeT,E : Ω→ Cn×n analyticΓ ⊂ Ω a simple contourT (z) + sE(z) nonsingular, all s ∈ [0, 1], z ∈ Γ.
Then T and T + E have the same number of eigenvalues inside Γ.
Proof:The winding number of det(T + sE) stays continuous for 0 ≤ s ≤ 1.
(Cornell University) Dissipative Spectral Theory 10 / 46
NEP perturbation theorems
A general recipe
Analyticity of T and E +Matrix nonsingularity test for T + sE =Inclusion region for Λ(T + E) +Eigenvalue counts for connected components of region
(Cornell University) Dissipative Spectral Theory 11 / 46
NEP perturbation theorems
Matrix Rouché
‖T (z)−1E(z)‖ < 1 on Γ =⇒ same eigenvalue count in Γ
Proof:‖T (z)−1E(z)‖ < 1 =⇒ T (z) + sE(z) invertible for 0 ≤ s ≤ 1.
(Gohberg and Sigal proved a more general version in 1971.)
(Cornell University) Dissipative Spectral Theory 12 / 46
NEP perturbation theorems
Nonlinear pseudospectra
Define the nonlinear ε-pseudospectrum as
Λε(T ) = z ∈ Ω : ‖T (z)−1‖ > ε−1
Let E = E : Ω→ Cn×n s.t. E analytic,maxz∈Ω ‖E(z)‖ < ε. Then
Λε(T ) =⋃E∈E
Λ(T + E).
If E0 = E ∈ Cn×n : ‖E0‖ < ε, we may also write
Λε(T ) =⋃
E0∈E0
Λ(T + E0).
(Cornell University) Dissipative Spectral Theory 13 / 46
NEP perturbation theorems
Nonlinear pseudospectra and backward error
Suppose λ, v an approximate eigenpair with ‖v‖ = 1,
T (λ)v = r, ‖r‖ small.
Then λ ∈ Λ‖r‖(T ), since(T (λ)− rv∗
)v = 0
(Cornell University) Dissipative Spectral Theory 14 / 46
NEP perturbation theorems
Nonlinear pseudospectra and dynamics
Suppose Ψ : [0,∞)→ CN×N , let
R(z) ≡∫ ∞
0e−ztΨ(t) dt.
Ψ bounded =⇒ R(z) defined in RHP and for any ε > 0,
supt>0‖Ψ(t)‖ ≥ αε
ε,
whereαε ≡ sup
‖R(λε)‖>ε−1
Re(λε)
(Similar proof to that for linear pseudospectra.)
(Cornell University) Dissipative Spectral Theory 15 / 46
NEP perturbation theorems
Pseudospectral counting
Let T,E analytic on Ω and define:
Ωε ≡ z ∈ Ω : ‖E(z)‖ < ε.
ThenΛ(T ) ∩ Ωε ⊂ Λε(T + E)
Also, ifU ⊂ Λε(T + E) a connected component.U ⊂ Ωε.
then U contains the same number of eigenvalues of T and T + E,of which there must be at least one.
(Cornell University) Dissipative Spectral Theory 16 / 46
NEP perturbation theorems
Weakly coupled problems
T (z) =
[L1(z) H(z)G(z) L2(z)
]is analytic over Ω, and
‖G(z)‖ ≤ γ, ‖H(z)‖ ≤ η, Λδ1(L1) ∩ Λδ2(L2) = ∅.
Assume γη < δ1δ2, boundary of Λδ1(L1) is strictly inside Ω. Then1 Λ(T ) ⊂ Λδ1(L1) ∪ Λδ2(L2)
2 T and L1 have same eigenvalue counts in Λδ1(L1)
3 For λ ∈ Λδ1(L1), eigenvector v satisfies ‖v2‖/‖v1‖ < γ/δ1.
4 For λ ∈ Λδ2(L2), eigenvector v satisfies ‖v2‖/‖v1‖ > γ/δ2.
(Cornell University) Dissipative Spectral Theory 17 / 46
Perturbing linear problems
Linear problems, nonlinear perturbations
Perturb linear problem with E analytic, “small” on Ω:
T (z) = A− zB + E(z).
Many linear perturbation theorems still hold!
(Cornell University) Dissipative Spectral Theory 18 / 46
Perturbing linear problems
Nonlinear perturbations + pseudospectra
T (z) = A− zI + E(z)
and suppose ‖E‖ < ε on Ω.
If U a connected component of Λε(A), U ⊂ Ω, thenA and T have the same eigenvalue counts in U .The eigenvalue count in U is at least one.
(Cornell University) Dissipative Spectral Theory 19 / 46
Perturbing linear problems
Nonlinear Gerschgorin
For D diagonal, consider
T (z) = D − zI + E(z)
such thatn∑j=1
|eij(z)| ≤ ρi
ThenΛ(T ) ⊂
⋃ni=1Gi where Gi = Bρi(dii)
U =⋃i∈I Gi a connected component, U ⊂ Ω
=⇒ U contains |I| eigenvalues.
(Cornell University) Dissipative Spectral Theory 20 / 46
Perturbing linear problems
Nonlinear Bauer-Fike bound
Suppose |E(z)| ≤ F componentwise on Ω,
T (z) = A− zI + E(z).
and A has eigentriples (λi, vi, w∗i ). Then
Λ(T ) ⊂n⋃i=1
Bφi(λi)
where φi = n‖F‖2 sec(θi) and
sec(θi) =‖wi‖‖vi‖|w∗i vi|
.
Can also count within connected components.
(Cornell University) Dissipative Spectral Theory 21 / 46
Application: DDEs
Application: Delay-differential equation
From NLEVP collection
T (λ) = A0 − λI +A1 exp(−λ)
Corresponding to
u′(t) = A0u(t) +A1u(t− 1)
Double non-semisimple eigenvalue λ = 3πi.
(Cornell University) Dissipative Spectral Theory 22 / 46
Application: DDEs
Pseudospectral plot
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
30
40
(Cornell University) Dissipative Spectral Theory 23 / 46
Application: DDEs
Pseudospectral plot
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
30
40
−1.5
−1.5
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
0
0
0
0
0.5
0.5
1
1
(Cornell University) Dissipative Spectral Theory 24 / 46
Application: DDEs
Gerschgorin applied
ConsiderV −1T (λ)V = D − λI + A1 exp(−λ)
Apply Gerschgorin-like bound
Λ(T ) ⊂3⋃i=1
Bρi(dii) ∪ | exp(−λ)| > exp(−σ)
where
ρi = exp(−σ)
∑j
(A1)ij
α∑j
(A1)ji
1−α
(Cornell University) Dissipative Spectral Theory 25 / 46
Application: DDEs
Example: Bounding the spectral abscissa
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
30
40
−1.5
−1.5
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
0
0
0
0
0.5
0.5
1
1
(Cornell University) Dissipative Spectral Theory 26 / 46
Application: DDEs
Example: Imaginary part of unstable eigenvalues
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
30
40
−1.5
−1.5
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
0
0
0
0
0.5
0.5
1
1
(Cornell University) Dissipative Spectral Theory 27 / 46
Application: DDEs
Switching terms
ConsiderV −1T (λ)V = D exp(−λ)− λ+ A0
Gerschgorin-like argument now bounds spectrum from left!
(Cornell University) Dissipative Spectral Theory 28 / 46
Application: DDEs
Example: Bounding spectrum from the left
−20 −15 −10 −5 0 5 10 15 20−40
−30
−20
−10
0
10
20
30
40
−1.5
−1.5
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
0
0
0
0
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0.5
1
1
(Cornell University) Dissipative Spectral Theory 29 / 46
Application: Resonances
Schrödinger resonances
(Cornell University) Dissipative Spectral Theory 30 / 46
Application: Resonances
Spectra and scattering
Spectrum for H = −∆ + V , supp(V ) compact.
(Cornell University) Dissipative Spectral Theory 31 / 46
Application: Resonances
Resonances and scattering
1.0 1.5 2.0 2.5 3.0 3.5 4.0k
50
100|φ
(k)|
For supp(V ) ⊂ Ω, consider a scattering experiment:
(H − k2)ψ = f on Ω
(∂n −B(k))ψ = 0 on ∂Ω
See resonance peaks (Breit-Wigner):
φ(k) ≡ w∗ψ ≈ C(k − k∗)−1.
(Cornell University) Dissipative Spectral Theory 32 / 46
Application: Resonances
1D resonances: a quadratic eigenvalue problem
(− d2
dx2+ V (x)− k2
)ψ = 0, x ∈ (a, b)(
d
dx− ik
)ψ = 0, x = b(
d
dx+ ik
)ψ = 0, x = a
Look for nontrivial solutions:Im(k) > 0: Bound statesIm(k) < 0: Resonances
See:
http://www.cs.cornell.edu/~bindel/sw/matscat/
(Cornell University) Dissipative Spectral Theory 33 / 46
Application: Resonances
Is it that easy?
−400 −200 0 200 400−40
−30
−20
−10
0
10
20
All eigenvaluesChecked eigenvalues
(Cornell University) Dissipative Spectral Theory 34 / 46
Application: Resonances
Is it that easy?
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2Potential
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0Pole locations
(Cornell University) Dissipative Spectral Theory 35 / 46
Application: Resonances
Sensitivity for resonances
Resonance solutions are stationary points with respect to ψ of
Φ(ψ, k) =
∫Ωψ[−∇2ψ + (V − k2)ψ
]dΩ−
∫∂Ωψ
(∂ψ
∂n−B(k)ψ
)dΓ
=
∫Ω
[(∇ψ)T (∇ψ) + ψ(V − k2)ψ
]dΩ−
∫∂ΩψB(k)ψ dΓ
If (ψ, k) a resonance pair, then Φ(ψ, k) = 0 and DψΦ(ψ, k) = 0.
(Cornell University) Dissipative Spectral Theory 36 / 46
Application: Resonances
Potential perturbations
If (ψ, k) a resonance pair, then Φ(ψ, k) = 0 and DψΦ(ψ, k) = 0.
Consider perturbed V :
δΦ = DψΦ · δψ +DV Φ · δV +DkΦ · δk = 0
Use DψΦ · δψ = 0:
δk = −DV Φ · δVDkΦ
(Cornell University) Dissipative Spectral Theory 37 / 46
Application: Resonances
Perturbation worked out
So look at how perturbations δV change k:
δk =
∫Ω δV ψ
2
2k∫
Ω ψ2 −
∫Γ ψB
′(k)ψ
Can also write in terms of a residual for ψ as a solution for the potentialV + δV :
δk =
∫Ω ψ(−∆ + (V + δV )− k2)ψ
2k∫
Ω ψ2 −
∫Γ ψB
′(k)ψ.
(Cornell University) Dissipative Spectral Theory 38 / 46
Application: Resonances
Backward error analysis in MatScat
1 Compute approximate solution (ψ, k).2 Map ψ to high-resolution quadrature grid to evaluate
δk =
∫Ω ψ(−∆ + V − k2)ψ
2k∫
Ω ψ2 −
∫Γ ψB
′(k)ψ.
3 If δk large, discard k; otherwise, accept k ≈ k + δk.
(Cornell University) Dissipative Spectral Theory 39 / 46
Application: Resonances
Nonlinear vs linear eigenproblems
Can also compute resonances byAdding a complex absorbing potentialComplex scaling methodsArtificial dampers
Both result in complex-symmetric ordinary eigenproblems:
(Kext − k2Mext)ψext =
([K11 K12
K21 K22
]− k2
[M11 M12
M21 M22
])[ψ1
ψ2
]= 0
where ψ2 correspond to extra variables (outside Ω).
(Cornell University) Dissipative Spectral Theory 40 / 46
Application: Resonances
Spectral Schur complement
0 5 10 15 20 25 30
0
5
10
15
20
25
30
0 5 10 15 20 25 30
0
5
10
15
20
25
30
Eliminate “extra” variables ψ2 to get
T (k)ψ1 =(K11 − k2M11 − C(k)
)ψ1 = 0
where
C(k) = (K12 − k2M12)(K22 − k2M22)−1(K21 − k2M21)
(Cornell University) Dissipative Spectral Theory 41 / 46
Application: Resonances
Apples to oranges?
T (k)ψ = (K − k2M − C(k))ψ = 0 (exact DtN map)
T (k)ψ = (K − k2M − C(k))ψ = 0 (spectral Schur complement)
Two ideas:Perturbation theory for NEP for local refinementComplex analysis to get more global analysis
(Cornell University) Dissipative Spectral Theory 42 / 46
Application: Resonances
Aside on spectral Schur complement
Inverse of a Schur complement is a submatrix of an inverse:
(Kext − z2Mext)−1 =
[T (z)−1 ∗∗ ∗
]So for reasonable norms,
‖T (z)−1‖ ≤ ‖(Kext − z2Mext)−1‖.
Or
Λε(T ) ⊂ Λε(Kext,Mext),
Λε(T ) ≡ z : ‖A(z)−1‖ > ε−1Λε(Kext,Mext) ≡ z : ‖(Kext − z2Mext)
−1‖ > ε−1
(Cornell University) Dissipative Spectral Theory 43 / 46
Application: Resonances
Nonlinear bounds from linear pseudospectra
Recall:
T (k)ψ = (K − k2M − C(k))ψ = 0 (exact DtN map)
A(k)ψ = (K − k2M − C(k))ψ = 0 (spectral Schur complement)
Let Ωε = z ∈ C : ‖C(z)− C(z)‖ < ε. Then:
Λ(T ) ∩ Ωε ⊂ Λε(T ) ⊂ Λε(Kext,Mext)
(Cornell University) Dissipative Spectral Theory 44 / 46
Application: Resonances
Assessing approximate resonances
-5
-4
-3
-2
-1
0
0 2 4 6 8 10
Im(k
)
Re(k)
CorrectSpurious0
0
-2
-2
-4
-4
-6
-8
-8
-8
-10
-10
-10
To get axisymmetric resonances in corral model, compute:Eigenvalues of a complex-scaled problemResiduals in nonlinear eigenproblemlog10 ‖T (k)− T (k)‖
(Cornell University) Dissipative Spectral Theory 45 / 46
Conclusion
Conclusion
Nonlinear eigenvalue problems are as natural as linear problemsLinear perturbation theorems with complex analytic proofs apply“Perturbation Theorems for Nonlinear Eigenvalue Problems”David Bindel and Amanda Hood
(Cornell University) Dissipative Spectral Theory 46 / 46