Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP...

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Some perturbation theorems for nonlinear eigenvalue problems David Bindel Department of Computer Science Cornell University 8 January 2013 (Cornell University) Dissipative Spectral Theory 1 / 46

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Page 1: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Some perturbation theorems for nonlineareigenvalue problems

David Bindel

Department of Computer ScienceCornell University

8 January 2013

(Cornell University) Dissipative Spectral Theory 1 / 46

Page 2: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Some favorite examples

Why nonlinear eigenvalue problems?

(Cornell University) Dissipative Spectral Theory 2 / 46

Page 3: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 4: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 5: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 6: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 7: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 8: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 9: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 10: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 11: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 12: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 13: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 14: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 15: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 16: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 17: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 18: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 19: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 20: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 21: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 22: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 23: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: DDEs

Pseudospectral plot

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(Cornell University) Dissipative Spectral Theory 23 / 46

Page 24: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: DDEs

Pseudospectral plot

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(Cornell University) Dissipative Spectral Theory 24 / 46

Page 25: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 26: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: DDEs

Example: Bounding the spectral abscissa

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(Cornell University) Dissipative Spectral Theory 26 / 46

Page 27: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: DDEs

Example: Imaginary part of unstable eigenvalues

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(Cornell University) Dissipative Spectral Theory 27 / 46

Page 28: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 29: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: DDEs

Example: Bounding spectrum from the left

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(Cornell University) Dissipative Spectral Theory 29 / 46

Page 30: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Schrödinger resonances

(Cornell University) Dissipative Spectral Theory 30 / 46

Page 31: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Spectra and scattering

Spectrum for H = −∆ + V , supp(V ) compact.

(Cornell University) Dissipative Spectral Theory 31 / 46

Page 32: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 33: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 34: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Is it that easy?

−400 −200 0 200 400−40

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All eigenvaluesChecked eigenvalues

(Cornell University) Dissipative Spectral Theory 34 / 46

Page 35: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Is it that easy?

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1.2Potential

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0Pole locations

(Cornell University) Dissipative Spectral Theory 35 / 46

Page 36: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 37: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 38: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 39: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 40: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 41: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Spectral Schur complement

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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

Page 42: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 43: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 44: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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

Page 45: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

Application: Resonances

Assessing approximate resonances

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-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

Page 46: Some perturbation theorems for nonlinear eigenvalue problemsbindel/present/2013-01-cardiff.pdfNEP perturbation theorems Beyond first order Suppose T;E: !Cn nanalytic ˆ a simple contour

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