Numerical Analysis of Resonances

David Bindel

Department of Computer ScienceCornell University

20 September 2012

My favorite applications

Resonance and anchor loss

PML region

Wafer (unmodeled)

Electrode

Resonating disk

The quantum corral and tunneling

Spectra and scattering

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

Resonances and scattering

1.0 1.5 2.0 2.5 3.0 3.5 4.0k

50

100|

(k)|

For supp(V ) , consider a scattering experiment:

(H k2) = f on (n B(k)) = 0 on

See resonance peaks (Breit-Wigner):

(k) w C(k k)1.

Resonances and scattering

Consider a scattering measurement (k)I Morally looks like = w(H E)1f?I w(H E)1f is well-defined off spectrum of HI Continuous spectrum of H is a branch cut for I Resonance poles are on a second sheet of definition for I Resonance wave functions blow up exponentially (not L2)

Common approach

1.0 1.5 2.0 2.5 3.0 3.5 4.0k

50

100|

(k)|

Goal: Understand localized leaky vibrationsI Far field infinite and homogeneousI Dynamics truncated resonance expansion

(Breit-Wigner):

(k) C(k k)1, k C

Reduce to a bounded domain and compute!

The 1D case: MatScat

http://www.cs.cornell.edu/~bindel/cims/matscat/

http://www.cs.cornell.edu/~bindel/cims/matscat/http://www.cs.cornell.edu/~bindel/cims/matscat/

MatScat

!3 !2 !1 0 1 2 3 4 5

0

50

100

Potential

!20 !15 !10 !5 0 5 10 15 20!0.8

!0.6

!0.4

!0.2

0Pole locations

Resonances and transients

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Scattering solutions

Schrdinger scattering from a potential V on [a,b]

H =( d

2

dx2+ V

) = E

For E = k2 > 0, get solutions

= eikx + scatter

where scatter satisfies outgoing BCs:(ddx ik

) = 0, x = b(

ddx

+ ik) = 0, x = a,

This is a Dirichlet-to-Neumann (DtN) map: (n B(k)) = 0

A quadratic eigenvalue problem

( d

2

dx2+ V (x) k2

) = 0, x (a,b)(

ddx ik

) = 0, x = b(

ddx

+ ik) = 0, x = a

Look for nontrivial solutions:I Im(k) > 0: Bound statesI Im(k) < 0: Resonances

Basic MatScat strategy

Pseudospectral collocation at Chebyshev points:(D2 + V (x) k2

) = 0, x (a,b)

(D ik) = 0, x = b(D + ik) = 0, x = a

Convert to linear problem with auxiliary variable = k.

Is it that easy?

400 200 0 200 40040

30

20

10

0

10

20

All eigenvaluesChecked eigenvalues

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 22

1.5

1

0.5

0Pole locations

Computational desiderata

I All resonances in some regionI and error estimatesI and sensitivity estimatesI and good computational complexity

Method 1: Prony and company

Extract resonances from time-domain data (or (k))

u(t)

k

ck exp(k t)

I This is a (modified) Prony problemI Long use both experimentally and computationally

(e.g. Wei-Majda-Strauss, JCP 1988 modified Prony applied to time-domain simulations)

I Variants like FDM still used (e.g. Johnsons harminv)

Computing resonances 2: complex scaling

Change coordinates to shift the branch cut:

H =( d

2

dx2+ V

) = E

where dx/dx = 1 + i(x) is deformed outside [a,b].

I Rotates the continuous spectrum to reveal resonancesI First used to define resonances (Simon 1979)I Also a computational method (aka PML):

I Truncate to a finite x domain.I Discretize using standard methodsI Solve a complex symmetric eigenvalue problem

One of my favorite computational tactics.

Computing resonances 3: a nonlinear eigenproblem

Can also define resonances via a NEP:

(H k2) = 0 on (n B(k)) = 0 on

Resonance solutions are stationary points with respect to of

(, k) =

[()T () + (V k2)

]d

B(k) d

Discretized equations (e.g. via finite or spectral elements) are

A(k) =(

K k2M C(k)) = 0

K and M are real symmetric and C(k) is complex symmetric.

Computational tradeoffs

I PronyI Relatively simple signal processingI Can be used with scattering experiment resultsI May require long simulationsI Numerically sensitive

I Complex scalingI Straightforward implementationI Yields a linear eigenvalue problemI How to choose scaling parameters, truncation?

I DtN map formulationI Bounded domain no artificial truncationI Yields a nonlinear eigenvalue problemI DtN map is spatially nonlocal except in 1D

(though diagonalized by Fourier modes on a circle)

Other options: complex absorbing potentials, approximate BCs(e.g. Engquist-Majda)

Computational desiderata

I All resonances in some regionI and error estimatesI and sensitivity estimatesI and good computational complexity

Forward and backward error analysis

Forward and backward error analysis

A simple example

Standard eigenvalue problem (A I)v = 0, v = 1:

(A I)v = r(A I)v = 0, A = A rvT

So (A) and (A), where (A) {A1 > 1}.

Or estimate by first-order sensitivity analysis

Sensitivity for resonances

Resonance solutions are stationary points with respect to of

(, k) =[2 + (V k2)

]d

(

n B(k)

)d

=

[()T () + (V k2)

]d

B(k) d

If (, k) a resonance pair, then (, k) = 0 and D(, k) = 0.

Potential perturbations

If (, k) a resonance pair, then (, k) = 0 and D(, k) = 0.

Consider perturbed V :

= D + DV V + Dk k = 0

Use D = 0:

k = DV VDk

Perturbation worked out

So look at how perturbations V change k :

k =

V

2

2k

2 B

(k)

Can also write in terms of a residual for as a solution for thepotential V + V :

k =

( + (V + V ) k

2)

2k

2 B

(k).

Backward error analysis in MatScat

1. Compute approximate solution (, k).2. Map to high-resolution quadrature grid to evaluate

k =

( + V k

2)

2k

2 B

(k).

3. If k large, discard k ; otherwise, accept k k + k .

Beyond 1D

1D was relatively easy:I Only small discretizations needed.I Worked with exact boundary conditionsI Could rewrite general NEP as a QEP

Nonlinear to linear eigenproblems

Can also compute resonances byI Adding a complex absorbing potentialI Complex scaling methodsI Artificial dampers

Both result in complex-symmetric ordinary eigenproblems:

(Kext k2Mext )ext =([

K11 K12K21 K22

] k2

[M11 M12M21 M22

])[12

]= 0

where 2 correspond to extra variables (outside ).

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 2 to get

A(k)1 =(

K11 k2M11 C(k))1 = 0

where

C(k) = (K12 k2M12)(K22 k2M22)1(K21 k2M21)

Apples to oranges?

A(k) = (K k2M C(k)) = 0 (exact DtN map)

A(k) = (K k2M C(k)) = 0 (spectral Schur complement)

Two ideas:I Perturbation theory for NEP for local refinementI Complex analysis to get more global analysis

Linear vs nonlinear

-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:I Eigenvalues of a complex-scaled problemI Residuals in nonlinear eigenproblemI log10 A(k) A(k)

Corrections two ways

A(k) = (K k2M C(k)) = 0 (exact DtN map)

A(k) = (K k2M C(k)) = 0 (spectral Schur complement)

I Plug (k , ) into true problem and correct:

k k T A(k)

T A(k)

I Write A(k) = A(k) + E(k) where E(k) = C(k) C(k).Interpret E(k) as a correction to Kext in linear problem.

Latter is promising for analysis beyond first-order sensitivity.

A little complex analysis

If A nonsingular on , analytic inside, count eigs inside by

W(det(A)) =1

2i

ddz

ln det(A(z)) dz

= tr(

12i

A(z)1A(z) dz)

E = A A also analytic inside . By continuity,

W(det(A)) = W(det(A + E)) = W(det(A))

if A + sE nonsingular on for s [0,1].

A general recipe

Analyticity of A and E +Matrix nonsingularity test for A + sE =Inclusion region for (A + E) +Eigenvalue counts for connected components of region

Application: Matrix Rouch

A(z)1E(z) < 1 on = same eigenvalue count in

Proof:A(z)1E(z) < 1 = A(z) + sE(z) invertible for 0 s 1.

(Gohberg and Sigal proved a more general version in 1971.)

Aside on spectral Schur complement

Inverse of a Schur complement is a submatrix of an inverse:

(Kext z2Mext )1 =[A(z)1

]So for reasonable norms,

A(z)1 (Kext z2Mext )1.

Or

(A) (Kext ,Mext ),

(A) {z : A(z)1 > 1}(Kext ,Mext ) {z : (Kext z2Mext )1 > 1}

Nonlinear bounds from linear pseudospectra

Recall:

A(k) = (K k2M C(k)) = 0 (exact DtN map)

A(k) = (K k2M C(k)) = 0 (spectral Schur complement)

Let S = {z C : C(z) C(z) < }. Then:

(A) S (A) (Kext,Mext)

Sensitivity and pseudospectra

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-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

TheoremLet S = {z : A(z) A(z) < }. Any connected component of(Kext ,Mext ) strictly inside S contains the same number ofeigenvalues for A(k) and A(k).

For more

More information at

http://www.cs.cornell.edu/~bindel/

I Links to tutorial notes on resonances with Maciej ZworskiI Matscat code for computing resonances for 1D problemsI These slides!

http://www.cs.cornell.edu/~bindel/