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Elliptic Eigenvalue Problems

Chiu-Yen Kao

Collaborators:

He Lin, Yuan Lou, Stanley Osher,

Fadil Santosa, Eli Yabolnovich, Eiji Yanagida

IPAM, Numerics and Dynamics for Optimal Transport April 17, 2008

Control the resonance frequencies of devices:

Maximize or minimize certain frequencies

Maximize the gap between adjacent frequencies

Maximize the ratio between real part and imaginary part of eigenvalues (quality factor optimization)

Minimize the principle eigenvalue

Application:

Vibration system control

Photonic crystal design

Optical Resonator

Population biology

MotivationGoal: Minimize a certain design objective

such that is the eigenvalue of

subjects to boundary condition on and is the elliptic differential operator.

Shape Optimization for Elliptic Eigenvalue ProblemMotivation I : Shape of the drum

Consider an open bounded , a positive ,

and satisfies the elliptic eigenvalue problem

The eigenvalues are

Q1:Shape problem:

Q2:Composition problem:

Theoretical Results

Q1:Shape problem:

- Rayleigh (1877) conjectured, and Faber (1923) and Krahn (1925) proved, that if you fix the area of a drum, the lowest eigenvalue is minimized uniquely by the disk.
- Payne, Pólya, and Weinberger conjecture (1955): The disk maximizes the ratio of to has been proved by Ashbaugh and Benguria (1992)

Q2:composition problem:

- Krein (1955) provided one dimensional optimal density distribution for maximal and minimal .
- Cox and McLaughling (1993): minimal for higher dimensions.

Shape Optimization of a drum head with a fixed domain

Let be a domain inside , and

Solve the optimization:

Subject to the constraint:

Eigenvalue Separation for DrumsFramework of Murat-Simon:

Let be a reference domain. Consider its variations

with

Definition: the shape derivative of at is the Frechet differential of at .

Shape Derivativesubject to

use Lagrange multiplier method:

the necessary condition for a minimizer is

together with the constraint

allows us, in principle, to find and .

Lagrange Multiplier MethodShape derivative

Gradient descent algorithm for the shape

The normal advection velocity of the shape is .

We solve the level set equation:

Because of , we need to have .

Then .

Gradient AscentShape Optimization of a photonic crystal with a fixed domain

Let be a domain inside , and

We begin with a shape with

and we want to maximize

Motivation II : Photonic Crystal

Design the material to have lower loss of energy

Mechanics Systems

Ex: damped mass spring

Electrical Systems

Ex: RLC circuit, quartz crystal

Optical Systems

Ex: photonic crystal

Pictures:

http://en.wikipedia.org/wiki/Quartz_clock

http://minty.stanford.edu/PBG/

Motivation III: Quality Factor OptimizationThe Mass Spring System (1)

The displacement satisfies: (1)

For small damping , the solution is

The total energy is and

where the period .

The Mass Spring System (2)

The quality factor is defined as :

1-D Schrödinger’s Equation

- Finite potential well:

Optical Resonator

- One-Dimensional Case
- Higher-Dimensional Case
- Goal: minimize the quality factor

1-D Forward Eigenvalue Solver (1)

- Solve
- by finite element method. Apply the test function
- By incorporating the boundary condition

1-D Forward Eigenvalue Solver (2)

Thus

The equation can be written as

It is a nonlinear Eigenvalue Problem !!

1-D Forward Eigenvalue Solver (3)

- The equation can be written as

2-D Forward Eigenvalue Solver (1)

- Boundary Integral Method
- It is a nonlinear Eigenvalue Problem !!

2-D Forward Eigenvalue Solver (3)

- Nonlinear Eigenvalue Problem: (Newton’s method) inverse iteration

Gradient Flow

In terms of a single mode damped oscillator, the quality factor is proportional

to the ratio between the real part and the imaginary part of the eigenvalue.

Our goal here is to maximize the quality factor subject to the wave equation

we discussed previously which can be written in the general eigenvalue problem

Suppose there is a small perturbation s.t.

We keep only the first order term

Premultiplying by the corresponding eigenvector leads to

Thus

Consider the elliptic eigenvalue problem with indefinite weight

Let be a domain inside , and

Solve the optimization:

Subject to the constraint:

IV: Eigenvalue with Indefinite Weightwhere represents the density of a species and is the growth rate.

1. If , uniformly as

2. If , uniformly as

The effect of dispersal and spatial heterogeneity in population dynamics.

Diffusive Logistic EquationTheorem: When is an interval, then there are exactly two global minimizers of . For the logistic model, this means that a single favorable region at one of the two ends of the whole habitat provides the best opportunity for the species to survive. Minimizers in 1D

In general domain: open question

Suppose

If then is not minimal. In particular, the strip at the end with much longer edge can’t be the optimal favorable region.

Minimizers in High DimensionsThe End

Thank you

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