Title: Spectral Optimization Problems Controlling Wave Phenomena
Description: Design problems seek a material arrangement or shape which fully harnesses the physical properties of the material(s) to create an environment in which a particular phenomena is most (or least) pronounced. Mathematically, design problems are formulated as PDE-constrained optimization problems to find the material arrangement that maximizes an objective function which expresses the desired behavior. The PDE constraint describes the relationship between the material and the phenomena of interest. The focus of this thesis is four design problems where the PDE constraint is a time-independent wave equation and the objective function governs some aspect of wave motion. We consider the shape optimization of functions of Dirichlet-Laplacian eigenvalues associated with the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The boundary of such a region is represented using a Fourier-cosine series and the optimization problem is solved numerically using a quasi-Newton method.

The method is applied to maximizing two particular nonsmooth functions of the eigenvalues: (a) the ratio of the n-th to first eigenvalues and (b) the ratio of the n-th eigenvalue gap to first eigenvalue. Both are generalizations of the Payne-Pólya-Weinberger ratio. The optimal values of these ratios and regions for which they are attained, for n ≤ 13, are presented and interpreted as a study of the range of the Dirichlet-Laplacian eigenvalues. For both spectral functions and each n, the optimal region has multiplicity two n-th eigenvalue. We consider a system governed by the wave equation with index of refraction n(x), taken to be variable within a bounded region of d-dimensional space and constant outside. The solution of the time-dependent wave equation with spatially-localized initial data spreads and decays with advancing time. The rate of spatially localized energy decay can be measured in terms of the eigenvalues of the scattering resonance problem, a non-selfadjoint eigenvalue problem consisting of the time-harmonic wave (Helmholtz) equation with outgoing radiation condition at infinity.

Specifically, the rate of energy escape is governed by the complex scattering eigenfrequency which is closest to the real axis. We study the structural design problem: Find a refractive index profile n* within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact support of n(x)-1 and pointwise upper and lower (material) bounds on n(x): 0 < n- ≤ n(x) ≤ n+ < ∞. We formulate this problem as a constrained optimization problem and prove that an optimal structure, n* exists. Furthermore, n*(x) is piecewise constant and achieves the material bounds, i.e., n*(x) is n- or n+ almost everywhere. In one dimension, we establish a connection between n*(x) and the well-known class of Bragg structures, where n(x) is constant on intervals whose length is one-quarter of the effective wavelength. Consider a system governed by the time-dependent Schroedinger equation in its ground state. When subjected to weak parametric forcing by an "ionizing field" (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes.

The decay-rate of this metastable state is governed by Fermi's Golden Rule (FGR), which depends on the potential V and the details of the forcing. We pose the potential design problem: find V* which minimizes FGR (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, the optimizing potentials appear to be interior points of the constraint set and to be smooth. The multi-scale structures that emerge incorporate the physical mechanisms of energy confinement via material contrast and interference effects. An analysis of locally optimal potentials reveals local optimality is attained via two mechanisms: (i) decreasing the density of states near a resonant frequency in the continuum and (ii) tuning the oscillations of extended states to make FGR, an oscillatory integral, small.

Finally, we explore the performance of optimal potentials via simulations of the time-evolution. We consider a general class of two-dimensional passive propagation media, represented as a planar graph where nodes are capacitors connected to a common ground and edges are inductors. Capacitances and inductances are fixed in time but vary in space. Kirchhoff's laws give the time dynamics of voltage and current in the system. By harmonically forcing input nodes and collecting the resulting steady-state signal at output nodes, we obtain a linear, analog device that transforms the inputs to outputs. We pose the lattice synthesis problem: given a linear transformation, find the inductances and capacitances for an inductor-capacitor circuit that can perform this transformation. Formulating this as an optimization problem, we numerically demonstrate its solvability using gradient-based methods. By solving the lattice synthesis problem for various desired transformations, we design several devices that can be used for signal processing and filtering.

In addition to these spectral optimization problems, we study several problems on wave propagation, diffraction, and scattering. The focus is on the behavior of time-harmonic solutions to continuous and discrete wave equations.