Thomas Trogdon

The Method of Fokas for Solving Linear Partial Differential Equations

The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method also allows for the equally explicit solution of problems for which no classical approach exists. In addition, it is possible to elucidate which boundary-value problems are well posed and which are not. We provide examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, and others do not. For the former, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.

Universality in numerical computations with random data.

Significance Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and also in an algorithm that models neural computation. The question of whether universality is present in all, or nearly all, computation is raised. The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time—i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components—the sample average and the sample variance—the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms as well as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own.

Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations

We solve the focusing and defocusing nonlinear Schrödinger (NLS) equations numerically by implementing the inverse scattering transform. The computation of the scattering data and of the NLS solution are both spectrally convergent. Initial conditions in a suitable space are treated. Using the approach of Biondini & Bui, we numerically solve homogeneous Robin boundary-value problems on the half line. Finally, using recent theoretical developments in the numerical approximation of Riemann–Hilbert problems, we prove that, under mild assumptions, our method of approximating solutions to the NLS equations is uniformly accurate in their domain of definition.

The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

We implement the new transform method for solving boundary-value problems developed by Fokas for periodic boundary conditions. The approach presented here is neither a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solved by the new transform method (which enhances further its scope and applicability), our implementation also has the advantage that it yields a new simpler approach to computing the limit from the periodic Cauchy problem to the Cauchy problem on the line.

Universality for Eigenvalue Algorithms on Sample Covariance Matrices

We prove a universal limit theorem for the halting time, or iteration count, of the power/inverse power methods and the QR eigenvalue algorithm. Specifically, we analyze the required number of iterations to compute extreme eigenvalues of random, positive definite sample covariance matrices to within a prescribed tolerance. The universality theorem provides a complexity estimate for the algorithms which, in this random setting, holds with high probability. The method of proof relies on recent results on the statistics of the eigenvalues and eigenvectors of random sample covariance matrices (i.e., delocalization, rigidity, and edge universality).

A numerical dressing method for the nonlinear superposition of solutions of the KdV equation

In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg?de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly decaying initial data. The second method is used to compute finite-genus solutions of the KdV equation. The combination of these numerical methods allows for the computation of exact solutions that are asymptotically (quasi-)periodic finite-gap solutions and are a nonlinear superposition of dispersive, soliton and (quasi-)periodic solutions in the finite (x, t)-plane. Such solutions are referred to as superposition solutions. We compute these solutions accurately for all values of x and t.

Sampling unitary ensembles

We develop a computationally efficient algorithm for sampling from a broad class of unitary random matrix ensembles that includes but goes well beyond the straightforward to sample Gaussian unitary ensemble (GUE). The algorithm exploits the fact that the eigenvalues of unitary ensembles (UEs) can be represented as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Consequently, our algorithm can be used to sample from UEs for which the associated orthogonal polynomials can be numerically computed efficiently. By facilitating high accuracy sampling of non-classical UEs, the algorithm can aid in the experimentation-based formulation or refutation of universality conjectures involving eigenvalue statistics that might presently be unamenable to theoretical analysis. Examples of such experiments are included.

Gibbs phenomenon for dispersive PDEs on the line

We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is expressed in terms of universal, computable special functions. We show that the leading-order behavior of the solution of dispersive PDEs near a discontinuity of the ICs is characterized by Gibbs-type oscillations and gives exactly the Wilbraham--Gibbs constants.

Numerical Inverse Scattering for the Toda Lattice

We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann–Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in