Thomas Trogdon

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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(More)
Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is(More)
We implement the new transform method for solving boundary value problems developed by Fokas for periodic boundary conditions. The approach presented here is not a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solve by the new transform method (which enhances(More)
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(More)
The effective and efficient numerical solution of Riemann–Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann– Hilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. The(More)
We derive a Riemann–Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We(More)
The authors present empirical universal distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of(More)
The transition between the standard snake instability of bright solitons of the hyperbolic nonlinear Schrödinger equation and the recently theoretically predicted oscillatory snake instability is experimentally demonstrated. The existence of this transition is proven on the basis of spatiotemporal spectral features of bright soliton laser beams propagating(More)
We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX∗, where X is an N × n matrix with iid standard complex normal entries. Under the scaling n = N + b √ 4cNc, c > 0 and N →∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices A are all(More)