arXiv subjects

Thomas Trogdon

Publications and source records attributed to Thomas Trogdon.

At least 19 recordsLinked to original sources

arXiv2026

A Riemann-Hilbert approach to the computation of transform pairs

We develop a unified methodology that integrates spectral theory, Riemann-Hilbert problems, and inverse scattering theory for the construction and numerical evaluation of transform pairs associated with linear variable-coefficient partial differential equations. The approach combines analytical formulae with numerical methods for ordinary differential equations and Riemann-Hilbert problems, yielding a hybrid analytical-numerical strategy for working with these transforms. Results are presented for transforms arising in the Dirac equation, demonstrating accurate computations, even in the presence of discontinuous coefficients.

math.CA↗

arXiv2026

Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice

We study the direct and inverse spectral theory for a class of finite Hermitian banded matrices. Using the theory of matrix orthogonal polynomials, we provide an explicit procedure for reconstructing a banded matrix from a matrix-valued measure that encodes its spectral data. We establish necessary and sufficient conditions for a measure to be the spectral measure of a matrix in the examined class. We further analyze the connections between this spectral analysis, block tridiagonalization algorithms, and the Toda lattice evolution on banded matrices.

math.SP↗

arXiv2025

Efficient computation of soliton gas primitive potentials

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $-\infty$. This accumulation results in an associated Riemann--Hilbert problem on a number of disjoint intervals. In the case where the jump matrices have specific square-root behavior, we describe an efficient and accurate numerical method to solve this Riemann--Hilbert problem and extract the potential. The keys to the method are, first, the deformation of the Riemann--Hilbert problem, making numerical use of the so-called $g$-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation.

nlin.SI↗

arXiv2025

A Lanczos-Based Algorithmic Approach for Spike Detection in Large Sample Covariance Matrices

We introduce a new approach for estimating the number of spikes in a general class of spiked covariance models without directly computing the eigenvalues of the sample covariance matrix. This approach is based on the Lanczos algorithm and the asymptotic properties of the associated Jacobi matrix and its Cholesky factorization. A key aspect of the analysis is interpreting the eigenvector spectral distribution as a perturbation of its asymptotic counterpart. The specific exponential-type asymptotics of the Jacobi matrix enables an efficient approximation of the Stieltjes transform of the asymptotic spectral distribution via a finite continued fraction. As a consequence, we also obtain estimates for the density of the asymptotic distribution and the location of outliers. We provide consistency guarantees for our proposed estimators, proving their convergence in the high-dimensional regime. We demonstrate that, when applied to standard spiked covariance models, our approach outperforms existing methods in computational efficiency and runtime, while still maintaining robustness to exotic population covariances.

math.ST↗

arXiv2025

The Akhiezer iteration and inverse-free solvers for Sylvester matrix equations

Two inverse-free iterative methods are developed for solving Sylvester matrix equations when the spectra of the coefficient matrices are on, or near, known disjoint subintervals of the real axis. Both methods use the recently-introduced Akhiezer iteration: one to address an equivalent problem of approximating the matrix sign function applied to a block matrix and the other to directly approximate the inverse of the Sylvester operator. In each case this results in provable and computable geometric rates of convergence. When the right-hand side matrix is low rank, both methods require only low-rank matrix-matrix products. Relative to existing approaches, the methods presented here can be more efficient and require less storage when the coefficient matrices are dense or otherwise costly to invert. Applications include solving partial differential equations and computing Fr\'echet derivatives.

math.NA↗

arXiv2024

Numerical inverse scattering transform for the defocusing nonlinear Schr\"odinger equation with box-type initial conditions on a nonzero background

We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schr\"{o}dinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude $q_o>0$ as $x\to \pm \infty$ by implementing numerically the corresponding Inverse Scattering Transform (IST). The Riemann--Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the $(x,t)$-plane depending on whether $|x/(2t)| < q_o$ or $|x/(2t)| > q_o$, as $t \to \infty$.

nlin.SI↗

arXiv2024

On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.

math.CA↗

arXiv2024

The ultraspherical rectangular collocation method and its convergence

We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our setting to prove convergence for the rectangular collocation method of Driscoll \& Hale for a restricted class of boundary conditions.

math.NA↗

arXiv2023

The Akhiezer iteration

We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint intervals. The iteration applies to shifted linear solves and can then be used for efficient matrix function approximation. Using the asymptotics of orthogonal polynomials, error bounds are provided. A key component in the efficiency of the method is the ability to compute the first $k$ orthogonal polynomial recurrence coefficients and the first $k$ weighted Stieltjes transforms of these orthogonal polynomials in $\mathrm{O}(k)$ complexity using a numerical Riemann--Hilbert approach. For a special class of orthogonal polynomials, the Akhiezer polynomials, the method can be sped up significantly, with the greatest speedup occurring in the two interval case where important formulae of Akhiezer are employed and the Riemann--Hilbert approach is bypassed.

math.NA↗

arXiv2023

On the convergence of Fourier spectral methods involving non-compact operators

Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

math.NA↗

arXiv2023

Computing the Tracy-Widom Distribution for Arbitrary $\beta>0$

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

math.NA↗

arXiv2023

GMRES, pseudospectra, and Crouzeix's conjecture for shifted and scaled Ginibre matrices

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted $N\times N$ matrix whose entries are independent complex Gaussians. When the right hand side of this linear system is independent of this random matrix, the $N\to\infty$ behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix's conjecture.

math.NA↗

arXiv2023

Stability of the Lanczos algorithm on matrices with regular spectral distributions

We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polynomials. We give a backwards stability result which can be upgraded to a forward stability result when the reference measure has a density supported on a single interval with square root behavior at the endpoints. Our analysis implies the Lanczos algorithm run on many large random matrix models is in fact forward stable, and hence nearly deterministic, even when computations are carried out in finite precision arithmetic. Since the Lanczos algorithm is not forward stable in general, this provides yet another example of the fact that random matrices are far from "any old matrix", and care must be taken when using them to test numerical algorithms.

math.NA↗

arXiv2023

A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.

math.NA↗

arXiv2023

An artificially-damped Fourier method for dispersive evolution equations

Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one side reappear on the other and for dispersive equations these are typically high-velocity, high-frequency waves. However, the fast Fourier transform is a very efficient numerical tool and it is important to find a way to damp these oscillations so that this transform can still be used. In this paper, we accurately model solutions to four nonlinear partial differential equations on an infinite domain by considering a finite interval and implementing two damping methods outside of that interval: one that solves the heat equation and one that simulates rapid exponential decay. Heat equation-based damping is best suited for small-amplitude, high-frequency oscillations while exponential decay is used to damp traveling waves and high-amplitude oscillations. We demonstrate significant improvements in the runtime of well-studied numerical methods when adding in the damping method.

math.NA↗

arXiv2022

Computation of Large-Genus Solutions of the Korteweg-de Vries Equation

We consider the numerical computation of finite-genus solutions of the Korteweg-de Vries equation when the genus is large. Our method applies both to the initial-value problem when spectral data can be computed and to dressing scenarios when spectral data is specified arbitrarily. In order to compute large genus solutions, we employ a weighted Chebyshev basis to solve an associated singular integral equation. We also extend previous work to compute period matrices and the Abel map when the genus is large, maintaining numerical stability. We demonstrate our method on four different classes of solutions. Specifically, we demonstrate dispersive quantization for "box" initial data and demonstrate how a large genus limit can be taken to produce a new class of potentials.

nlin.SI↗

arXiv2022

Randomized matrix-free quadrature: unified and uniform bounds for stochastic Lanczos quadrature and the kernel polynomial method

We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our analysis of spectrum approximation unifies and simplifies several one-off analyses for these algorithms which have appeared over the past decade. In addition, we derive bounds for spectral sum approximation which guarantee that, with high probability, the algorithms are simultaneously accurate on all bounded analytic functions. Finally, we provide comprehensive and complimentary numerical examples. These examples illustrate some of the qualitative similarities and differences between the algorithms, as well as relative drawbacks and benefits to their use on different types of problems.

math.NA↗

arXiv2022

Growth factors of random butterfly matrices and the stability of avoiding pivoting

Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.

math.NA↗