Automatic Deformation of Riemann-Hilbert Problems with Applications to the Painlev\'e II Transcendents

TL;DR

This paper introduces an automated algorithm for deforming contours in Riemann-Hilbert problems, enhancing the stability and efficiency of numerical solutions, demonstrated on Painlevé II transcendents.

Contribution

It presents a novel, fully algorithmic approach to contour deformation in RHPs, replacing manual, asymptotic analysis with a graph-based optimization method.

Findings

Automated contour deformation improves numerical stability.

The method effectively solves Painlevé II RHPs.

Contour deformations are computed as shortest paths in a weighted graph.

Abstract

The stability and convergence rate of Olver's collocation method for the numerical solution of Riemann-Hilbert problems (RHPs) is known to depend very sensitively on the particular choice of contours used as data of the RHP. By manually performing contour deformations that proved to be successful in the asymptotic analysis of RHPs, such as the method of nonlinear steepest descent, the numerical method can basically be preconditioned, making it asymptotically stable. In this paper, however, we will show that most of these preconditioning deformations, including lensing, can be addressed in an automatic, completely algorithmic fashion that would turn the numerical method into a black-box solver. To this end, the preconditioning of RHPs is recast as a discrete, graph-based optimization problem: the deformed contours are obtained as a system of shortest paths within a planar graph weighted by the relative strength of the jump matrices. The algorithm is illustrated for the RHP representing the Painlev\'e II transcendents.