Automatic Deformation of Riemann–Hilbert Problems with Applications to the Painlevé II Transcendents
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The stability and convergence rate of Olver’s collocation method for the numerical solution of Riemann–Hilbert problems (RHPs) are 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 Painlevé II transcendents.
KeywordsRiemann–Hilbert problems Contour deformation Preconditioning Discrete optimization Greedy algorithm Painlevé II
This research was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics.”