On computing high-dimensional Riemann theta functions

Chimmalgi, S.; Wahls, S.

doi:10.1016/j.cnsns.2023.107266

On computing high-dimensional Riemann theta functions

Title

On computing high-dimensional Riemann theta functions

Author

Chimmalgi, S. (Karlsruhe Institut für Technologie)
Wahls, S. (TU Delft Team Sander Wahls)

Date

2023

Abstract

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.

Subject

Finite-genus solutions
Korteweg–de-Vries equation
Nonlinear Fourier transform
Nonlinear Schrödinger equation
Riemann theta function
Tensor-train decomposition

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http://resolver.tudelft.nl/uuid:87752975-84ee-44d6-8ba4-973ea55ecc73

DOI

https://doi.org/10.1016/j.cnsns.2023.107266

ISSN

1007-5704

Source

Communications in Nonlinear Science and Numerical Simulation, 123

Part of collection

Institutional Repository