6,595 research outputs found
On computing high-dimensional Riemann theta functions
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
Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions
This is the second in a series of papers on the numerical treatment of
hyperelliptic theta-functions with spectral methods. A code for the numerical
evaluation of solutions to the Ernst equation on hyperelliptic surfaces of
genus 2 is extended to arbitrary genus and general position of the branch
points. The use of spectral approximations allows for an efficient calculation
of all characteristic quantities of the Riemann surface with high precision
even in almost degenerate situations as in the solitonic limit where the branch
points coincide pairwise. As an example we consider hyperelliptic solutions to
the Kadomtsev-Petviashvili and the Korteweg-de Vries equation. Tests of the
numerics using identities for periods on the Riemann surface and the
differential equations are performed. It is shown that an accuracy of the order
of machine precision can be achieved.Comment: 16 pages, 8 figure
Hyperelliptic Theta-Functions and Spectral Methods
A code for the numerical evaluation of hyperelliptic theta-functions is
presented. Characteristic quantities of the underlying Riemann surface such as
its periods are determined with the help of spectral methods. The code is
optimized for solutions of the Ernst equation where the branch points of the
Riemann surface are parameterized by the physical coordinates. An exploration
of the whole parameter space of the solution is thus only possible with an
efficient code. The use of spectral approximations allows for an efficient
calculation of all quantities in the solution with high precision. The case of
almost degenerate Riemann surfaces is addressed. Tests of the numerics using
identities for periods on the Riemann surface and integral identities for the
Ernst potential and its derivatives are performed. It is shown that an accuracy
of the order of machine precision can be achieved. These accurate solutions are
used to provide boundary conditions for a code which solves the axisymmetric
stationary Einstein equations. The resulting solution agrees with the
theta-functional solution to very high precision.Comment: 25 pages, 12 figure
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
Schottky Algorithms: Classical meets Tropical
We present a new perspective on the Schottky problem that links numerical
computing with tropical geometry. The task is to decide whether a symmetric
matrix defines a Jacobian, and, if so, to compute the curve and its canonical
embedding. We offer solutions and their implementations in genus four, both
classically and tropically. The locus of cographic matroids arises from
tropicalizing the Schottky-Igusa modular form.Comment: 17 page
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