Related papers: The nonlinear steepest descent method for Riemann-…
Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…
Under investigation in this work is the coupled Sasa-Satsuma equation, which can describe the propagations of two optical pulse envelopes in birefringent fibers. The Riemann-Hilbert problem for the equation is formulated on the basis of the…
In this paper, a descent method for nonsmooth multiobjective optimization problems on complete Riemannian manifolds is proposed. The objective functions are only assumed to be locally Lipschitz continuous instead of convexity used in…
In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the…
We provide new complexity information for the convergence of the Hybrid Steepest Descent Method for solving the Variational Inequality Problem for a strict contraction on Hilbert space over a closed convex set C given either as the fixed…
This is the first in a series of papers which deal with the development of novel methods for solving a system of linear algebraic equations with a time complexity lower than existing algorithms. The NxN system of linear equations, Ax = b,…
We investigate the Cauchy problem of a new higher-order nonlinear Schr\"{o}dinger equation (NHNSE) with weighted Sobolev initial data which is derived by ourselves. By applying $\bar{\partial}$-steepest descent method, we derive the…
We study the Cauchy problem for the reduced Maxwell-Bloch equations with initial data for the electric field in weighted Sobolev spaces, assuming that all atoms initially reside in their ground state. Using the d-bar steepest descent…
Gradient descent methods are fundamental first-order optimization algorithms in both Euclidean spaces and Riemannian manifolds. However, the exact gradient is not readily available in many scenarios. This paper proposes a novel inexact…
In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta…
The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of…
We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous…
In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic…
In this paper, we consider a class of nonconvex-linear minimax problems on Riemannian manifolds, which find wide applications in machine learning and signal processing. For solving this class of problems, we develop a flexible Riemannian…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.
The limited memory steepest descent method (LMSD) proposed by Fletcher is an extension of the Barzilai-Borwein "two-point step size" strategy for steepest descent methods for solving unconstrained optimization problems. It is known that the…
We obtain rigorous large time asymptotics for the Landau-Lifshitz equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous…
We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kru\v{z}kov are obtained as the - a posteriori unique - limit points of the JKO variational…
In a Hilbert framework, for convex differentiable optimization, we consider accelerated gradient methods obtained by combining temporal scaling and averaging techniques with Tikhonov regularization. We start from the continuous steepest…
This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$…