Convergence of a Normalized Gradient Algorithm for Computing Ground States

TL;DR

This paper proves the convergence and provides error estimates for a normalized gradient algorithm used to approximate ground states of the 1D cubic nonlinear Schrödinger equation, combining theoretical analysis with numerical discretization.

Contribution

It establishes the convergence and error bounds of the imaginary time evolution method with finite difference discretization for ground state computation.

Findings

The algorithm converges exponentially to a discretized ground state.

Error estimates depend on discretization parameters.

The method is proven to be convergent with quantifiable accuracy.

Abstract

We consider the approximation of the ground state of the one-dimensional cubic nonlinear Schr{\"o}dinger equation by a normalized gradient algorithm combined with linearly implicit time integrator, and finite difference space approximation. We show that this method, also called imaginary time evolution method in the physics literature, is con-vergent, and we provide error estimates: the algorithm converges exponentially towards a modified solitons that is a space discretization of the exact soliton, with error estimates depending on the discretization parameters.