Convergence of a Normalized Gradient Algorithm for Computing Ground States
This provides rigorous convergence guarantees for a widely used numerical method in physics, addressing a gap in theoretical understanding for a specific class of problems.
The paper proves convergence and provides error estimates for a normalized gradient algorithm (imaginary time evolution method) combined with linearly implicit time integration and finite difference spatial discretization for computing the ground state of the one-dimensional cubic nonlinear Schrödinger equation. The algorithm converges exponentially to a modified soliton, with error estimates depending on discretization parameters.
We consider the approximation of the ground state of the one-dimensional cubic nonlinear Schr{ö}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.