[Paper Review] Convergence of a Normalized Gradient Algorithm for Computing Ground States

Erwan Faou, Tiphaine Jézéquel|arXiv (Cornell University)|Mar 8, 2016

Advanced Mathematical Physics Problems14 references19 citations

TL;DR

This paper proves the convergence of a normalized gradient algorithm—known as the imaginary time method—for computing ground states of the one-dimensional cubic nonlinear Schrödinger equation. Using a linearly implicit time integrator with finite difference spatial discretization, the method converges exponentially to a discrete soliton that approximates the exact ground state, with error bounds depending on spatial mesh size $h$ and domain truncation $K$. The key contribution is a rigorous convergence analysis with explicit error estimates in the $H^1$ norm.

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.

Motivation & Objective

• To rigorously establish convergence of the normalized gradient (imaginary time) method for computing ground states of the one-dimensional cubic nonlinear Schrödinger equation.
• To analyze the error between the computed solution and the exact soliton in the presence of spatial and temporal discretization.
• To demonstrate that the linearly implicit time scheme preserves the discrete ground state exactly, ensuring stability and accuracy.
• To provide quantitative error estimates for the fully discrete algorithm, including dependence on mesh size $h$, time step $\tau$, and domain truncation $K$.

Proposed method

• The algorithm uses a linearly implicit time integrator to solve the parabolic equation $\partial_t \psi = \frac{1}{2}\Delta\psi + |\psi|^2\psi$, which corresponds to the negative $L^2$-gradient of the energy functional.
• The scheme is defined by $\psi_n^* = \psi_n - \tau \widehat{\nabla H}(\psi_n, \psi_n^*)$, where $-\widehat{\nabla H}(\psi_n, \psi_n^*) = \frac{1}{2}\Delta_h \psi_n^* + |\psi_n|^2 \psi_n^*$, ensuring unconditional stability and exact preservation of the ground state.
• After each time step, the solution is normalized in $L^2$ via $\psi_{n+1} = \psi_n^* / \|\psi_n^*\|_{L^2}$ to maintain unit mass.
• Spatial discretization is performed using central finite differences on a truncated domain $|x| \leq Kh$, with zero boundary conditions, and the discrete $H^1$ norm is used to measure convergence.
• The discrete ground state $\eta_{h,K}$ is defined as the unique minimizer of the energy functional under unit $L^2$ norm, satisfying a discrete eigenvalue equation.
• A local coordinate system is constructed around $\eta_{h,K}$ to analyze stability and convexity, enabling the proof of exponential convergence.

Experimental results

Research questions

• RQ1 Does the normalized gradient algorithm with linearly implicit time integration converge to the true ground state of the 1D cubic NLS equation under spatial and temporal discretization?
• RQ2 What is the rate of convergence of the algorithm, and how does it depend on the discretization parameters $h$, $\tau$, and $K$?
• RQ3 Can the linearly implicit scheme preserve the discrete ground state exactly, and does this property enhance convergence and stability?
• RQ4 How close is the computed discrete soliton $\eta_{h,K}$ to the exact continuous soliton $\eta$ in the $H^1$ norm?
• RQ5 Do alternative schemes (semi-explicit or fully implicit) also converge, and what are their error bounds in the discrete setting?

Key findings

• The algorithm converges exponentially in time to the discrete ground state $\eta_{h,K}$, with $\|\psi_n - \eta_{h,K}\|_h \leq C e^{-cn\tau}$ for constants $C, c > 0$ independent of $h$, $K$, and $\tau$.
• The discrete ground state $\eta_{h,K}$ satisfies a discrete eigenvalue equation $\frac{1}{2}(\Delta_h \eta_{h,K})^\ell + |\eta_{h,K}^\ell|^2 \eta_{h,K}^\ell = \lambda_h \eta_{h,K}^\ell$, ensuring exact preservation by the scheme.
• The error between the continuous soliton $\eta$ and the discrete soliton $\eta_{h,K}$ is bounded by $\|i_h \eta_{h,K} - \eta\|_{H^1} \leq C(h + \frac{1}{h^2} e^{-C_1 h K})$, showing spectral convergence in $K$ and algebraic in $h$.
• The fully discrete solution $\psi_n$ satisfies $\|i_h \psi_n - \eta\|_{H^1} \leq C(e^{-cn\tau} + h + \frac{1}{h^2} e^{-C_1 h K})$, combining temporal exponential decay and spatial discretization error.
• The linearly implicit scheme preserves the ground state exactly, a property not shared by semi-explicit or fully implicit alternatives, which enhances numerical stability and accuracy.
• For alternative schemes (semi-explicit or fully implicit), the solution converges to a modified discrete ground state $\eta_{\tau,h,K}$ with error $\|i_h \eta_{\tau,h,K} - \eta\|_{H^1} \leq C(\tau + h + \frac{1}{h^2} e^{-C_1 h K})$, showing that time discretization introduces an additional $\tau$-dependent error.