Efficient Nehari manifold optimization algorithms for computing ground state solutions of nonlinear elliptic systems

TL;DR

This paper introduces a novel Riemannian accelerated gradient algorithm on the Nehari manifold for efficiently computing ground state solutions of nonlinear elliptic systems, improving convergence and stability over existing methods.

Contribution

The paper develops a new Riemannian accelerated gradient algorithm on the Nehari manifold that combines nonlinear extrapolation and nonmonotone step-size strategies for better efficiency and robustness.

Findings

Significantly reduces iteration count compared to previous methods.

Effectively computes ground states for multi-component elliptic systems.

Demonstrates robustness under various coupling and potential conditions.

Abstract

This paper presents a class of efficient manifold optimization algorithms for computing the ground state solutions of a semilinear elliptic system, which are unstable saddle points of the variational functional. Variational arguments show that these unstable saddle points can be characterized as the local minimizers of the variational functional constrained to the Nehari manifold $\mathcal{N}$. The Nehari manifold optimization method (NMOM) proposed in [Z. Chen, W. Liu, Z. Xie, and W. Yi. SIAM J. Sci. Comput., 47(4): A2098-A2126, 2025] provides a Riemannian gradient descent framework on $\mathcal{N}$ for such constrained minimization problems. To deal with both the intrinsic instability of the solutions and the increased computational complexity introduced by the coupling between components, we combine the ideas from the NMOM and the Nesterov-type acceleration to develop a new efficient Riemannian accelerated gradient algorithm on $\mathcal{N}$ (RAG-$\mathcal{N}$). The key idea is to perform an easy-to-implement nonlinear extrapolation step on $\mathcal{N}$, followed by a Riemannian steepest-descent update at the extrapolated point. To enhance the robustness, we further incorporate a nonmonotone step-size search strategy into the RAG-$\mathcal{N}$ algorithm, obtaining a variant with improved stability. Numerical experiments show that the RAG-$\mathcal{N}$ algorithms substantially reduce the number of iterations compared with the Riemannian steepest descent algorithm of NMOM. Finally, we apply the RAG-$\mathcal{N}$ algorithms to compute the ground state solutions of semilinear elliptic systems with two, three and four components, and investigate their behavior under different coupling coefficients and various settings, including Gaussian-type external potentials and singular diffusion coefficients.