A spectral Levenberg-Marquardt-Deflation method for multiple solutions of semilinear elliptic systems

TL;DR

The paper develops a spectral Levenberg-Marquardt-Deflation framework for finding multiple solutions of semilinear elliptic systems. It combines Legendre-Galerkin discretization with a trust-region Levenberg-Marquardt iteration and a deflation mechanism to sequentially uncover distinct solutions without relying on Hessian information. The approach demonstrates strong accuracy and efficiency across ODE and PDE examples, uncovering new solutions not previously reported (e.g., in the Gray-Scott model) and handling high-dimensional discrete systems. This method offers a robust, scalable avenue for exploring nonlinear elliptic systems with multiple steady states and has potential for broad applicability in physics, chemistry, and biology.

Abstract

Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve some applications. Developing an efficient numerical method for finding multiple solutions is very necessary due to the nonlinearity and multiple solutions of these equations. Moreover, providing an efficient iteration plays an important role in successfully obtaining multiple solutions with fast and stable convergence. In the current paper, an efficient algorithm for finding multiple solutions of semilinear elliptic systems is proposed, where the trust region Levenberg-Marquardt method is firstly used to iterate the resulted nonlinear algebraic system. When the nonlinear term in these equations has only the first derivative, our algorithm can efficiently find multiple solutions as well. Several numerical experiments are tested to show the efficiency of our algorithm, and some solutions which have not been shown in the literature are also found and shown.

Figures (21)

• Figure 2.1: The simple flow configuration of our algorithm.
• Figure 3.1: Multiple solutions of the Schnakenberg model with $d_2 = 50.$
• Figure 3.2: Multiple solutions of the Schnakenberg model with $d_2 = 70.$
• Figure 3.3: Multiple solutions of the Gray-Scott model (Red: $u_{24}(x)$; Blue: $v_{24}(x)$).
• Figure 3.4: New solutions of the Gray-Scott model (Red: $u_{24}(x)$; Blue: $v_{24}(x)$).