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Numerical analysis of spatiotemporal high-index saddle dynamics for finding multiple solutions of semilinear elliptic problems

Lei Zhang, Xiangcheng Zheng, Shangqin Zhu

TL;DR

The paper addresses computing multiple solutions of semilinear elliptic PDEs by extending high-index saddle dynamics (HiSD) to a spatiotemporal setting. It introduces a fully discrete, retraction-free orthonormality-preserving scheme that operates in full space-time and avoids retraction-based updates on the Stiefel manifold, while maintaining the orthogonality of auxiliary modes. The authors establish gradient stability, derive error estimates, and prove Morse index preservation for computed saddles, and extend the framework to semilinear advection-reaction-diffusion models. Numerical experiments in one and two dimensions demonstrate the method’s effectiveness in locating multiple solutions and constructing the solution landscape, validating both accuracy and index preservation. This work provides the first rigorous full space-time accuracy analysis of HiSD and strengthens the connection between saddle-search algorithms and PDE numerical methods, with broad implications for complex infinite-dimensional systems.

Abstract

This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold constraint without introducing the analytical complications of retraction-based updates, we design a fully discrete retraction-free orthonormality-preserving scheme for spatiotemporal HiSD. This scheme exhibits favorable structural properties that substantially reduce the difficulties arising from coupling and gradient nonlinearities in spatiotemporal HiSD. Exploiting these properties, we establish gradient stability and error estimates, which consequently ensure the preservation of the Morse index for the computed saddle points. The framework is further extended to the semilinear advection-reaction-diffusion equation. Numerical experiments demonstrate the efficiency of the proposed method in finding multiple solutions and constructing the solution landscape of semilinear elliptic problems. To the best of our knowledge, this work presents the first rigorous full space--time accuracy analysis of the HiSD system. It reveals intrinsic connections between saddle-search algorithms and numerical methods for PDEs, enhancing their mutual compatibility for a broad range of problems.

Numerical analysis of spatiotemporal high-index saddle dynamics for finding multiple solutions of semilinear elliptic problems

TL;DR

The paper addresses computing multiple solutions of semilinear elliptic PDEs by extending high-index saddle dynamics (HiSD) to a spatiotemporal setting. It introduces a fully discrete, retraction-free orthonormality-preserving scheme that operates in full space-time and avoids retraction-based updates on the Stiefel manifold, while maintaining the orthogonality of auxiliary modes. The authors establish gradient stability, derive error estimates, and prove Morse index preservation for computed saddles, and extend the framework to semilinear advection-reaction-diffusion models. Numerical experiments in one and two dimensions demonstrate the method’s effectiveness in locating multiple solutions and constructing the solution landscape, validating both accuracy and index preservation. This work provides the first rigorous full space-time accuracy analysis of HiSD and strengthens the connection between saddle-search algorithms and PDE numerical methods, with broad implications for complex infinite-dimensional systems.

Abstract

This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold constraint without introducing the analytical complications of retraction-based updates, we design a fully discrete retraction-free orthonormality-preserving scheme for spatiotemporal HiSD. This scheme exhibits favorable structural properties that substantially reduce the difficulties arising from coupling and gradient nonlinearities in spatiotemporal HiSD. Exploiting these properties, we establish gradient stability and error estimates, which consequently ensure the preservation of the Morse index for the computed saddle points. The framework is further extended to the semilinear advection-reaction-diffusion equation. Numerical experiments demonstrate the efficiency of the proposed method in finding multiple solutions and constructing the solution landscape of semilinear elliptic problems. To the best of our knowledge, this work presents the first rigorous full space--time accuracy analysis of the HiSD system. It reveals intrinsic connections between saddle-search algorithms and numerical methods for PDEs, enhancing their mutual compatibility for a broad range of problems.
Paper Structure (13 sections, 11 theorems, 59 equations, 4 figures, 1 table)

This paper contains 13 sections, 11 theorems, 59 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and challenges
  3. Spatiotemporal HiSD
  4. Contributions of the work
  5. Preliminaries
  6. Scheme and orthonormality preservation
  7. Numerical scheme
  8. Orthonormality preservation
  9. Estimates of gradients
  10. Error estimates and index preservation
  11. Extension to advection-reaction-diffusion model
  12. Numerical experiments
  13. Concluding remarks

Key Result

Theorem 1

\newlabelorth0 If the initial values $\{v_{i,0}\}_{i=1}^k$ of (SD) satisfy $(v_{i,0},v_{j,0})=\delta_{ij}$ for $1\leq i,j \leq k$, then $(v_{i},v_{j})=\delta_{ij}$ for $1\leq i,j \leq k$ and for any $t>0$.

Figures (4)

  • Figure 1: Numerical solutions of $u$ at different time instants for the orthonormality-preserving scheme (\ref{['app']}) (red) and its non-orthonormal variant (green). Here, $F:= \|F(u_h^n)\|_{l^\infty}$ (cf. Section \ref{['sec7']}) measures the degree of convergence and $K$ denotes the number of negative eigenvalues of the Hessian at each state.
  • Figure 1: Plots of $u_h(T)$ and $F(u_h(T))$ in Example 1.
  • Figure 2: The first six smallest eigenvalues of the Hessian at $u_h(T)$ in Example 1.
  • Figure 3: Solution landscape for (\ref{['elliptic']}) in Example 2.

Theorems & Definitions (23)

  • Remark 2.1
  • Theorem 1
  • Proof 1
  • Remark 3.1
  • Lemma 1
  • Remark 3.2
  • Proof 2
  • Theorem 2
  • Proof 3
  • Theorem 1
  • ...and 13 more