Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation

2026-01-20

AI Summary

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

Technology Category

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• Reasoning under Uncertainty: Stochastic Optimization

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Abstract

Inverse parallel schemes remain indispensable tools for computing the roots of nonlinear systems, yet their dynamical behavior can be unexpectedly rich, ranging from strong contraction to oscillatory or chaotic transients depending on the choice of algorithmic parameters and initial states. A unified analytical-data-driven methodology for identifying, measuring, and reducing such instabilities in a family of uni-parametric inverse parallel solvers is presented in this study. On the theoretical side, we derive stability and bifurcation characterizations of the underlying iterative maps, identifying parameter regions associated with periodic or chaotic behavior. On the computational side, we introduce a micro-series pipeline based on kNN-driven estimation of the local largest Lyapunov exponent (LLE), applied to scalar time series derived from solver trajectories. The resulting sliding-window Lyapunov profiles provide fine-grained, real-time diagnostics of contractive or unstable phases and reveal transient behaviors not captured by coarse linearized analysis. Leveraging this correspondence, we introduce a Lyapunov-informed parameter selection strategy that identifies solver settings associated with stable behavior, particularly when the estimated LLE indicates persistent instability. Comprehensive experiments on ensembles of perturbed initial guesses demonstrate close agreement between the theoretical stability diagrams and empirical Lyapunov profiles, and show that the proposed adaptive mechanism significantly improves robustness. The study establishes micro-series Lyapunov analysis as a practical, interpretable tool for constructing self-stabilizing root-finding schemes and opens avenues for extending such diagnostics to higher-dimensional or noise-contaminated problems.

Problem

• inverse parallel schemes
• nonlinear systems
• dynamical instability
• Lyapunov exponents
• chaotic transients

Innovation

• Lyapunov exponent
• kNN-LLE estimation
• inverse parallel solvers
• bifurcation analysis
• adaptive parameter selection