Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation

Mudassir Shams, Andrei Velichko, Bruno Carpentieri

TL;DR

The paper tackles instability in uni-parametric inverse parallel root-finding schemes, where convergence can range from contraction to chaotic transients depending on parameter choices. It couples rigorous stability/bifurcation analysis with a data-driven micro-series Lyapunov profiling pipeline that estimates local exponents $\lambda_1(t)$ from solver trajectories via a kNN-based estimator, enabling adaptive parameter selection. Key contributions include a fractional-order inverse parallel scheme INVM^{\alpha} with memory via the Caputo derivative, a reproducible micro-series Lyapunov workflow, explicit parameter-tuning criteria based on $\lambda_1(t)$, and substantial empirical validation showing robustness and efficiency gains over baselines and the ZHM method. The approach provides a practical, interpretable mechanism for constructing self-stabilizing root-finding schemes and offers a pathway to extend the diagnostics to higher-dimensional and noisy problems.

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.

Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation

TL;DR

The paper tackles instability in uni-parametric inverse parallel root-finding schemes, where convergence can range from contraction to chaotic transients depending on parameter choices. It couples rigorous stability/bifurcation analysis with a data-driven micro-series Lyapunov profiling pipeline that estimates local exponents from solver trajectories via a kNN-based estimator, enabling adaptive parameter selection. Key contributions include a fractional-order inverse parallel scheme INVM^{\alpha} with memory via the Caputo derivative, a reproducible micro-series Lyapunov workflow, explicit parameter-tuning criteria based on , and substantial empirical validation showing robustness and efficiency gains over baselines and the ZHM method. The approach provides a practical, interpretable mechanism for constructing self-stabilizing root-finding schemes and offers a pathway to extend the diagnostics to higher-dimensional and noisy problems.

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.
Paper Structure (18 sections, 1 theorem, 58 equations, 9 figures, 12 tables)

This paper contains 18 sections, 1 theorem, 58 equations, 9 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Development, Methodology, and Implementation of Lyapunov Profiling
  3. Local Convergence Analysis
  4. Mathematical Description of the Method
  5. Data-Driven Lyapunov Profiling for the Parallel Scheme
  6. Ensemble Generation and Observables for the Test Problem
  7. Base cases and initial vectors.
  8. Parallel scheme and scalar observables.
  9. kNN-Based Lyapunov Estimation from Micro-Series
  10. Micro-series construction.
  11. kNN micro-series Lyapunov estimator.
  12. Sliding-Window Lyapunov Profiles and Parameter Selection
  13. Selection criteria for $\alpha$ based on $\lambda_1(t_{\mathrm{end}})$.
  14. Numerical Results and Implementation Details
  15. Experimental Setup and Data
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\zeta_{1},\ldots,\zeta_{\upsilon}$ be simple roots of a nonlinear equation $f(x)=0$, and assume that the initial approximations $x_{1}^{[0]},\ldots,x_{\upsilon}^{[0]}$ are sufficiently close to the corresponding exact roots. Then the fractional-order INVM$^{\alpha}$ method defined in (1ss) conv

Figures (9)

  • Figure 1: Flowchart of the data-driven Lyapunov profiling pipeline used to diagnose and tune the inverse parallel scheme INVM$^{\alpha}$. The five numbered steps correspond to the stages listed in Section 2.3.
  • Figure 2: Example of the logarithmic error curve and linear regression fit used in the kNN--LLE estimator for Example 1 ($f(x)=x^{3}-1$). The plot corresponds to the observable $\|s_k\|_2$ in Case II with $\alpha = 0$. The slope of the first segment is interpreted as the short-horizon Lyapunov indicator $\lambda_1$, while an optional second segment can be used to characterise longer-horizon behaviour.
  • Figure 3: (a-f): Residual error of the scheme INVM$^{\alpha}$ for utilizing random initial vector for solving (\ref{['1g']})
  • Figure 4: (a-h): Log-error evolution of the residual vector $\|r_k\|_2$ and the correction vector $\|s_k\|_2$ for different values of the parameter $\alpha$. Each panel shows the behavior of the log-error with respect to the window end index $k_{\text{end}}$.
  • Figure 5: (a--e) Residual error of the INVM$^{\alpha}$ scheme using kNN--LLE estimation combined with Lyapunov profiling for the selection of $\alpha=3$ and $\beta=0.1,0.3,0.5,0.7,0.9$ in solving (\ref{['1g']}).
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof