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Direct Finite-Time Contraction (Step-Log) Profiling--Driven Optimization of Parallel Schemes for Nonlinear Problems on Multicore Architectures

Mudassir Shams, Andrei Velichko, Bruno Carpentieri

TL;DR

The paper tackles the challenge of efficiently computing all distinct roots of nonlinear equations using high-order parallel schemes whose performance hinges on parameter choices. It introduces a training-free tuning framework based on Direct finite-time contraction (step-log) profiling, producing two compact scores, $S_{\min}$ and $S_{mom}$, to guide parameter selection for a bi-parametric third-order Weierstrass-type scheme SAB$^{[3]}$. Through ensemble-based trajectory analysis, the approach yields consistent improvements in convergence speed, stability, and robustness across diverse nonlinear problems, including biomedical models, while remaining implementation-light and reproducible. This profiling paradigm provides a practical alternative to analytical or bifurcation-based tuning, with potential for extension to other parallel root-finding families and adaptive strategies in large-scale or real-time contexts.

Abstract

Efficient computation of all distinct solutions of nonlinear problems is essential in many scientific and engineering applications. Although high-order parallel iterative schemes offer fast convergence, their practical performance is often limited by sensitivity to internal parameters and the lack of reproducible tuning procedures. Classical parameter selection tools based on analytical conditions and dynamical-system diagnostics can be problem-dependent and computationally demanding, which motivates lightweight data-driven alternatives. In this study, we propose a parameterized single-step bi-parametric parallel Weierstrass-type scheme with third-order convergence together with a training-free tuning framework based on Direct finite-time contraction (step-log) profiling. The approach extracts Lyapunov-like finite-time contraction information directly from solver trajectories via step norms and step-log ratios, aggregates the resulting profiles over micro-launch ensembles, and ranks parameter candidates using two compact scores: the stability minimum S_min and the stability moment S_mom. Numerical results demonstrate consistent improvements in convergence rate, stability, and robustness across diverse nonlinear test problems, establishing the proposed profiling-based strategy as an efficient and reproducible alternative to classical parameter tuning methods.

Direct Finite-Time Contraction (Step-Log) Profiling--Driven Optimization of Parallel Schemes for Nonlinear Problems on Multicore Architectures

TL;DR

The paper tackles the challenge of efficiently computing all distinct roots of nonlinear equations using high-order parallel schemes whose performance hinges on parameter choices. It introduces a training-free tuning framework based on Direct finite-time contraction (step-log) profiling, producing two compact scores, and , to guide parameter selection for a bi-parametric third-order Weierstrass-type scheme SAB. Through ensemble-based trajectory analysis, the approach yields consistent improvements in convergence speed, stability, and robustness across diverse nonlinear problems, including biomedical models, while remaining implementation-light and reproducible. This profiling paradigm provides a practical alternative to analytical or bifurcation-based tuning, with potential for extension to other parallel root-finding families and adaptive strategies in large-scale or real-time contexts.

Abstract

Efficient computation of all distinct solutions of nonlinear problems is essential in many scientific and engineering applications. Although high-order parallel iterative schemes offer fast convergence, their practical performance is often limited by sensitivity to internal parameters and the lack of reproducible tuning procedures. Classical parameter selection tools based on analytical conditions and dynamical-system diagnostics can be problem-dependent and computationally demanding, which motivates lightweight data-driven alternatives. In this study, we propose a parameterized single-step bi-parametric parallel Weierstrass-type scheme with third-order convergence together with a training-free tuning framework based on Direct finite-time contraction (step-log) profiling. The approach extracts Lyapunov-like finite-time contraction information directly from solver trajectories via step norms and step-log ratios, aggregates the resulting profiles over micro-launch ensembles, and ranks parameter candidates using two compact scores: the stability minimum S_min and the stability moment S_mom. Numerical results demonstrate consistent improvements in convergence rate, stability, and robustness across diverse nonlinear test problems, establishing the proposed profiling-based strategy as an efficient and reproducible alternative to classical parameter tuning methods.
Paper Structure (29 sections, 1 theorem, 48 equations, 16 figures, 9 tables)

This paper contains 29 sections, 1 theorem, 48 equations, 16 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Proposed Parallel Schemes and Convergence Methodology
  3. Local Convergence Analysis
  4. Mathematical Description of the Method
  5. Methodology and Parameter Optimization Framework
  6. Why data-driven tuning is needed
  7. Direct finite-time contraction (step-log) profiling
  8. Profile-based scores $S_{\min}$ and $S_{mom}$
  9. Parameter scan and selection rule
  10. Numerical Experiments and Results
  11. General Numerical Methodology
  12. Implementation Framework
  13. Computational Environment
  14. Numerical Outcomes
  15. Overall Assessment.
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\zeta _{1},\ldots ,\zeta _{\upsilon }$ be simple zeros of a nonlinear equation $f(x)=0$. If the initial approximations $x_{1}^{[0]},\ldots ,x_{\upsilon }^{[0]}$ are sufficiently close to their corresponding exact roots, then the order SAB$^{[3]}$ method defined above converges with order three.

Figures (16)

  • Figure 1: Example of an aggregated step-log contraction profile and the extraction of the profile-based scores. The solid curve shows the ensemble mean $\bar{\lambda}^{(s)}_{W}(t_{\mathrm{end}})$ and the dashed curves indicate mean$\pm$std across $N=50$ micro-launches (here $W=10$; parameters are shown in the plot title).
  • Figure 2: Residual error histories of the scheme for $(\alpha,\beta)=(-16.5,\,19)$.
  • Figure 3: Two-dimensional maps of the proposed profile-based scores over the $(\alpha,\beta)$ plane. Brighter colors correspond to earlier and stronger contractive behavior in the step-log profile.
  • Figure 4: Aggregated step-log Lyapunov profile for $(\alpha,\beta)=(13.15,\,0.4615)$ ($W=10$, $N=50$). The solid curve shows the ensemble-averaged profile, while dashed curves indicate mean$\pm$std across micro-launches.
  • Figure 5: Step-log profile for $(\alpha,\beta)=(6.385,\,0.4615)$ with metric annotations. The detected minimum defines $S_{\min}$, while the negative mass and its centroid determine $S_{\mathrm{mom}}$.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof