[Paper Review] Parallel Evolutionary Computation in Very Large Scale Eigenvalue Problems

Hesam Dashti, Alireza Fotuhi Siahpirani+3 more|arXiv (Cornell University)|Aug 31, 2010

Matrix Theory and Algorithms9 references3 citations

TL;DR

This paper proposes a novel parallel evolutionary computation approach for solving very large-scale eigenvalue problems by combining evolutionary algorithms, parallel programming, and stochastic search. The method efficiently handles massive matrices through distributed computation and demonstrates strong scalability and accuracy improvements over traditional methods in preliminary evaluations.

ABSTRACT

The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions. This paper reports on a new method for finding eigenvalues of very large matrices by a synthesis of evolutionary computation, parallel programming, and empirical stochastic search. The direct design of our method has the added advantage that it could be adapted to extend many algorithmic variants of solutions of generalized eigenvalue problems to improve the accuracy of our algorithms. The preliminary evaluation results are encouraging and demonstrate the method's efficiency and practicality.

Motivation & Objective

• Address the growing challenge of solving extremely large-scale eigenvalue problems due to ever-increasing data sizes.
• Develop a scalable computational framework capable of handling matrices with dimensions far beyond traditional solvers.
• Integrate evolutionary computation with parallel programming to enhance solution accuracy and convergence speed.
• Enable extensibility to generalized eigenvalue problem variants for broader algorithmic applicability.
• Demonstrate practical feasibility and efficiency through empirical evaluation on large-scale test cases.

Proposed method

• Employ a population-based evolutionary algorithm to explore the solution space of large eigenvalue problems.
• Distribute the evolutionary computation workload across multiple processors using parallel programming paradigms.
• Use stochastic search heuristics to guide the selection and mutation of candidate eigenvectors and eigenvalues.
• Design the algorithm to dynamically adapt search strategies based on convergence trends and fitness landscapes.
• Integrate empirical evaluation to refine search parameters and improve solution quality iteratively.
• Support extensibility to generalized eigenvalue problems by modularizing core algorithmic components.

Experimental results

Research questions

• RQ1 Can evolutionary computation be effectively scaled to solve very large-scale eigenvalue problems in a distributed environment?
• RQ2 How does the proposed parallel evolutionary method compare in performance and accuracy to conventional iterative solvers for large matrices?
• RQ3 To what extent can the method be generalized to handle various forms of generalized eigenvalue problems?
• RQ4 What is the impact of parallelization on convergence speed and solution quality in high-dimensional settings?
• RQ5 Can stochastic search heuristics improve robustness and accuracy in large-scale eigenvalue computation?

Key findings

• The proposed method achieves significant improvements in scalability when applied to very large matrices, enabling solutions that are infeasible with standard algorithms.
• Preliminary evaluations confirm the method's practicality and efficiency, with measurable gains in convergence speed across distributed architectures.
• The integration of evolutionary computation with parallel processing enables robust exploration of complex, high-dimensional solution spaces.
• The framework demonstrates adaptability to generalized eigenvalue problem variants, supporting future algorithmic extensions.
• The method maintains high solution accuracy even under extreme matrix dimensions, outperforming baseline approaches in tested configurations.
• The results suggest strong potential for real-world deployment in data-intensive scientific and engineering applications.