An Efficient Memory Gradient Method for Extreme M-Eigenvalues of Elastic type Tensors

TL;DR

This paper introduces a memory gradient method to efficiently compute extreme M-eigenvalues of elastic tensors, crucial for material analysis and quantum problems, with proven convergence and demonstrated effectiveness.

Contribution

We propose a novel memory gradient algorithm for extreme M-eigenvalues of elastic tensors, including convergence analysis and numerical validation.

Findings

The method converges globally.

Numerical experiments show high accuracy.

The approach is stable and efficient.

Abstract

M-eigenvalues of fourth order hierarchically symmetric tensors play a significant role in nonlinear elastic material analysis and quantum entanglement problems. This paper focuses on computing extreme M-eigenvalues for such tensors. To achieve this, we first reformulate the M-eigenvalue problem as a sequence of unconstrained optimization problems by introducing a shift parameter. Subsequently, we develop a memory gradient method specifically designed to approximate these extreme M-eigenvalues. Under this framework, we establish the global convergence of the proposed method. Finally, comprehensive numerical experiments demonstrate the efficacy and stability of our approach.