Fast automatically differentiable matrix functions and applications in molecular simulations

TL;DR

This paper introduces efficient methods for differentiating complex matrix functions like the matrix logarithm and roots, enabling accurate gradient computations crucial for applications such as analyzing defects in silicon crystals.

Contribution

The authors develop contour integral-based differentiation techniques for matrix functions, improving computational efficiency and accuracy over traditional finite difference methods.

Findings

Methods achieve high numerical accuracy

Computational complexity is optimized

Applied to defect analysis in silicon crystals

Abstract

We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log(A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal maps as described by (Hale et al., SIAM J. Numer. Anal. 2008) for evaluation and differentiation and analyze the computational complexity as well as numerical accuracy compared to high accuracy finite difference methods. As a demonstrator application we compute properties of structural defects in silicon crystals at positive temperatures, requiring efficient and accurate gradients of matrix trace-logarithms.