Derivatives for Eigenvalues and Eigenvectors via Analytic Reverse Algorithmic Differentiation

Abstract

Eigenvalue and eigenvectors are essential metrics to characterize dynamic system behavior and stability. When performing gradient-based design optimization, derivatives of these metrics are required. Analytic forward algorithmic differentiation (FAD) for a self-adjoint generalized eigenproblem has been a useful technique. However, reverse algorithmic differentiation (RAD) is preferred over FAD because it scales more favorably with the number of design variables. We propose two RAD formulas based on their mode-based FAD counterparts that project the derivative onto a reduced eigenvector space. One challenge for the mode-based derivative is that the reduced eigenvectors yield inexact gradient results. An approximation technique mitigates this issue. We verify the proposed methods by implementing a reverse Lanczos iteration and the adjoint of an Euler–Bernoulli beam test case.