Mode-based derivation of adjoint equations - a lazy man's approach

TL;DR

This paper introduces a novel modal-based method for deriving adjoint equations using an extended Arnoldi factorization, offering an alternative to traditional continuous, discrete, and automatic differentiation approaches.

Contribution

It presents the dynamical Arnoldi method (DAM) for approximating operators in adjoint derivation, applicable to non-symmetric coupled PDEs, expanding the toolkit for adjoint computations.

Findings

Successfully applied to Burgers and Euler equations.

Demonstrated effectiveness in an optimization problem.

Provides a third alternative for adjoint derivation methods.

Abstract

A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative, third method. It is based on a modal representation of the linearized operator of the governing (primal) system. To approximate the operator an extended version of the Arnoldi factorization, the dynamical Arnoldi method (DAM) is introduced. The DAM allows to derive approximations for operators of non-symmetric coupled equations, which are inaccessible by the classical Arnoldi factorization. The approach is applied to the Burgers equation and to the Euler equations on periodic and non-periodic domains. Finally, it is tested on an optimization problem.