Mode-based derivation of adjoint equations - a lazy man's approach
This work provides an alternative approach to adjoint derivation for PDEs, potentially useful for researchers in computational optimization, though it is an incremental method compared to existing continuous and discrete adjoint techniques.
The paper introduces a third method for deriving adjoint equations, based on a modal representation and the dynamical Arnoldi method (DAM), and demonstrates its application to Burgers and Euler equations, showing feasibility for optimization problems.
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.