A numerical procedure and unified formulation for the adjoint approach in hyperbolic PDE-constrained optimal control problems

TL;DR

This paper introduces a unified numerical approach for PDE-constrained optimization using the adjoint method, employing a non-conservative hyperbolic system and finite volume scheme for efficient forward and backward evolution.

Contribution

It presents a novel unified formulation and finite volume scheme for hyperbolic PDE-constrained optimization problems with the adjoint approach.

Findings

The method effectively handles the constraint PDE and adjoint model in a single framework.

Stable time stepping is achieved with a single parameter per iteration.

Numerical tests demonstrate the approach's applicability and stability.

Abstract

The present paper aims at providing a numerical strategy to deal with PDE-constrained optimization problems solved with the adjoint method. It is done through out a unified formulation of the constraint PDE and the adjoint model. The resulting model is a non-conservative hyperbolic system and thus a finite volume scheme is proposed to solve it. In this form, the scheme sets in a single frame both constraint PDE and adjoint model. The forward and backward evolutions are controlled by a single parameter $\eta$ and a stable time step is obtained only once at each optimization iteration. The methodology requires the complete eigenstructure of the system as well as the gradient of the cost functional. Numerical tests evidence the applicability of the present technique