Domenico Lahaye and Wouter Mulckhuyse
The purpose of this paper is to provide a framework for the implementation of an adjoint sensitivity formulation for least‐squares partial differential equations constrained…
Abstract
Purpose
The purpose of this paper is to provide a framework for the implementation of an adjoint sensitivity formulation for least‐squares partial differential equations constrained optimization problems exploiting a multiphysics finite elements package. The estimation of the diffusion coefficient in a Poisson‐type diffusion equation is used as an example.
Design/methodology/approach
The authors derive the adjoint formulation in a continuous setting allowing to attribute to the direct and adjoint states the role of different fields to be solved for. They are one‐way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. Having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post‐processing procedure. This sensitivity can then be used to efficiently solve the least‐squares problem.
Findings
The authors derived the adjoint formulation in a continuous setting allowing the direct and adjoint states to be attributed the role of different fields to be solved. They are one‐way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. It is found that, having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post‐processing procedure.
Research limitations/implications
This paper implies that modern multiphysics finite elements packages provide a flexible and extendable software environment for the experimentation with different adjoint formulations. Such tools are therefore expected to become increasingly important in solving notoriously difficult partial differential equation (PDE)‐constrained least‐squares problems. The framework also provides the possibility of experimentation with different regularization techniques (total variation and multiscale techniques for instance) to handle the ill‐posedness of the problem.
Originality/value
In this paper the adjoint sensitivity computation is casted as a multiphysics problem allowing for a flexible and extendable implementation.