A penalty method for PDE-constrained optimization in inverse problems

TL;DR

This paper introduces a quadratic penalty method for PDE-constrained inverse problems that balances the all-at-once and reduced approaches, reducing non-linearity and initial sensitivity.

Contribution

It proposes a novel penalty-based algorithm that combines advantages of existing methods, improving efficiency and robustness in PDE-constrained optimization.

Findings

Reduces non-linearity of inverse PDE problems

Less sensitive to initial guesses

Comparable computational complexity to reduced methods

Abstract

Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand-sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the paramaters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the non-linearity of the problem and is less sensitive the initial iterate.