A penalty-free approach to PDE constrained optimization: Application to an inverse wave problem

Inverse wave problems (IWPs) amount in non-linear optimization problems where a certain distance between a state variable and some observations of a wavefield is to be minimized. Additionally, we require the state variable to be the solution of a model equation that involves a set of parameters to be optimized. Typical approaches to solve IWPs includes the adjoint method, which generates a sequence of parameters and strictly enforces the model equation at each iteration, and, the wavefield reconstruction inversion (WRI) method, which jointly generates a sequence of parameters and state variable but does not strictly enforce the model. WRI is considered to be an interesting approach because, by virtue of not enforcing the model at each iteration, it expands the search space, and can thus find solutions that may not be found by a typical adjoint method. However, WRI techniques generally requires the tuning of a penalty parameter until the model equation is considered satisfied. Alternatively, a fixed penalty parameter can be chosen but, in such case, it is impossible for the algorithm to find a solution that satisfies the model equation exactly. In the present work, we present a, to our knowledge, novel technique of WRI type which jointly generates a sequence of parameters and state variable, and which loosely enforces the model. The method is based on a TR-SQP method which aims at minimizing, at each iteration, both the residual relative to the linearized model and a quadratic approximation of the cost functional. Our method approximately solves a sequence of quadratic subproblems by using a Krylov method. The Hessian-vector product is computed using the second-order adjoint method. The method is demonstrated on a synthetic case, with a configuration relevant to medical imaging.