Solving forward and inverse Helmholtz equations via controllability methods

Abstract

Waves are useful for probing an unknown medium by illuminating it with a source. To infer the characteristics of the medium from (boundary) measurements, for instance, one typically formulates inverse scattering problems in frequency domain as a PDE-constrained optimization problem. Finding the medium, where the simulated wave field matches the measured (real) wave field, the inverse problem requires the repeated solutions of forward (Helmholtz) problems. Typically, standard numerical methods, e.g. direct solvers or iterative methods, are used to solve the forward problem. However, large-scaled (or high-frequent) scattering problems are known being competitive in computation and storage for standard methods. Moreover, since the optimization problem is severely ill-posed and has a large number of local minima, the inverse problem requires additional regularization akin to minimizing the total variation. Finding a suitable regularization for the inverse problem is critical to tackle the ill-posedness and to reduce the computational cost and storage requirement. In my thesis, we first apply standard methods to forward problems. Then, we consider the controllability method (CM) for solving the forward problem: it instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period with the conjugate gradient (CG) method, the CMCG method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. Based on the original CM, introduced in 1994 by Bristeau et al., for sound-soft scattering problems, we extend the CMCG method to general boundary-value problems governed by the Helmholtz equation. Numerical results not only show the usefulness, robustness, and efficiency of the CMCG method for solving the forward problem, but also demonstrate remarkably accurate solutions. Second, we formulate the PDE-constrained optimization problem governed by the inverse scattering problem to reconstruct the unknown medium. Instead of a grid-based discrete representation combined with standard Tikhonov-type regularization, the unknown medium is projected to a small finite-dimensional subspace, which is iteratively adapted using dynamic thresholding. The adaptive (spectral) space is governed by solving several Poisson-type eigenvalue problems. To tackle the ill-posedness that the Newton-type optimization method converges to a false local minimum, we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy. Numerical examples illustrate the usefulness of the ASI approach, which not only efficiently and remarkably reduces the dimension of the solution space, but also yields an accurate and robust method.