Hessian Approximations for Large-Scale Inverse Problems Governed By Partial Differential Equations
Inverse problems abound in all areas of science, engineering, andbeyond. These can be seen as tools that can be used to refine mathematical models using measurement data. Here by refine we mean, estimate unknown or uncertain input parameters that cannot directly be measured. This is an important task, since the quality and predictability of the mathematical models relies on the ability to estimate these parameters as accurately as possible. In this thesis, we focus on a particular class of inverse problems, namely on inverse problems governed by partial differential equations (PDEs). These inverse problems are formulated as nonlinear least squares optimization problems constrained by PDEs. The major part of the thesis is devoted to developing efficient computational strategies to solve these optimization problems. To this end, we focus on derivative-based optimization methods, e.g., quasi-Newton and Newton. The first- and second-order (when applicable) derivative information is derived using adjoint-based techniques. For quasi-Newton, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods. For Newtons' method, we aim to reduce the computational cost (measured in PDE solves) per Newton iteration. There are a number of existing approaches in the literature that target this goal. For instance, via efficient preconditioning of the underlying Newton system, inexact Newton-CG solves, via low-rank approximations of the second-order derivative (Hessian) of the optimization objective, and via inexact Hessian-vector products (i.e., inexact second-order adjoint solves). In this thesis we focus on the latter and derive bounds for tolerances for inexact PDE solves required by the Hessian apply. We apply these tolerances for an inverse problem governed by a Poisson problem and show that relaxing the Hessian apply can lead to an overall reduced number of PDE solves.
In the last part of the thesis, we go beyond a deterministic setup andquantify the uncertainties in the solution of inverse problems. To this end, we adopt the framework of Bayesian inference which allows us to systematically take into account noisy observations, uncertain models and prior knowledge about the unknown. The problem of interest is the estimation of the basal sliding coefficient field for an uncertain thermally-dependent nonlinear Stokes ice sheet model. The novelty in this inverse problem is the uncertainty in the forward model in addition to the uncertain basal sliding coefficient field. This additional uncertainty stems from the unknown temperature distribution within the ice, which is dictated by both the unknown thermal conductivity and unknown geothermal heat flux. To account for model uncertainties, we use the Bayesian approximation error (BAE) approach combined with a variance reduction technique. Preliminary results indicate that the BAE approach can be used to account for model uncertainties induced by the unknown thermal properties of the ice, and that failure to take into account these uncertainties can lead to erroneous estimates. In addition, we show that BAE combined with a variance reduction technique has the potential to reduce the offline costs of the BAE approach.