Inverse problems governed by partial differential equations (PDEs) is a means to learn, from data, unknown or uncertain aspects of PDE-based mathematical models. PDE-based models often are constructed from scientific principles and are pervasive in science and engineering. The improvement of such a model can be accomplished by determining a ``best'' set of model parameters, or by reducing the uncertainty of the parameters that characterize the model. Ultimately, the goal is to improve the predictive capacity of such models and quantitatively understand the limitations of model-based predictions.

In this dissertation, we describe algorithmic approaches for the Newton-based solution of large-scale computational inverse problems governed by PDEs. We present and motivate hierarchical matrix approximations of the Hessian, a key-component of Newton's method, as a means to exploit localized sensitivities of underlying elliptic PDE operators and which perform well for inverse problems with highly-informative data. To circumvent the computational challenges associated with generating hierarchical matrix approximations of matrix-free operators, such as the Hessian, we describe a local point spread function methodology, whose computational cost is independent of the problem discretization. It is numerically demonstrated that hierarchical matrix approximations of the Hessian can be effective for large-scale ice sheet inverse problems with highly informative data and thin ice sheets. Lastly, we present a scalable means to solve PDE- and bound-constrained optimization problems by an interior-point filter line-search strategy that leverages performant algebraic multigrid linear solvers. Bound constraints arise naturally in many inverse problems as a means to enforce sign-definiteness as dictated by e.g., physical principles. The inclusion of bound constraints does add particular computational challenges such as non-smooth complementarity conditions as part of the conditions for optimality. The Newton linear system solve strategy described in Chapter 4 is accelerated due to the inclusion of bound-constraints, wherein the performance of many other strategies is negatively impacted by such bound-constraints. A theoretical analysis is provided which demonstrates the algorithmic scaling of the approach. In addition, we demonstrate algorithmic and parallel scaling of the described approach by applying the framework to an example elliptic PDE- and bound-constrained problem.

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.