Optimal a priori discretization error bounds for geodesic finite elements

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an H1-type Finsler norm and with the H1-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem

Numerical methods for optimal transport and optimal information transport on the sphere

The primary contribution of this dissertation is in developing and analyzing efficient, provably convergent numerical schemes for solving fully nonlinear elliptic partial differential equation arising from Optimal Transport on the sphere, and then applying and adapting the methods to two specific engineering applications: the reflector antenna problem and the moving mesh methods problem. For these types of nonlinear partial differential equations, many numerical studies have been done in recent years, the vast majority in subsets of Euclidean space. In this dissertation, the first major goal is to develop convergent schemes for the sphere. However, another goal of this dissertation is application-centered, that is evaluating whether the partial differential equation techniques using Optimal Transport are actually the best methods for solving such problems. The reflector antenna is an optics inverse problem where one finds the shape of a reflector surface in order to refocus light into a prescribed far-field output intensity. This problem can be solved using Optimal Transport. The moving mesh methods problem is an adaptive mesh technique where one redistributes the density of the vertices of a mesh without tangling the edges connecting the vertices. Both Optimal Transport and Optimal Information Transport approaches can be used in solving this problem. The Monge Problem of Optimal Transport is concerned with computing the “optimal” mapping between two probability distributions. This actually can define a Riemannian distance between probability measures in a probability space. An-other choice of Riemannian metric on this space, the infinite-dimensional Fisher-Rao metric, gives an “information geometric” structure to the space of probability measures. It turns out that a simple partial differential equation can be solved for a mapping that relates to the underlying information geometry given by the Fisher-Rao metric. Solving for such an “information geometric” mapping is known as Optimal Information Transport. In this dissertation, a convergence framework is first established for com-puting the solution to the partial differential equation formulation of Optimal Transport on the sphere. This convergence framework uses geodesic normal coor-dinates to perform computations in local tangent planes. The numerical scheme also has a control on the Lipschitz constant of the discrete solution, which allows a convergence theorem for consistent and monotone discretizations to be proved in the absence of a comparison principle for the partial differential equation. Then, a finite-difference scheme for the partial differential equation formulation of Opti-mal Transport on the sphere is constructed which satisfies the hypotheses of the convergence theorem. An explicit formula for the mixed Hessian term is derived for two different cost functions. In order to construct a monotone discretization, discrete Laplacian terms are carefully added into the scheme. Current work has established convergence rates for solutions of monotone discretizations of linear elliptic partial differential equations on compact 2D manifolds without boundary. The goal is to then generalize these linearized arguments for the Optimal Transport case on the sphere. Computations are performed for the reflector antenna problem. Other ad hoc schemes exist for computing the reflector antenna problem, but the proposed scheme is the most efficient provably convergent scheme. Further adaptations are made that allow for the scheme to deal with non-smooth cases more explicitly. For the moving mesh methods problem, a comparison of computations via Optimal Transport and Optimal Information Transport is performed for the sphere using provably convergent monotone schemes for both computations. These comparisons show the merits of using Optimal Information Transport for some challenging computations. Optimal Information Transport also seems like a natural generalization to other compact 2D surfaces beyond the sphere

A Nash-Hormander iteration and boundary elements for the Molodensky problem

We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after m steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.Comment: 32 pages, 14 figures, to appear in Numerische Mathemati

Error bounds for PDE-regularized learning

In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that the target function satisfies an unknown PDE, the PDE error term quantifies how well this PDE is approximated by the auxiliary PDE used for regularization. It is shown that this error term decreases if more data is provided. The data error term quantifies the accuracy of the given data. Furthermore, the PDE-regularized learning problem is discretized by generalized Galerkin discretizations solving the associated minimization problem in subsets of the infinite dimensional functions space, which are not necessarily subspaces. For such discretizations an error bound in terms of the PDE error, the data error, and a best approximation error is derived