Probabilistic Numerical Solutions of Partial Differential Equations on Riemannian 2-Manifolds

Specialeforsvar: Theo Rüter Würtzen

Titel: Probabilistic Numerical Solutions of Partial Differential Equations on Riemannian 2-Manifolds

Abstract: Partial differential equations are ubiquitous in physics and simulations, and generally have to be solved numerically.
The field of probabilistic numerics focuses on framing computation as probabilistic inference and has led to the development of efficient and competitive probabilistic numerical solvers of ordinary differential equations, which give the solution as a stochastic process. In this thesis, we extend the use of these solvers to partial differential equations on Riemannian manifolds. We give introductions to the touched upon topics, including but not limited to discretization of Riemannian manifolds, the associated discrete differential operators, and the use of stochastic differential equations as priors. We give an algorithm to build an
intrinsic triangulation of a manifold, and empirically demonstrate how certain priors can accelerate the process of solving nonlinear PDEs. We have structured the thesis as a high level guide on how to implement and follow the results, and we try to motivate the steps to the best of our efforts. We teach and explain with a focus on giving intuition through examples and
analogies, and provide pointers to further reading.

Vejledere: Sebastian Weichwald
                  Peter Nicholas Krämer, Søren Hauberg, DTU
Censor:      Jes Frellsen, DTU