Geometric Numerical Methods: From Random Fields to Shape Matching

Doctoral thesis, 2025

Geometry is central to many applied problems, though its influence varies. Some problems are inherently geometric, requiring numerical methods that preserve the underlying structure to remain accurate. Others are well understood in Euclidean space but demand different techniques when extended to curved settings. This thesis addresses such geometric challenges through studying numerical methods for two main types of problems: matching problems and stochastic (partial) differential equations. It is based on seven papers: the first three focus on SPDEs and SDEs, while the remaining consider matching problems and related differential equations. The first develops a numerical method for fractional SPDEs on the sphere, combining a recursive splitting scheme with surface finite elements. The second studies a Chebyshev–Galerkin approach for simulating non-stationary Gaussian random fields on hypersurfaces. The third introduces a geometric integrator for stochastic Lie–Poisson systems, derived via a reduction of the implicit midpoint method for canonical Hamiltonian systems. The fourth explores sub-Riemannian shape matching, where shapes are matched using constrained motions, and shows how this problem can be interpreted as a neural network. The fifth studies the convergence of a gradient flow for the Gaussian Monge problem. The sixth adapts geometric shape matching to recover protein conformations from single-particle Cryo-EM data by using rigid deformations of chains of particles. The seventh investigates the numerical signature of blow-up in hydrodynamic equations, showing that numerical solutions can be used to detect the onset in a class of hydrodynamic equations.