@phdthesis{Richter2021, author = {Richter, Lorenz}, title = {Solving high-dimensional PDEs, approximation of path space measures and importance sampling of diffusions}, doi = {10.26127/BTUOpen-5864}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-58647}, school = {BTU Cottbus - Senftenberg}, year = {2021}, abstract = {Motivated by computing functionals of high-dimensional, potentially metastable diffusion processes, this thesis studies robustness issues appearing in the numerical approximation of expectation values and their gradients. A major challenge being high variances of corresponding estimators, we investigate importance sampling of stochastic processes for improving statistical properties and provide novel nonasymptotic bounds on the relative error of corresponding estimators depending on deviations from optimality. Numerical strategies that aim to come close to those optimal sampling strategies can be encompassed in the framework of path space measures, and minimizing suitable divergences between those measures suggests a variational formulation that can be addressed in the spirit of machine learning. A key observation is that while several natural choices of divergences have the same unique minimizer, their finite sample properties differ vastly. We provide the novel log-variance divergence, which turns out to have favorable robustness properties that we investigate theoretically and apply in the context of path space measures as well as in the context of densities, for instance offering promising applications in Bayesian variational inference. Aiming for optimal importance sampling of diffusions is (more or less) equivalent to solving Hamilton-Jacobi- Bellman PDEs and it turns out that our numerical methods can be equally applied for the approximation of rather general high-dimensional semi-linear PDEs. Motivated by stochastic representations of elliptic and parabolic boundary value problems we refine variational methods based on backward SDEs and provide the novel diffusion loss, which can be related to other state-of-the-art attempts, while offering certain numerical advantages.}, subject = {Approximation of high dimensional PDEs; Stochastic optimal control; Importance sampling; Machine learning; Robustness; Approximation hochdimensionaler PDEs; Stochastische optimale Steuerung; Importance Sampling; Machine Learning; Robustheit; Sequenzielle Monte-Carlo-Methode; Stochastisches Modell; Maschinelles Lernen; Stochastische optimale Kontrolle}, language = {en} }