Abstract: The Hamiltonian Monte Carlo (HMC) method allows sampling from continuous densities. Favorable scaling with dimension has led to wide adoption of HMC by the statistics community. Modern auto-differentiating software should allow more widespread usage in Bayesian inverse problems. This presentation analyzes the two major difficulties encountered using HMC for inverse problems: poor conditioning and multi-modality. Novel results on preconditioning and replica exchange Monte Carlo parameter selection are presented.  Recommendations are analyzed rigorously in the Gaussian case, and shown to generalize in a fusion plasma reconstruction.

Bio: Ian Langmore studied the mathematics of inverse problems under Gunther Uhlmann at the University of Washington. After his PhD, he worked on probabilistic methods for inverse problems in transport with Guillaume Bal at Columbia. At Google, Ian worked on Bayesian modeling for local search algorithms, before moving on to the Climate & Energy team, where he works on a variety of problems. This places him happily at the intersection of applied probability, software engineering, and physics, all with a real-world purpose. His secret goal is to use rigorous mathematics in engineering.