This thesis studies novel and efficient computational sampling methods for appli cations in three types of stochastic inversion problems: seismic waveform inversion, filtering problems, and static parameter estimation.
A primary goal of a large class of seismic inverse problems is to detect parameters that characterize an earthquake. We are interested to solve this task by analyzing the full displacement time series at a given set of seismographs, but approaching the full waveform inversion with the standard Monte Carlo (MC) method is prohibitively expensive. So we study tools that can make this computation feasible. As part of the inversion problem, we must evaluate the misfit between recorded and synthetic seismograms efficiently. We employ as misfit function the Wasserstein metric origi nally suggested to measure the distance between probability distributions, which is becoming increasingly popular in seismic inversion. To compute the expected values of the misfits, we use a sampling algorithm called MultiLevel Monte Carlo (MLMC). MLMC performs most of the sampling at a coarse spacetime resolution, with only a few corrections at finer scales, without compromising the overall accuracy.
We further investigate the Wasserstein metric and MLMC method in the context of filtering problems for partially observed diffusions with observations at periodic time intervals. Particle filters can be enhanced by considering hierarchies of discretizations to reduce the computational effort to achieve a given tolerance. This methodology is called MultiLevel Particle Filter (MLPF). However, particle filters, and consequently MLPFs, suffer from particle ensemble collapse, which requires the implementation of a resampling step. We suggest for onedimensional processes a resampling procedure
based on optimal Wasserstein coupling. We show that it is beneficial in terms of computational costs compared to standard resampling procedures.
Finally, we consider static parameter estimation for a class of continuoustime statespace models. Unbiasedness of the gradient of the loglikelihood is an important property for gradient ascent (descent) methods to ensure their convergence. We propose a novel unbiased estimator of the gradient of the loglikelihood based on a doublerandomization scheme. We use this estimator in the stochastic gradient ascent method to recover unknown parameters of the dynamics.
Date of Award  Feb 2 2022 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Raul Tempone (Supervisor) 

 multilevel Monte Carlo
 PDE with Random coefficients
 particles filters
 diffusions
 score function
 coupled conditional particle filter