SIAM/ASA Journal of Uncertainty Quantification Invited Paper Session

We present an optimal transport framework for conditional sampling of probability measures. Conditional sampling is a fundamental task of solving Bayesian inverse problems and generative modeling. Optimal transport provides a flexible methodology to sample target distributions appearing in these problems by constructing a deterministic coupling that maps samples from a reference distribution (e.g., a standard Gaussian) to the desired target. To extend these tools for conditional sampling, we first develop the theoretical foundations of block triangular transport in a Banach space setting by drawing connections between monotone triangular maps and optimal transport. To learn these block triangular maps, I will then present a computational approach, called monotone generative adversarial networks (MGANs). Our algorithm uses only samples from the underlying joint probability measure and is hence likelihood-free, making it applicable to inverse problems where likelihood evaluations are inaccessible or computationally prohibitive. We will demonstrate the accuracy of MGAN for sampling the posterior distribution in Bayesian inverse problems involving ordinary and partial differential equations and for probabilistic image in-painting. 

In this talk, we will present a family of sparsity promoting Bayesian hierarchical models based on combining Gaussian distributions with the deterministic effects of sparsity-promoting regularization like $l_1$ norms, total variation and/or constraints. Unlike Bayesian hierarchical models based on conditional continuous distributions, for example, conditional Gaussian distributions, using regularized Gaussian distributions results in sparse samples without needing large hierarchical models. We will show how to derive approximate Gibbs samplers for these hierarchical models and discuss advantages and disadvantages of the presented method with regard to theory, modeling and computation. 

Least squares regression is a ubiquitous tool for building emulators of problems across science and engineering for purposes such as design space exploration and uncertainty quantification. When the regression data are generated using an experimental design process (e.g. a quadrature grid) involving computationally expensive models, or when the data size is large, sketching techniques have shown promise at reducing the cost of the construction of the regression model while ensuring accuracy comparable to that of the full data. However, random sketching strategies, such as those based on leverage scores, lead to regression errors that are random and may exhibit large variability. To mitigate this issue, we present a novel boosting approach that leverages cheaper, lower-fidelity data of the problem at hand to identify the best sketch among a set of candidate sketches. This in turn specifies the sketch of the intended high-fidelity model and the associated data. We provide theoretical analyses of this bifidelity boosting (BFB) approach and discuss the conditions the low- and high-fidelity data must satisfy for successful boosting. In doing so, we derive a bound on the residual norm of the BFB sketched solution relating it to its ideal, but computationally expensive, high-fidelity boosted counterpart. Empirical results on both manufactured and PDE data corroborate the theoretical analyses and illustrate the efficacy of the BFB solution in reducing the regression error, as compared to the nonboosted solution. 

Filtering involves the real-time estimation of a dynamical system's state from incomplete and noisy observations. For high-dimensional systems, ensemble Kalman filters are often the preferred method. These filters use an ensemble of interacting particles to sequentially estimate the system's state as new observations come in. While ensemble Kalman filters are widely successful in practice, their theoretical analysis is complicated by the complex dependencies between particles. This presentation introduces ensemble Kalman filters that include an additional resampling step to break these dependencies. The resulting algorithm allows for a non-asymptotic, dimension-free theoretical analysis that improves and extends existing results for filters without resampling, while maintaining comparable performance in various numerical examples.