Novel Monte Carlo Methods for Large-Scale Linear Algebra Operations

Linear algebra operations play an important role in scientific computing and data analysis. With increasing data volume and complexity in the Big Data era, linear algebra operations are important tools to process massive datasets. On one hand, the advent of modern high-performance computing architectures with increasing computing power has greatly enhanced our capability to deal with a large volume of data. One the other hand, many classical, deterministic numerical linear algebra algorithms have difficulty to scale to handle large data sets. Monte Carlo methods, which are based on statistical sampling, exhibit many attractive properties in dealing with large volume of datasets, including fast approximated results, memory efficiency, reduced data accesses, natural parallelism, and inherent fault tolerance. In this dissertation, we present new Monte Carlo methods to accommodate a set of fundamental and ubiquitous large-scale linear algebra operations, including solving large-scale linear systems, constructing low-rank matrix approximation, and approximating the extreme eigenvalues/ eigenvectors, across modern distributed and parallel computing architectures. First of all, we revisit the classical Ulam-von Neumann Monte Carlo algorithm and derive the necessary and sufficient condition for its convergence. To support a broad family of linear systems, we develop Krylov subspace Monte Carlo solvers that go beyond the use of Neumann series. New algorithms used in the Krylov subspace Monte Carlo solvers include (1) a Breakdown-Free Block Conjugate Gradient algorithm to address the potential rank deficiency problem occurred in block Krylov subspace methods; (2) a Block Conjugate Gradient for Least Squares algorithm to stably approximate the least squares solutions of general linear systems; (3) a BCGLS algorithm with deflation to gain convergence acceleration; and (4) a Monte Carlo Generalized Minimal Residual algorithm based on sampling matrix-vector products to provide fast approximation of solutions. Secondly, we design a rank-revealing randomized Singular Value Decomposition (R3SVD) algorithm for adaptively constructing low-rank matrix approximations to satisfy application-specific accuracy. Thirdly, we study the block power method on Markov Chain Monte Carlo transition matrices and find that the convergence is actually depending on the number of independent vectors in the block. Correspondingly, we develop a sliding window power method to find stationary distribution, which has demonstrated success in modeling stochastic luminal Calcium release site. Fourthly, we take advantage of hybrid CPU-GPU computing platforms to accelerate the performance of the Breakdown-Free Block Conjugate Gradient algorithm and the randomized Singular Value Decomposition algorithm. Finally, we design a Gaussian variant of Freivalds’ algorithm to efficiently verify the correctness of matrix-matrix multiplication while avoiding undetectable fault patterns encountered in deterministic algorithms

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Optimal low-rank approximations of Bayesian linear inverse problems

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space