Deflation-based preconditioners for stochastic models of flow in porous media

Abstract

Numerical analysis is a powerful mathematical tool that focuses on finding approximate solutions to mathematical problems where analytical methods fail to produce exact solutions. Many numerical methods have been developed and enhanced through the years for this purpose, across many classes, with some methods proven to be well-suited for solving certain equations. The key in numerical analysis is, then, choosing the right method or combination of methods for the problem at hand, with the least cost and highest accuracy possible (while maintaining efficiency). In this thesis, we consider the approximate solution of a class of 2-dimensional differential equations, with random coefficients. We aim, through using a combination of Krylov methods, preconditioners, and multigrid ideas to implement an algorithm that offers low cost and fast convergence for approximating solutions to these problems. In particular, we propose to use a "training" phase in the development of a preconditioner, where the first few linear systems in a sequence of similar problems are used to drive adaptation of the preconditioning strategy for subsequent problems. Results show that our algorithms are successful in effectively decreasing the cost of solving the model problem from the cost shown using a standard AMG-preconditioned CG method.