AbstractsMathematics

Solving linear systems of equations is an integral part of most scientific simulations. In recent years, there has been a considerable interest in large scale scientific simulation of complex physical processes. Iterative solvers are usually preferred for solving linear systems of such magnitude due to their lower computational requirements. Currently, computational scientists have access to a multitude of iterative solver options available as "plug-and- play" components in various problem solving environments. Choosing the right solver configuration from the available choices is critical for ensuring convergence and achieving good performance, especially for large complex matrices. However, identifying the "best" preconditioned iterative solver and parameters is challenging even for an expert due to issues such as the lack of a unified theoretical model, complexity of the solver configuration space, and multiple selection criteria. Therefore, it is desirable to have principled practitioner-centric strategies for identifying solver configuration(s) for solving large linear systems. The current dissertation presents a general practitioner-centric framework for (a) problem independent retrospective analysis, and (b) problem-specific predictive modeling of performance data. Our retrospective performance analysis methodology introduces new metrics such as area under performance-profile curve and conditional variance-based finetuning score that facilitate a robust comparative performance evaluation as well as parameter sensitivity analysis. We present results using this analysis approach on a number of popular preconditioned iterative solvers available in packages such as PETSc, Trilinos, Hypre, ILUPACK, and WSMP. The predictive modeling of performance data is an integral part of our multi-stage approach for solver recommendation. The key novelty of our approach lies in our modular learning based formulation that comprises of three sub problems: (a) solvability modeling, (b) performance modeling, and (c) performance optimization, which provides the flexibility to effectively target challenges such as software failure and multiobjective optimization. Our choice of a "solver trial" instance space represented in terms of the characteristics of the corresponding "linear system", "solver configuration" and their interactions, leads to a scalable and elegant formulation. Empirical evaluation of our approach on performance datasets associated with fairly large groups of solver configurations demonstrates that one can obtain high quality recommendations that are close to the ideal choices.