Iterative Solvers for Saddle-Point Systems

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This proposal concerns the investigation, analysis and implementation of numerical solution methods for a family of linear systems that arise from problems that can be generally posed as minimization problems with constraints. Such problems are extremely important, and they frequently appear in a surprisingly large scope of scientific models. Examples of relevant applications include computer graphics, data mining, image processing, medical imaging, fluid flow, electromagnetics, and many more instances. The proposal puts forward a list of objectives that include the design of new solvers for constrained optimization problems and constrained differential equations, the derivation of new algorithms that combine various solution methodologies into one optimized solver, and the investigation of questions pertaining to the best way to solve the problem, with focus on the question to what extent various reformulations affect the robustness and reliability of solution procedures. Long-term objectives include the pursuit of a unified framework that sheds light on connections among different solution methodologies and among the different applications that lead to the systems that are considered. The issues addressed in this proposal are not just of theoretical interest but also of a high practical value, including the implementation of various solvers and the involvement of highly qualified personnel. The timeliness of this proposal is driven by the continual development of new disciplines and problems that involve the solution of large-scale linear algebra systems, and the new frontiers that processing speed, parallel computing architectures and high performance computing have reached, pushing the envelope in terms of the size of problems that can be tackled.