Numerical Solution of Large-Scale Sparse Linear Systems Arising from Problems with Constraints

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***Recent advances in computational science and engineering, combined with the availability of increased computing power and new computational paradigms, require experts in the scientific computing domain to develop new algorithms and methodologies for solving problems that are larger than ever before, and whose features must be exploited in order to be able to generate robust, efficient, and reliable numerical solution techniques. This proposal aims to take on these challenges for the numerical solution of structured linear systems that arise throughout the computation of a large variety of problems with constraints.******Linear systems that arise from problems with constraints are highly structured, yet they are notorious for being difficult to solve. Modern methods are based on model reduction techniques (namely, the attempt to turn the original problem into a smaller problem that maintains most of the original properties). For linear algebra problems arising from problems with constraints, preserving the structure and other attributes is critical and it presents a significant challenge.******My primary goal will be to develop fast and robust solvers, which are scalable and respect the structure. To accomplish this goal, I will aim to generate a unified framework for a variety of problems from the area of constrained optimization and numerical solution of partial differential equations. Attributes such as symmetry, matrix rank property, and spectral distribution, to name just a few, should be explored and exploited for this mission to be successful. An equally important goal will be to develop numerical software for scalable, flexible, and portable solvers that allow for solving large-scale problems.******There are many relevant applications here and many connections to other areas. Primarily, any problem that can be posed as a constrained optimization problem requires solving linear systems of the form discussed in this proposal. Problems in machine learning, computer graphics, robotics, medical imaging, and many other computational areas, provide a rich source. In addition, the numerical solution of partial differential equations provides a large collection of problems with constraints, too; computational electromagnetics or fluid dynamics are just two examples. Altogether, the large collection of problems to be solved and the importance of dealing efficiently with large-scale problems, make this area of research extremely active and important.******Given the large number of applications that lead to mathematical models with constraints, and given the increasing amounts of data that our society needs to deal with, advances in fast numerical solvers for the problems concerned in this proposal may have a significant positive impact in computational science and engineering.