Preconditioning for block matrices with square blocks

Abstract: Linear systems of equations appear in one way or another in almost every scientific and engineering problem. They are so ubiquitous that, in addition to solving linear problems, also non-linear problems are typically reduced to a sequence of linear ones. The field of numerical solution methods for linear systems is rich, but we can broadly classify the methods into two classes: direct solvers and iterative solvers. The availability of modern large-scale computational resources motivates the development and the use of well parallelizable efficient solvers with a limited memory footprint. For many problems, these properties can be achieved by the employment of iterative solution methods combined with preconditioning techniques. This work focuses on the design of preconditioners for block-matrices with square blocks. This form of matrices occurs in many applications, encountered for instance when numerically solving partial differential equations, ordinary differential equations and others.The work in this thesis can broadly be divided into two types of problems, one being optimal control problems within the PDE-constrained optimization framework, and the other being fully implicit Runge-Kutta time-stepping schemes. These necessitate the solution of large and sparse linear systems, for which we employ iterative solution methods. Principal attention is given to Krylov subspace methods. In order to obtain a solution within practical time and memory usage, such methods generally necessitate the use of preconditioners in order to be efficient. The main topic of the thesis is thus the design of preconditioners, although the entire solution procedure is explored.

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