The Induced Dimension Reduction Method Applied to Convection-Diffusion-Reaction Problems

Discretization of (linearized) convection-diusion-reaction problems yields<br/>a large and sparse non symmetric linear system of equations,<br/>Ax = b: (1)<br/>In this work, we compare the computational behavior of the Induced Dimension<br/>Reduction method (IDR(s)) [10], with other short-recurrences Krylov methods,<br/>specically the Bi-Conjugate Gradient Method (Bi-CG) [1], restarted Generalized<br/>Minimal Residual (GMRES(m)) [4], and Bi-Conjugate Gradient Stabilized method<br/>(Bi-CGSTAB) [11].<br/

La familia de métodos IDR (Induced Dimension Reduction) para resolver grandes sistemas lineales

Los métodos IDR e IDR(s) son métodos iterativos de espacios de Krylov para resolver grandes sistemas no simétricos de ecuaciones lineales. El enfoque de estos métodos es completamente diferente del de los métodos de espacios de Krylov más tradicionales y exigen un esfuerzo adicional para familiarizarse con ellos y entender las conexiones y diferencias con los métodos de espacios de Krylov más conocidos. El trabajo es una puesta al día de esta clase de métodos, dando las ideas fundamentales de su análisis y experimentando su aplicación en sistemas provenientes de la discretización de ecuaciones en derivadas parciales.Departamento de Matemática AplicadaMáster en Investigación en Matemática

Scharz Preconditioners for Krylov Methods: Theory and Practice

Several numerical methods were produced and analyzed. The main thrust of the work relates to inexact Krylov subspace methods for the solution of linear systems of equations arising from the discretization of partial di#11;erential equa- tions. These are iterative methods, i.e., where an approximation is obtained and at each step. Usually, a matrix-vector product is needed at each iteration. In the inexact methods, this product (or the application of a preconditioner) can be done inexactly. Schwarz methods, based on domain decompositions, are excellent preconditioners for thise systems. We contributed towards their under- standing from an algebraic point of view, developed new ones, and studied their performance in the inexact setting. We also worked on combinatorial problems to help de#12;ne the algebraic partition of the domains, with the needed overlap, as well as PDE-constraint optimization using the above-mentioned inexact Krylov subspace methods