Incremental Incomplete LU factorizations with applications to time-dependent PDEs

A common problem which arises in many complex applications is to solve a sequence of linear systems of the form: Akxk = bk, for k = 1, 2, .. In these applications Ak does not generally change too much from one step k to the next, as it is often the result of a continuous process (e.g. Ak can represent the discretization of some problem at time tk.) We are faced with the problem of solving each consecutive system effectively, by taking advantage of earlier systems if possible. This problem arises for example in computational fluid dynamics, when the equations change only slightly possibly in some parts of the domain. In such situations it is wastful to recompute entirely any LU or ILU factorizations computed for the previous coefficient matrix. Though much is known about finding effective preconditioners to solve general sparse linear systems which arise in real-life applications, little has been done so far to address the issue of updating such preconditioners. In our presentation we will consider a number of techniques for computing incremental ILU factorizations. We will also discuss the mathematical properties of the new methods as well as of Université Sciences et Technologies de Lille, Laboratoire Paul Painlevé, UMR 8524, France and INRIA Lille Nord Europe, EPI SIMPAF, France. e-mail:[email protected] Université de Picardie Jules Verne, LAMFA, UMR 6140, Amiens, France and INRIA Lille Nord Europe, EPI SIMPAF, France. e-mail: [email protected] University of Minnesota, Computer Science, USA e-mail: [email protected]