Improving Triangular Preconditioner Updates for Nonsymmetric Linear Systems Jurjen Duintjer Tebbens and Miroslav T˚ uma Institute of Computer Science, Czech Academy of Sciences, Pod Vod´arenskou vˇ eˇ z´ ı 2, 18207 Praha 8, Czech Republic {tebbens,tuma}@cs.cas.cz Abstract. We present an extension of an update technique for precondi- tioners for sequences of non-symmetric linear systems that was proposed in [5]. In addition, we describe an idea to improve the implementation of the update technique. We demonstrate the superiority of the new ap- proaches in numerical experiments with a model problem. 1 Introduction Sequences of linear systems with large and sparse matrices arise in many ap- plications like computational fluid dynamics, structural mechanics, numerical optimization as well as in solving non-PDE problems. In many cases, one or more systems of nonlinear equations are solved by a Newton or Broyden-type method [6], and each nonlinear equation leads to a sequence of linear systems. The solution of sequences of linear systems is the main bottleneck in many of the above mentioned applications. For example, some solvers need strong pre- conditioners to be efficient and computing preconditioners for individual systems separately may be very expensive. In recent years, a few attempts to update preconditioners for sequences of large sparse systems have been made. If a sequence of linear systems arises from a quasi-Newton method, straightforward approximate small rank updates can be useful (this has been done in the SPD case in [9,3]). For shifted SPD lin- ear systems, an update technique was proposed in [8] and a different one can be found in [2]. The latter technique, based on approximate diagonal updates, has been extended to sequences of parametric complex symmetric linear sys- tems (see [4]). This technique, in turn, was generalized to approximate (possibly permuted) triangular updates for nonsymmetric sequences [5]. In addition, recy- cling of Krylov subspaces by using adaptive information generated during pre- vious runs has been used to update both preconditioners and Krylov subspace iterations (see [7,10,1]). In this paper we address two ways to improve the triangular updates of pre- conditioners for nonsymmetric sequences of linear systems from [5]. It was dis- cussed in [5] that triangular updates may be particularly beneficial under three types of circumstances: first, if preconditioner recomputation is for some rea- son expensive (e.g. in parallel computations, matrix-free environment); second, I. Lirkov, S. Margenov, and J. Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp. 737–744, 2008. c  Springer-Verlag Berlin Heidelberg 2008