Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

This talk will focus on preconditioning techniques for solving sequences of symmetric indefinite systems Aixi = bi, where the matrices Ai are supposed to slightly change. This case often arises in Computational Science and Engineering, for example when a Newton-type method is used to solve a nonlinear problem. With this aim in mind, we study the case of a sequence with a given matrix A ∈ Rn×n and multiple right-hand sides that are not known simultaneously. The idea is to generalize the class of Limited Memory Preconditioners, initially motivated by the BFGS method, analysed in [1] for A symmetric positive definite. The extension to the symmetric indefinite case leads us to study the following preconditioner H, constructed from a small set of k linearly independent vectors forming the columns of a matrix S: