A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

Abstract

Many applications in structural mechanics require the numerical solution of sequences of linear systems typically issued from a finite element discretization of the governing equations on fine meshes. The method of Lagrange multipliers is often used to take into account mechanical constraints. The resulting matrices then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. A popular strategy is then to combine preconditioning and deflation to yield an efficient method. We propose an alternative that is applicable to the general case and not only to matrices with a saddle point structure. In this approach, we consider to update an existing algebraic or application-based preconditioner, using specific available information exploiting the knowledge of an approximate invariant subspace or of matrix-vector products. The resulting preconditioner has the form of a limited memory quasi-Newton matrix and requires a small number of linearly independent vectors. Numerical experiments performed on three large-scale applications in elasticity highlight the relevance of the new approach. We show that the proposed method outperforms the deflation method when considering sequences of linear systems with varying matrices.

On preconditioner updates for sequences of saddle-point linear systems

“…Limited Memory Preconditioners (LMPs) are based on the idea of improving a first-level preconditioner, henceforth referred to as seed precon-ditioner, with a technique inspired by limited-memory quasi-Newton methods [15]. Initially developed for symmetric positive definite matrices [10,11], LMPs have been extended to saddle-point and to general indefinite matrices, and have been used in different applications, either to improve a preconditioner for a single system or to obtain preconditioners for sequences of linear systems with fixed or varying matrices [2,4,16].…”

“…where H is the preconditioned matrix P seed M [4]. Note that the choice of a small value for q is driven by the computational cost of computing and applying S T MS or S T H T H S. Details on this issue are given in [2,11,16].…”

“…In [2,4] updates are used to precondition sequences of saddle-point linear systems arising in structural mechanics applications. The saddle-point matrices are either constant or varying through the sequence, with C i = 0 in both cases.…”

“…These sequences appear, e.g., in interior-point methods for quadratic programming and in Lagrangian approaches for the solution of PDE problems, with applications to optimal control, elasticity, polycrystalline aggregates, etc. [1][2][3][4][5].…”

“…where F i and W i are suitable symmetric positive definite matrices and κ is a scalar [6,7], and to sparse approximate factorizations of M i . This updating technique is inspired by limited-memory quasi-Newton methods and the resulting preconditioners are called limited-memory preconditioners [2,4,10,11]. The reader is referred to [1,6,7] for details on the blocks of P c i , P d i and P t i and the spectral properties of the preconditioned matrices.…”