A Rayleigh–Ritz preconditioner for the iterative solution to large scale nonlinear problems

The approximation to the solution of large sparse symmetric linear problems arising from nonlinear systems of equations is considered. We are focusing herein on reusing information from previous processes while solving a succession of linear problems with a Conjugate Gradient algorithm. We present a new Rayleigh–Ritz preconditioner that is based on the Krylov subspaces and superconvergence properties, and consists of a suitable reuse of a given set of Ritz vectors. The relevance and the mathematical foundations of the current approach are detailed and the construction of the preconditioner is presented either for the unconstrained or the constrained problems. A corresponding practical preconditioner for iterative domain decomposition methods applied to nonlinear elasticity is addressed, and numerical validation is performed on a poorly-conditioned large-scale practical problem.

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