This paper describes the implementation of a parallel variant of GMRES on Paragon. This variant builds an orthonormal Krylov basis in two steps: it first computes a Newton basis then orthogonalises it. The first step requires matrix-vector products with a general sparse unsymmetric matrix and the second step is a QR factorisation of a rectangular matrix with few long vectors. The algorithm has ...

The ILU factorization is one of the most popular preconditioners for the Krylov subspace method, alongside the GMRES. Properties of the preconditioner derived from the ILU factorization are relayed onto the dropping rules. Recently, Zhang et al. [Numer. Linear. Algebra. Appl., Vol. 19, pp. 555–569, 2011] proposed a Flexible incomplete Cholesky (IC) factorization for symmetric linear systems. Th...

Abstract: Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) deals with nonlinear models and/or constraints. A Continuation/GMRESMethod for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation Ax = b of the original NMPC equations on every time step. We have previousl...

We consider the solution of linear systems of algebraic equations that arise from elliptic nite element problems We study a two level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative Schwarz methods This method combines the advantages of the two methods It converges faster than the additive Schwarz algorithm and is more parallelizabl...

The parallel strong-scaling of iterative methods is often determined by the number global reductions at each iteration. Low-synch Gram–Schmidt algorithms are applied here to Arnoldi algorithm reduce and therefore improve solvers for nonsymmetric matrices such as GMRES Krylov–Schur methods. In context, QR factorization “left-looking” processes one column a time. Among generating an orthogonal ba...

The purpose of this paper is to derive new computable convergence bounds for GMRES. The new bounds depend on the initial guess and are thus conceptually different from standard “worst-case” bounds. Most importantly, approximations to the new bounds can be computed from information generated during the run of a certain GMRES implementation. The approximations allow predictions of how the algorit...

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase, and variations of these which adapt the restart value k, are analyzed with respect to scalability (their...

In this paper we describe a fast multilevel algorithm for the solution of a system of nonlinear integrodifferential equations that model steady-state combined conductive-radiative heat transfer in two space dimensions. This extends our previous work in one space dimension. We formulate the equations as a compact fixed point problem with the temperature as the unknown. The fixed point map requir...