Abstract
Efficient choice of the initial guess for the iterative solution of series of systems is considered. The series of systems are typical for unsteady nonlinear fluid flow problems. The history of iterative solution at previous time steps is used for computing a better initial guess. This strategy is applied for two iterative linear system solvers (GCR and GMRES). A reduced model technique is developed for implicitly discretized nonlinear evolution problems. The technique computes a better initial guess for the inexact Newton method. The methods are successfully tested in parallel CFD simulations. The latter approach is suitable for GRID computing as well.
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References
S. Eisenstat and H. Walker, Globally convergent inexact Newton methods. SIAM J. Optim. 4, 1994, 393-422.
S. Eisenstat and H. Walker, Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput., 17, 1996, 16-32.
G. Golub and G. Meurant, Matrices, moments and quadrature II: how to compute the norm of the error in iterative methods. BIT, 37-3, 1997, 687-705.
P. Gosselet and Ch. Rey, On a selective reuse of Krylov subspaces in Newton-Krylov approaches for nonlinear elasticity. Domain decomposition methods in science and engineering, Natl. Auton. Univ. Mex., Mexico, 2003, 419-426.
C. T. Kelley, Iterative Methods for Optimization. Frontiers in Applied Mathematics 18. SIAM, Philadelphia, 1999
I. Keshtiban, F. Belblidia and M. Webster Compressible flow solvers for Low Mach number flows – a review. Technical report CSR2, Institute of Non-Newtonian Fluid Mechanics, University of Wales, Swansea, UK, 2004.
G. Meurant, Computer solution of large linear systems. Amsterdam, North-Holland, 1999.
G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm.Numerical Algorithms, 16, 1998, 77-87.
M. Pernice and H. Walker, NITSOL: a Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput., 19, 1998, 302-318.
M.Rathinam and L.Petzold, Dynamic iteration using reduced order models: a method for simulation of large scale modular systems. SIAM J. Numer.Anal., 40, 2002, 1446-1474.
M. Rathinam and L. Petzold, A new look at proper orthogonal decomposition. SIAM J. Numer.Anal., 41, 2003, 1893-1925.
F. Risler and Ch. Rey, Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems. Numer. Algorithms 23, 2000, 1-30.
M. Schefer and S. Turek, Benchmark Computations of Laminar Flow around a Cylinder. In: Flow Simulation with High-Performance Computers II (E.H.Hirschel ed.), Notes on Numerical Fluid Mechanics,52, Vieweg, 1996, 547-566.
D. Tromeur-Dervout and Y. Vassilevski, POD acceleration of fully implicit solver for unsteady nonlinear flows and its application on GRID architecture. Proc. Int. Conf. PCFD05, 2006, 157-160.
D. Tromeur-Dervout and Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. J.Comput.Phys. 219, 2006, 210-227.
D. Tromeur-Dervout and Y. Vassilevski, POD acceleration of fully implicit solver for unsteady nonlinear flows and its application on grid architecture.Adv. Eng. Softw. 38, 2007, 301-311.
D. Tromeur-Dervout, Résolution des Equations de Navier-Stokes en Formulation Vitesse Tourbillon sur Systèmes multiprocesseurs à Mémoire Distribuée. Thesis, Univ. Paris VI, 1993.
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Tromeur-Dervout, D., Vassilevski, Y. (2010). Acceleration of iterative solution of series of systems due to better initial guess. In: Tromeur-Dervout, D., Brenner, G., Emerson, D., Erhel, J. (eds) Parallel Computational Fluid Dynamics 2008. Lecture Notes in Computational Science and Engineering, vol 74. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14438-7_3
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DOI: https://doi.org/10.1007/978-3-642-14438-7_3
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