Preconditioned Linear Systems of Time-dependent Pdes. Properties and Performances

In this talk, we will survey some properties of the preconditioners introducedin [1,2] for the solution of the linear systems arising in time-dependent PDEs.Moreover, we will give theoretical results on the convergence rate of the under-lying preconditioned iterations using various Krylov subspace methods. Moreprecisely, if s is the size of the matrices related to the time-step discretiza-tion, we will see that, asymptotically, the number of iterations required forconvergence is at most O(log s), under suitable conditions. Therefore, if thepreconditioned Krylov accelerator requires O(1) (O(log s)) iterations to con-verge, the computational complexity will be O(s log s) (O(s log s)), e.g., forPDEs whose Jacobian matrices, after semidiscretization, have a few nonzerodiagonals. Details can be found in [3] using CG for the normal equations andin a forthcoming paper using GMRES.Numerical experiments will confirm the theoretical analysis. References[1] D. Bertaccini. A circulant preconditioner for the systems of LMF-basedODE codes, SIAM J. Sci. Comput., 22-3 (2000), pp. 767-786.[2] D. Bertaccini. Reliable preconditioned iterative linear solvers for somenumerical integrators, Numer. Linear Algebra Appl., 8-2 (2001), pp.111–125.[3] D. Bertaccini, M. K. Ng. The Convergence rate of Block PreconditionedSystems Arising from LMF-based ODE codes, BIT 41-3 (2001), pp. 433–450. Acknowledgement: This research was partially supported by the ItalianMinistry of Scientific Research under Project “Ateneo Giovani 2000”, Univer-sità di Roma “La Sapienza”.