On the perturbation of the Incomplete Cholesky factorization

Gerard

CEA/DIF DCSA/EDD BP12 91680 Bruyeres le Chatel France


Abstract

In this paper we are concerned with the incomplete Cholesky decomposition of symmetric M--matrices C=e I +A where A arises from the discretization of an elliptic partial differential equation, e being a ``small'' positive real parameter. Such problem arise, for instance, from discretizing parabolic equations. As a model problem we can use the two--dimensional heat equation in the unit square with Dirichlet boundary conditions and an initial condition u(x,0)=u_0(t) We discretize in space with finite differences with a stepsize h and a time implicit scheme. Then, we obtain (I/k+(1\over h^2)A) u^{n+1}=u^n/ k+f^{n+1}, where k is the time step and A the matrix of the corresponding elliptic problem. For some problems it makes sense to choose k=h. After multiplication by h^2 the matrix of the problem is C=kI+A where k is ``small''. The matrix C being symmetric positive definite, we would like to solve the linear system at each time step with the preconditioned conjugate gradient algorithm. A very popular preconditioner is the incomplete Cholesky decomposition without any fill-in IC(1,1). Usually the time step k is small. Therefore, it is interesting to know if one can compute an approximation of the incomplete decomposition of C knowing the one of A. Moreover, most of the time the time step is not constant, therefore one cannot compute the decomposition of C once for all. It has to be recomputed at each time step. Hence, it would be interesting to find a way to cheaply update the incomplete decomposition from one time step to the next. We will describe answers to this problem and algorithms corresponding to perturbations of order 0 and 1 and give numerical examples for several problems with comparisons between our perturbed preconditioners and the incomplete decomposition of C, the one of A and the SSOR preconditioning showing that our perturbed factorizations give better results. Then we will apply the previous results to the solution of the heat equation. Finally we will deal with another problem where C=I+e A.