Approximate eigen-decomposition preconditioners for solving numerical PDE problems

Approximate matrix inverse has played key role in constructing preconditioners since last 1990s, because of their solid theoretical background. Sparse approximate inverse preconditioners (SAIPs) have attracted much attention recently, because their potential usefulness in a parallel environment. As a general matrix approach, in this paper, we propose an approximate eigen-decomposition preconditioners by combining a FFT-like multiplication algorithm. Some numerical tests are given to show this algorithm is more effective than the traditional method such as ILU with PETSc for solving a wide class of discrete elliptic problems.