Approximate and Incomplete Factorizations

Abstract

In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g., computed via Gaussian elimination) by disallowing certain fill-ins. As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners is primarily algebraic in nature and can in principle be applied to general sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system grows as the mesh size h is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coefficients and strong convection terms.We will describe the basic ILU and (modified) MILU preconditioners. Then we will review briefly several variants: more fill, relaxed ILU, shifted ILU, ILQ, as well as block and multilevel variants. We will also touch on a related class of approximate factorization methods which arise more directly from approximating a partial differential operator by a product of simpler operators.Finally, we will discuss parallelization aspects, including reordering, series expansion, and domain decomposition techniques. Generally, this class of preconditioner does not possess a high degree of parallelism in its original form. Re-ordering and approximation by truncating certain series expansion will increase the parallelism, but usually with a deterioration in convergence rate. Domain decomposition offers a compromise.