Incomplete Factorizations
J. A. Scott, Miroslav Tůma · Nečas center series · 2023
Incomplete FactorizationsThey [incomplete factorizations] 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 are primarily algebraic in nature and can in principle be applied to any sparse matrices.When applied to PDE problems, they are usually not optimal ... On the other hand, they are often quite robust.-Chan & van der Vorst (1997).Having introduced incomplete factorization preconditioners in the previous chapter, the focus in this chapter is on different ways to compute such factorizations and their relationship to the complete factorizations used in sparse direct methods.We denote the incomplete factors by L and U ; in the SPD case, U = L T .We assume that the sparsity patterns of A and its incomplete factors always include the positions of the diagonal entries.