Parallel Iterative Solution Methods for Linear Systems arising from Discretized PDE's

نویسنده

Henk A. Van der Vorst

چکیده

In these notes we will present an overview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and they include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, CGS, Bi-CGSTAB, QMR, LSQR and GMRES.We will show how these methods can be derived from simple basic iteration formulas. We will not give convergence proofs, but we will refer for these, as far as available, to litterature. Iterative methods are often used in combination with so-called preconditioning operators (approximations for the inverses of the operator of the system to be solved). Since these preconditioners are not essential in the derivation of the iterative methods, we will not give much attention to them in these notes. However, in most of the actual iteration schemes, we have included them in order to facilitate the use of these schemes in actual computations. For the application of the iterative schemes one usually thinks of linear sparse systems, e.g., like those arising in the nite element or nite di erence approximations of (systems of) partial di erential equations. However, the structure of the operators plays no explicit role in any of these schemes, and these schemes might also successfully be used to solve certain large dense linear systems. Depending on the situation that might be attractive in terms of numbers of oating point operations. It will turn out that all of the iterative are parallelizable in a straight forward manner. However, especially for computers with a memory hierarchy (i.e., like cache or vector registers), and for distributed memory computers, the performance can often be improved signi cantly through rescheduling of the operations. We will discuss parallel implementations, and occasionally we will report on experimental ndings.

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