Schwarz Splitting and Template Operators.
Abstract
Schwarz alternating method is an old mathematical technique dating from 1869. It was commonly believed that SAM was a useful tool for theoretical analysis but not a very practical approach for computations. The earlier experiences showed that SAM converged slowly. In this thesis, SAM is reexamined and generalized. The governing factors of convergence of SAM are explored through the analysis for the model problem. Based on this knowledge, many acceleration schemes can be combined with SAM to yield a new type of iterative method for large sparse matrix problems. In particular, when these techniques are applied to the solution of the model problem, an optimal complexity can be achieved. Some generalizations of SAM, namely Schwarz splittings (SS), are presented here. For many important applications, such as performing parallel computations in a non-shared memory environment, using composite grids and also applying fast solvers in an irregular region, (SS)s are found to be useful techniques. In order to identify the types of problems for which (SS)s are most suitable, a new structure for the linear operators called template operators has been developed. Some decay results for the elements of the inverses of sparse operators are given. These results provide a theoretical basis for determining when these SS techniques can be used successfully.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 1987
- Accession Number
- ADA187956
Entities
People
- Wei-pai Tang
Organizations
- Stanford University