On the interplay between inner and outer iterations for a class of iterative methods

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:223867

Giladi, E ^[1]

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1. Stanford Univ., CA (United States)

Iterative algorithms for solving linear systems of equations often involve the solution of a subproblem at each step. This subproblem is usually another linear system of equations. For example, a preconditioned iteration involves the solution of a preconditioner at each step. In this paper, the author considers algorithms for which the subproblem is also solved iteratively. The subproblem is then said to be solved by {open_quotes}inner iterations{close_quotes} while the term {open_quotes}outer iteration{close_quotes} refers to a step of the basic algorithm. The cost of performing an outer iteration is dominated by the solution of the subproblem, and can be measured by the number of inner iterations. A good measure of the total amount of work needed to solve the original problem to some accuracy c is then, the total number of inner iterations. To lower the amount of work, one can consider solving the subproblems {open_quotes}inexactly{close_quotes} i.e. not to full accuracy. Although this diminishes the cost of solving each subproblem, it usually slows down the convergence of the outer iteration. It is therefore interesting to study the effect of solving each subproblem inexactly on the total amount of work. Specifically, the author considers strategies in which the accuracy to which the inner problem is solved, changes from one outer iteration to the other. The author seeks the `optimal strategy`, that is, the one that yields the lowest possible cost. Here, the author develops a methodology to find the optimal strategy, from the set of slowly varying strategies, for some iterative algorithms. This methodology is applied to the Chebychev iteration and it is shown that for Chebychev iteration, a strategy in which the inner-tolerance remains constant is optimal. The author also estimates this optimal constant. Then generalizations to other iterative procedures are discussed.

• Research Organization: Front Range Scientific Computations, Inc., Boulder, CO (United States); USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• OSTI ID: 223867
• Report Number(s): CONF-9404305--Vol.1; ON: DE96005735
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ITERATIVE METHODS
MATHEMATICS
MATRICES
OPTIMIZATION