This new parallel domain decomposition algorithm, introduced by Garbey [M. Garbey, A direct solver for the heat equation with domain decomposition in space and time, in: Springer Ulrich Langer et al. (Ed)., Domain Decomposition in Science and Engineering XVII, vol. 60, 2007, pp. 501 508], generalizes the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems.
Moreover the role of the regularization parameter ε has been investigated; For practical purposes one can choose ε (almost) arbitrarily small and solve the discrete problem with a direct solver or by using the preconditioned conjugate gradient method.
The values of ϵ are limited by a lower bound of 10−5, and in case of divergence of the iterative solver for (epsilonle10^{-5}) a direct solver is used instead.
For mid-sized circuits, if the number of coupling nodes is less than a few hundreds and a direct solver does not produce significant additional fill-ins, a direct solver works the best also for the coupling solver.
The optimization framework consisted of a model generator, a direct solver, and a numerical optimizer.
Clearly, the discretization matrix A gets denser with increasing p. Therefore, the cost of a direct solver, particularly for large problems, becomes prohibitively expensive.
In error diagnosis mutation operator, first we copy the nucleotide s iparent and connection information c iparent of a parent solution to the s ichild and c ichild of a child solution for all i (1 ≤ i ≤ N ), respectively; here we copy the c iparent, which is assigned to the parent solution by a direct problem solver, to the child solution.
The resulting fast method reduces the computational work from O N3) required by a commonly used direct solver to O(NlogN) per iteration and the memory requirement from O N2) to O(N).
The approach is developed by integrating a direct problem solver with an optimizer.
