Parallel interval Newton-like Schwarz methods for almost linear parabolic problems

1Overlapping Schwarz for Linear and Nonlinear Parabolic Problems

The basic ideas underlying waveform relaxation were rst suggested in the late 19th century by Picard and Lindell of ((Lin94], Lin93]). However much recent interest in waveform relaxation as a practical parallel method for the solution of stii ordinary diierential equations (ODE's) has been generated after the publication of a paper by Lelarasmee and coworkers LRSV82] in the VLSI literature, and...

Overlapping Schwarz for Parabolic Problems

Schwarz Methods for the Time-Parallel Solution of Parabolic Control Problems

Suppose we are interested in the following distributed control problem: given a system governed by the parabolic PDE ẏ+L y = u on the time interval [0,T ] (where ẏ denotes the time derivative of y), we wish to choose the forcing term u = u(t) to minimize the discrepancy between the trajectory and the desired state ŷ= ŷ(t). After semi-discretization in space, we obtain for a given choice of para...

Comparison of parallel implementation of some multi-level Schwarz methods for singularly perturbed parabolic problems

Parallel multi-level algorithms combining a time discretization and an overlapping domain decomposition technique are applied to the numerical solution of singularly perturbed parabolic problems. Two methods based on the Schwarz alternating procedure are considered: a two-level method with auxiliary "correcting" subproblems as well as a three-level method with auxiliary "predicting" and "correc...

Newton Two-stage Parallel Iterative Methods for Nonlinear Problems

Two-stage parallel Newton iterative methods to solve nonlinear systems of the form F (x) = 0 are introduced. These algorithms are based on the multisplitting technique and on the two-stage iterative methods. Convergence properties of these methods are studied when the Jacobian matrix F ′(x) is either monotone or an H-matrix. Furthermore, in order to illustrate the performance of the algorithms ...