An efficient algorithm for the parallel solution of high-dimensional differential equations

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

AbstractFor linear constant-coefficient differential-algebraic equations, we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval. In particular, we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions, which may have wider and more useful application extent. Numerical simulations demonstrate the validity of the theory

Krylov's methods in function space for waveform relaxation.

by Wai-Shing Luk.Thesis (Ph.D.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 104-113).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Functional Extension of Iterative Methods --- p.2Chapter 1.2 --- Applications in Circuit Simulation --- p.2Chapter 1.3 --- Multigrid Acceleration --- p.3Chapter 1.4 --- Why Hilbert Space? --- p.4Chapter 1.5 --- Parallel Implementation --- p.5Chapter 1.6 --- Domain Decomposition --- p.5Chapter 1.7 --- Contributions of This Thesis --- p.6Chapter 1.8 --- Outlines of the Thesis --- p.7Chapter 2 --- Waveform Relaxation Methods --- p.9Chapter 2.1 --- Basic Idea --- p.10Chapter 2.2 --- Linear Operators between Banach Spaces --- p.14Chapter 2.3 --- Waveform Relaxation Operators for ODE's --- p.16Chapter 2.4 --- Convergence Analysis --- p.19Chapter 2.4.1 --- Continuous-time Convergence Analysis --- p.20Chapter 2.4.2 --- Discrete-time Convergence Analysis --- p.21Chapter 2.5 --- Further references --- p.24Chapter 3 --- Waveform Krylov Subspace Methods --- p.25Chapter 3.1 --- Overview of Krylov Subspace Methods --- p.26Chapter 3.2 --- Krylov Subspace methods in Hilbert Space --- p.30Chapter 3.3 --- Waveform Krylov Subspace Methods --- p.31Chapter 3.4 --- Adjoint Operator for WBiCG and WQMR --- p.33Chapter 3.5 --- Numerical Experiments --- p.35Chapter 3.5.1 --- Test Circuits --- p.36Chapter 3.5.2 --- Unstructured Grid Problem --- p.39Chapter 4 --- Parallel Implementation Issues --- p.50Chapter 4.1 --- DECmpp 12000/Sx Computer and HPF --- p.50Chapter 4.2 --- Data Mapping Strategy --- p.55Chapter 4.3 --- Sparse Matrix Format --- p.55Chapter 4.4 --- Graph Coloring for Unstructured Grid Problems --- p.57Chapter 5 --- The Use of Inexact ODE Solver in Waveform Methods --- p.61Chapter 5.1 --- Inexact ODE Solver for Waveform Relaxation --- p.62Chapter 5.1.1 --- Convergence Analysis --- p.63Chapter 5.2 --- Inexact ODE Solver for Waveform Krylov Subspace Methods --- p.65Chapter 5.3 --- Experimental Results --- p.68Chapter 5.4 --- Concluding Remarks --- p.72Chapter 6 --- Domain Decomposition Technique --- p.80Chapter 6.1 --- Introduction --- p.80Chapter 6.2 --- Overlapped Schwarz Methods --- p.81Chapter 6.3 --- Numerical Experiments --- p.83Chapter 6.3.1 --- Delay Circuit --- p.83Chapter 6.3.2 --- Unstructured Grid Problem --- p.86Chapter 7 --- Conclusions --- p.90Chapter 7.1 --- Summary --- p.90Chapter 7.2 --- Future Works --- p.92Chapter A --- Pseudo Codes for Waveform Krylov Subspace Methods --- p.94Chapter B --- Overview of Recursive Spectral Bisection Method --- p.101Bibliography --- p.10