Waveform Relaxation Research Articles

Optimization of Schwarz waveform relaxation over short time windows

Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.

Delay-Extraction-Based Waveform Relaxation Algorithm for Fast Transient Analysis of Power Distribution Networks

In this paper, a waveform relaxation algorithm is presented for the efficient transient analysis of large power distribution networks. The network is modeled as an orthogonal grid of transmission lines where a delay-extraction-based macromodel is used to represent each line segment in the time domain. Novel partitioning and iterative techniques are proposed for fast convergence and improved scalability of the proposed relaxation algorithm. The overall algorithm is highly parallelizable and exhibits good scaling with both the size of the circuit matrices involved and the number of CPUs available. Numerical examples are presented to illustrate the validity and efficiency of the proposed algorithm.

Waveform relaxation methods for fractional differential equations with the Caputo derivatives

In this paper, we report new waveform relaxation methods for fractional differential equations with the Caputo derivatives. The convergence properties of the waveform relaxation methods are studied under linear and nonlinear conditions for the right-hand side of equations. Some convergent splittings are given. This is the first time for studying the waveform relaxation methods for fractional differential equations in references.

Overlapping Partitioning Techniques for Simulation of Strongly Coupled Distributed Interconnects

The large number of interconnects in high-speed circuits is a major bottleneck for fast simulation of such circuits. Recently, waveform relaxation methods based on transverse partitioning (WR-TP) were proposed to address this issue. It was shown that the complexity of WR-TP grows only linearly with the number of lines. However, as the coupling between the lines becomes stronger, the WR-TP algorithm either fails to converge or the number of iterations required for convergence increases. In this paper, an overlapping partitioning method for WR-TP is presented, which overcomes the effect of strong coupling between the lines. Numerical examples are presented which demonstrate the accuracy and efficiency of the proposed method for tightly coupled lines.

A parareal algorithm based on waveform relaxation

We report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (WR) technique with the classical parareal algorithm. The parallelism can be simultaneously exploited in both sub-systems by WR and time by parareal. We also provide a sharp estimation on errors for the new algorithm. The iterations of parareal and WR are balanced to optimize the performance of the algorithm. Furthermore, the parallel speedup and efficiency of the new approach are analyzed by comparing with the classical parareal algorithm and the WR technique, respectively. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations

We report a new parallel iterative algorithm for semi-linear parabolic partial differential equations (PDEs) by combining a kind of waveform relaxation (WR) techniques into the classical parareal algorithm. The parallelism can be simultaneously exploited by WR and parareal in different directions. We provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain, respectively. The iterations of the parareal and the WR are balanced to optimize the performance of the algorithm. Furthermore, the speedup and the parallel efficiency of the new approach are analyzed. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

A new domain decomposition waveform relaxation algorithm with local time-stepping

We propose a new domain decomposition waveform relaxation algorithm, which enables using different time steps across subdomains for parallel solving initial-boundary-value problems of linear parabolic equations. Distinct with the classical Schwarz waveform relaxation, the algorithm includes preconditioners to accelerate convergence and has greatly reduced memory requirements. A specific implementation of the local time stepping is also presented and its well-posedness is proved. The preconditioned system is analyzed at the continuous level. Finally, the efficiency of the algorithm is shown through numerical experiments.

Longitudinal-Partitioning-Based Waveform Relaxation Algorithm for Efficient Analysis of Distributed Transmission-Line Networks

In this paper, a waveform relaxation algorithm is presented for efficient transient analysis of large transmission-line networks. The proposed methodology represents lossy transmission lines as a cascade of lumped circuit elements alternating with lossless line segments, where the lossless line segments are modeled using the method of characteristics. Partitioning the transmission lines at the natural interfaces provided by the method of characteristics allows the resulting subcircuits to be weakly coupled by construction. The subcircuits are solved independently using a proposed hybrid iterative technique that combines the advantages of both traditional Gauss-Seidel and Gauss-Jacobi algorithms. The overall algorithm is highly parallelizable and exhibits good scaling with both the size of the network involved and the number of CPUs available. Numerical examples have been presented to illustrate the validity and efficiency of the proposed work.

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.

Higher-Order Cosimulation of Field/Circuit Coupled Problems

This paper discusses the weak coupling of nonlinear magnetoquasistatic field models (e.g., of induction motors) to an external electric network model (e.g., of frequency converters). A piece-wise linear inductance (as lumped field model) is not sufficiently accurate if the magnetic field exhibits strong eddy current effects. Here, we propose to represent the field model in the circuit equations by a more complex surrogate model. It is fitted by dynamic iteration (`waveform relaxation') during the cosimulation procedure. We demonstrate the link between dynamic iterations and the achievable convergence order of the time integrator numerically.

Using Backward-Traversing Waveform Relaxation Algorithm to Perform Incremental Simulation and Simulation-on-Demand for Circuit Simulation

This paper considers the repeated-executing characteristics of circuit simulators in practical situations, and raises according methods. These methods are incremental simulation and simulation-on-demand (SOD, just simulate the portion of circuit contributing to user-wanted outputs). A specialized relaxation-based algorithm, Backward-traversing Waveform Relaxation (BTWR), is adopted as the fundamental simulation algorithm. All proposed methods have been implemented and tested. Experimental results justify the values of proposed methods.

ITERATIVE METHODS FOR SCALABLE UNCERTAINTY QUANTIFICATION IN COMPLEX NETWORKS

In this paper we address the problem of uncertainty management for robust design, and verification of large dynamic networks whose performance is affected by an equally large number of uncertain parameters. Many such networks (e.g., power, thermal, and communication networks) are often composed of weakly interacting subnetworks. We propose intrusive and nonintrusive iterative schemes that exploit such weak interconnections to overcome the dimensionality curse associated with traditional uncertainty quantification methods (e.g., generalized polynomial chaos, probabilistic collocation) and accelerate uncertainty propagation in systems with a large number of uncertain parameters. This approach relies on integrating graph theoretic methods and waveform relaxation with generalized polynomial chaos, and probabilistic collocation, rendering these techniques scalable. We introduce an approximate Galerkin projection that based on the results of graph decomposition computes "strong" and "weak" influence of parameters on states. An appropriate order of expansion, in terms of the parameters, is then selected for the various states. We analyze convergence properties of this scheme and illustrate it in several examples.

Incremental Circuit Simulation and Simulation-on-demand based on Backward-traversing Waveform Relaxation and Portion-simulating Algorithms

Abstract In this paper, the practical behavior of circuit simulator users are considered and utilized to speed up the circuit simulation. The used methods are incremental simulation and simulation-on-demand (SOD, just simulate the portion of circuit contributing to user-wanted outputs). A specialized relaxation-based algorithm, Backward-traversing Waveform Relaxation (BTWR), is adopted as the fundamental simulation algorithm. All proposed methods have been implemented and tested. Experimental results justify the values of proposed methods.

Optimized Schwarz Waveform Relaxation and Discontinuous Galerkin Time Stepping for Heterogeneous Problems

We design and analyze a Schwarz waveform relaxation algorithm for domain decomposition of advection-diffusion-reaction problems with strong heterogeneities. The interfaces are curved, and we use optimized Ventcell transmission conditions. We analyze the semidiscretization in time with discontinuous Galerkin as well. We also show two-dimensional numerical results using generalized mortar finite elements in space.

On the Convergence of Continuous‐Time Waveform Relaxation Methods for Singular Perturbation Initial Value Problems

This paper extends the continuous‐time waveform relaxation method to singular perturbation initial value problems. The sufficient conditions for convergence of continuous‐time waveform relaxation methods for singular perturbation initial value problems are given.

Waveform Relaxation-Based Large-Scale Circuit Simulation for MOSFET and Lossy Coupled Transmission Line Circuits

In this paper, we propose a modified Waveform Relaxation algorithm to perform large-scale circuit simulation for MOSFET circuits containing lossy coupled transmission lines that have been encountered in modern circuit design community, in which a full time-domain transmission line calculation algorithm based on the Method of Characteristic is adopted. New software techniques are proposed to enhance the robustness as well as efficiency of the simulation process. All proposed methods have been implemented and executed to justify the claimed advantages.

Transient Simulation of Complex High-Speed Channels via Waveform Relaxation

This paper presents a class of numerical schemes for fast transient simulation of electrically long and complex linear channels terminated by linear or nonlinear networks. The common denominator of all schemes is waveform relaxation. Domain decomposition approaches based on longitudinal and transverse partitioning are pursued, leading to various iterative methods characterized by different properties and numerical efficiency. For each scheme, we present a detailed convergence analysis and a set of numerical results obtained on industrial benchmarks. The main contribution of this paper is a novel theoretical framework based on a combination of longitudinal and transverse relaxation, leading to particularly efficient simulations when combined with compact channel representations based on delay-rational macromodels. Our prototypal implementation outperforms SPICE solvers, with speedup of up to two orders of magnitude.

An iterative splitting method via waveform relaxation

This paper explores a new numerical strategy for a closed formulation of iterative splitting methods and their embedding in classical waveform-relaxation methods. Since iterative splitting has been developed in several papers, an abstract framework that relates these methods to other classical splitting methods would be useful and is needed. Here, we present an embedding of the iterative splitting method in the waveform-relaxation and exponential splitting methods. While we can use the theoretical background of the classical schemes, a simpler iterative splitting analysis is obtained. This is achieved by basing the analysis on semigroup and fixpoint schemes. Our approach is illustrated with numerical results obtained on differential equations with constant and time-dependent coefficients.

Multi-Splitting Waveform Relaxation Methods for Solving the Initial Value Problem of Linear Integral-Differential-Algebraic Equations

We present the multi-splitting waveform relaxation (MSWR) methods for solving the initial value problem of linear integral-differential-algebraic equations. Based on the spectral radius of the derived operator by decoupled process, a convergent condition is proposed for the MSWR method. Finally we discussed the convergent condition of discrete-time case of MSWR.

Quadrature methods for highly oscillatory linear and non-linear systems of ordinary differential equations: part II

This work presents methods of efficient numerical approximation for linear and nonlinear systems of highly oscillatory ordinary differential equations. We show how an appropriate choice of quadrature rule improves the accuracy of approximation as the frequency of oscillation grows. We present asymptotic and Filon-type methods to solve highly oscillatory linear systems of ODEs, and WRF method, representing a special combination of Filon-type methods and waveform relaxation methods, for nonlinear systems. Numerical examples support this paper.