Waveform Relaxation Iterations Research Articles

Discrete-time analysis of optimized Schwarz waveform relaxation with Robin parameters depending on the targeted iteration count

We propose a new approach that provides new results in the convergence analysis of optimized Schwarz waveform relaxation (OSWR) iterations for parabolic problems, and allows to define efficient optimized Robin parameters that depend on the targeted iteration count, a property that is shared by the actual observed optimal parameters, while traditional Fourier analysis in the time direction leads to iteration independent parameters. This new approach is based on the exact resolution of the time semi-discrete error equations. It allows to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. While the general ideas may apply to an arbitrary space dimension, the analysis is first presented in the one dimensional case. Numerical experiments illustrate the performance obtained with such iteration-dependent optimized Robin parameters.

Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique

We present in this paper a time parallel algorithm for $${\dot{u}}=f(t,u)$$ with initial-value $$u(0)=u_0$$ , by using the waveform relaxation (WR) technique, and the diagonalization technique. With a suitable parameter $$\alpha $$ , the WR technique generates a functional sequence $$\{u^k(t)\}$$ via the dynamic iterations $${\dot{u}}^k=f(t,u^k)$$ , $$u^k(0)=\alpha u^k(T)-\alpha u^{k-1}(T)+u_0$$ , and at convergence we get $$u^{\infty }(t)=u(t)$$ . Each WR iterate represents a periodic-like differential equation, which is very suitable for applying the diagonalization technique yielding direct parallel-in-time computation. The parameter $$\alpha $$ controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the WR iterations, and we perform a detailed analysis for the influence of the parameter $$\alpha $$ on the method. We show that the roundoff error is proportional to $$\epsilon (2N+1)\max \{|\alpha |^2, |\alpha |^{-2}\}$$ ( $$N=T/\varDelta t$$ and $$\epsilon $$ is the machine precision), and the convergence factor can be bounded by $$|\alpha |e^{-TL}/(1-|\alpha |e^{-TL})$$ , where $$L\ge 0$$ is the one-sided Lipschitz constant of f. We also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored. Our analysis includes the heat and wave equations as special cases. Numerical results are given to support our findings.

A multigrid waveform relaxation method for solving the poroelasticity equations

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.

The WR–HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem

We consider the numerical method for non-self-adjoint positive definite linear differential equations, and its application to the unsteady discrete elliptic problem, which is derived from spatial discretization of the unsteady elliptic problem with Dirichlet boundary condition. Based on the idea of the alternating direction implicit (ADI) iteration technique and the Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation (WR) iteration method for solving the non-self-adjoint positive definite linear differential equations, called the WR–HSS method. We analyze the convergence property of the WR–HSS method, and prove that the WR–HSS method is unconditionally convergent to the solution of the system of linear differential equations. In addition, we derive the upper bound of the contraction factor of the WR–HSS method in each iteration which is only dependent on the Hermitian part of the corresponding non-self-adjoint positive definite linear differential operator. Finally, the applications of the WR–HSS method to the unsteady discrete elliptic problem demonstrate its effectiveness and the correctness of the theoretical results.

Waveform relaxation for the computational homogenization of multiscale magnetoquasistatic problems

This paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems. In the monolithic heterogeneous multiscale method, the nonlinear macroscale problem is solved using the Newton–Raphson scheme. The resolution of many mesoscale problems per Gauß point allows to compute the homogenized constitutive law and its derivative by finite differences. In the proposed approach, the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations. The exchange of information between both problems is still carried out using the heterogeneous multiscale method. However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gauß point.

Parallel methods for weakly singular Volterra integral equations on GPUs

The purpose of this paper is to employ graphics processing units for the numerical solution of large systems of weakly singular Volterra Integral Equations (VIEs), by means of Waveform Relaxation (WR) methods. A CUDA solver based on different kinds of WR iterations is developed. Numerical results on large systems of VIEs arising from the semi-discretization in space of fractional diffusion-wave equations are presented, showing the obtained speed-up.

Electromagnetic Interference Analysis of Multiconductor Transmission Line Networks Using Longitudinal Partitioning-Based Waveform Relaxation Algorithm

With the use of low powered devices, susceptibility of high-speed interconnects to electromagnetic interference (EMI) is becoming a critical aspect of signal integrity analysis. For modeling the EMI in time domain, commercial circuit simulators like SPICE typically use longitudinal segmentation methodologies to discretize the interconnect network. For long lines as found in printed circuit board or cables, a large number of longitudinal segments are required to capture the response of the network leading to inefficient simulations. In this study, a waveform relaxation (WR) algorithm for the efficient EMI analysis of multiconductor transmission line networks is presented. Techniques to compress the size of the subcircuits, reduce communication overheads, and accelerate the convergence of the WR iterations are provided. The overall algorithm is demonstrated to be highly parallelizable and exhibits good scaling with both the size of the network involved and the number of central processing units available.

Two stage waveform relaxation method for the initial value problems of differential-algebraic equations

In this paper, we consider a two stage strategy for waveform relaxation (WR) iterations, applied to initial value problems for differential-algebraic equations (DAEs) in the form Aẏ(t)+By(t)=f(t). Outer iterations of TSWR are defined by MAẏ(k+1)(t)+M1y(k+1)(t)=N1y(k)(t)+NAẏ(k)(t)+f(t), where A=MA−NA, B=M1−N1, and each iteration y(k+1)(t) is computed using an inner iterative process, based on another splitting M1=M2−N2. Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when MA is an Hermitian positive semi-definite matrix with P-regular splittings, the convergence and the comparison theorems of TSWR are analyzed. Finally, the numerical experiments are presented.

Parallel and Scalable Transient Simulator for Power Grids via Waveform Relaxation (PTS-PWR)

This paper presents a fast algorithm for transient simulation of power grids in very large scale integration systems using waveform relaxation (WR) techniques. Novel partitioning methods and convergence accelerators are developed for fast convergence of WR iterations when applied to power grid networks. Unlike the direct solvers, the new method is highly parallelizable and scales well with the increasing number of CPUs, leading to significant speed-ups. Numerical examples are presented to demonstrate the validity and efficiency of the proposed method.

A Modified Discrete-Time Jacobi Waveform Relaxation Iteration

In this paper, we investigate an accelerated version of the discrete-time Jacobi waveform relaxation iteration method. Based on the well known Chebyshev polynomial theory, we show that significant speed up can be achieved by taking linear combinations of earlier iterates. The convergence and convergence speed of the new iterative method are presented and it is shown that the convergence speed of the new iterative method is sharper than that of the Jacobi method but blunter than that of the optimal SOR method. Moreover, at every iteration the new iterative method needs almost equal computation work and memory storage with the Jacobi method, and more important it can completely exploit the particular advantages of the Jacobi method in the sense of parallelism. We validate our theoretical conclusions with numerical experiments.

An analysis of convergence for nonstationary two-stage multisplitting waveform relaxation methods

In this paper, we consider a nonstationary multisplitting two-stage strategy for waveform relaxation iterations, solving the initial value problem of ordinary differential equations. Convergence properties will be given when the coefficient matrix is an M-matrix or a Hermitian positive definite matrix. Finally numerical experiments are presented. In Memory of Professor D.J. Evans

Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge–Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval.

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.

An iterated pseudospectral method for delay partial differential equations

The Chebyshev pseudospectral semi-discretization preconditioned by a transformation in space is applied to delay partial differential equations. The Jacobi waveform relaxation method is then applied to the resulting semi-discrete delay systems, which gives simple systems of ordinary equations d d t U k ( t ) = M α U k ( t ) + f α ( t , U t k − 1 ) . Here, M α is a diagonal matrix, which depends on a parameter α ∈ [ 0 , 1 ] , which is used in the transformation in space, k is the index of waveform relaxation iterations, U t k is a functional argument computed from the previous iterate and the function f α , like the matrix M α , depends on the process of semi-discretization. Jacobi waveform relaxation splitting has the advantage of straightforward (because M α is diagonal) application of implicit numerical methods for time integration. Another advantage of Jacobi waveform relaxation is that the resulting systems of ordinary differential equations can be efficiently integrated in a parallel computing environment. The spatial transformation is used to speed up the convergence of waveform relaxation by preconditioning the Chebyshev pseudospectral differentiation matrix. We study the relationship between the parameter α and the convergence of waveform relaxation with error bounds derived here for the iteration process. We find that convergence of waveform relaxation improves as α increases, with the greatest improvement at α = 1 . These results are confirmed by numerical experiments for hyperbolic, parabolic and mixed hyperbolic-parabolic problems with and without delay terms.

An analysis of convergence for two-stage waveform relaxation methods

This paper consider a two-stage (or inner/outer) strategy for waveform relaxation (WR) iterations, applied to initial value problems for linear systems of ordinary differential equations (ODEs) in the form y ̇ (t)+Qy(t)=f(t) . Outer WR iterations are defined by y ̇ k+1(t)+Dy k+1(t)=N 1y k(t)+f(t) , where Q= D− N 1, and each iteration y k+1 ( t) is computed using an inner iterative process, based on an other splitting D= M− N 2. Each ODE is then discretized by means of Theta method. For an M-matrix Q we prove that the method converges under the assumption that the whole splitting Q= M− N 1− N 2 is an M-splitting, independently of the number of inner iterations. Moreover, some comparison results are given in order to relate the ratio of convergence of the whole inner/outer process both to the number of inner iterations actually done and to discretization parameters h and θ. Finally numerical experiments are presented.

Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wa′ e Sa + g(t) w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W e W1−W2 and S e S1 + S2 and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system a′ e W−1Sa + g(t) W−1w are somewhat faster but less convenient to implement since the matrix W−1S is no longer sparse. The pseudospectral methods lead to the system U′ e \tilde DU + g(t) w with a differentiation matrix \tilde D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).

Circulant preconditioned WR-BVM methods for ODE systems

We consider the solution of a system of ordinary differential equations (ODEs) by waveform relaxation (WR) iterations in conjunction with boundary value methods (BVMs). The WR method is a continuous-in-time analogue of the stationary method and it iterates with functions. In each WR iteration, we use BVMs to discretize systems of ODEs. BVMs are relatively new ODE solvers based on linear multistep formulae. In this paper, we discuss the use of the generalized minimal residual (GMRES) method with block-circulant–circulant-block preconditioners for solving the linear systems arising from the application of BVMs in each WR iteration. These preconditioners are effective in speeding up the convergence rate of the GMRES method. Numerical experiments are presented to illustrate the effectiveness of our methods.

Preconditioned WR–LMF-based method for ODE systems

The waveform relaxation (WR) method was developed as an iterative method for solving large systems of ordinary differential equations (ODEs). In each WR iteration, we are required to solve a system of ODEs. We then introduce the boundary value method (BVM) which is a relatively new method based on the linear multistep formulae to solve ODEs. In particular, we apply the generalized minimal residual method with the Strang-type block-circulant preconditioner for solving linear systems arising from the application of BVMs to each WR iteration. It is demonstrated that these techniques are very effective in speeding up the convergence rate of the resulting iterative processes. Numerical experiments are presented to illustrate the effectiveness of our methods.

Frequency analysis of preconditioned waveform relaxation iterations

The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the err