A Direct Time Parallel Solver by Diagonalization for the Wave Equation

Abstract

With the advent of very large scale parallel computers, it has become more and more important to also use the time direction for parallelization when solving evolution problems. While there are many successful algorithms for diffusive problems, only some of them are also effective for hyperbolic problems. We present here a mathematical analysis of a new method based on the diagonalization of the time stepping matrix proposed by Maday and Rønquist in 2007. Like many time-parallelization methods, at first this does not seem to be a very promising approach: the matrix is essentially triangular, or, for equidistant time steps, actually a Jordan block, and thus not diagonalizable. If one chooses however different time steps, diagonalization is possible, and one has to trade off between the accuracy due to necessarily having different time steps, and numerical errors in the diagonalization process of these almost nondiagonalizable matrices. We present for the first time such a diagonalization technique for the Newmark scheme for solving wave equations, and derive a mathematically rigorous optimization strategy for the choice of the parameters in the special case when the Newmark scheme becomes Crank-Nicolson. Our analysis shows that small to medium scale time parallelization is possible with this approach. We illustrate our results with numerical experiments for model wave equations in various dimensions and also an industrial test case for the elasticity equations with variable coefficients.

A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

“…A large condition number results in large roundoff error in the implementation of step-(a) and step-(c) of (1.3) due to floating point operations, which could seriously pollute the accuracy of the obtained numerical solution. This issue was carefully justified by Gander et al in [19] and in particular…”

“…where ǫ is the machine precision. In [19], the authors considered the geometrically increasing step-sizes {∆t j = ∆t 1 τ n−j } n j=1 and with this choice an explicit diagonalization of B can be written down, where τ > 1 is a parameter. However, it is very difficult to make a good choice of τ : if τ tends to 1 the matrix B tends to non-normal and thus the condition number of the eigenvector matrix V becomes very large; if τ is far larger than 1 the global discretization error will be an issue, because the step-sizes grows rapidly as n increases.…”

“…By a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for these three matrices V , V −1 and D. With the given formulas of V and V −1 , we prove that the condition number of V satisfies Cond 2 (V ) = O(n 2 ) and this implies that the roundoff error arising from the diagonalization procedure only increases moderately as n grows. Hence, compared to the algorithm in [19] a much larger n can be used to yield satisfactory parallelism in practice. We mention that the spectral decomposition algorithm developed in this paper is much faster than the benchmarking algorithm as implemented in MATLAB's eig function.…”

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

“…One obvious difficulty is how to address the underlying regularization treatment in the framework of PinT algorithms, which seems to be highly dependent on the regularized problem structure and discretization schemes. Inspired by several recent works [13,25,26,29,30,44] on diagonalization-based PinT algorithms, we propose to redesign the existing quasi-boundary value methods in a structured manner such that the diagonalization-based PinT direct solver can be successfully employed. Such a PinT direct solver can greatly speed up the quasi-boundary value methods while achieving a comparable reconstruction accuracy.…”

Toward error estimates for general space-time discretizations of the advection equation

“…There are however also other PinT algorithms especially designed for hyperbolic problems, see e.g. ParaExp [13,14], the diagonalization technique [18], and waveform relaxation [16,17,39,40], and combinations thereof, see e.g. [21], based on earlier work in [43].…”