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Parsimonious convolution quadrature

Jens M. Melenk, Jörg Nick

Abstract

We present a method to rapidly approximate convolution quadrature (CQ) approximations, based on a piecewise polynomial interpolation of the Laplace domain operator, which we call the \emph{parsimonious} convolution quadrature method. For implicit Euler and second order backward difference formula based discretizations, we require $O(\sqrt{N}\log N)$ evaluations in the Laplace domain to approximate $N$ time steps of the convolution quadrature method to satisfactory accuracy. The methodology proposed here differentiates from the well-understood fast and oblivious convolution quadrature \cite{SLL06}, since it is applicable to Laplace domain operator families that are only defined and polynomially bounded on a positive half space, which includes acoustic and electromagnetic wave scattering problems. The methods is applicable to linear and nonlinear integral equations. To elucidate the core idea, we give a complete and extensive analysis of the simplest case and derive worst-case estimates for the performance of parsimonious CQ based on the implicit Euler method. For sectorial Laplace transforms, we obtain methods that require $O(\log^2 N)$ Laplace domain evaluations on the complex right-half space. We present different implementation strategies, which only differ slightly from the classical realization of CQ methods. Numerical experiments demonstrate the use of the method with a time-dependent acoustic scattering problem, which was discretized by the boundary element method in space.

Parsimonious convolution quadrature

Abstract

We present a method to rapidly approximate convolution quadrature (CQ) approximations, based on a piecewise polynomial interpolation of the Laplace domain operator, which we call the \emph{parsimonious} convolution quadrature method. For implicit Euler and second order backward difference formula based discretizations, we require evaluations in the Laplace domain to approximate time steps of the convolution quadrature method to satisfactory accuracy. The methodology proposed here differentiates from the well-understood fast and oblivious convolution quadrature \cite{SLL06}, since it is applicable to Laplace domain operator families that are only defined and polynomially bounded on a positive half space, which includes acoustic and electromagnetic wave scattering problems. The methods is applicable to linear and nonlinear integral equations. To elucidate the core idea, we give a complete and extensive analysis of the simplest case and derive worst-case estimates for the performance of parsimonious CQ based on the implicit Euler method. For sectorial Laplace transforms, we obtain methods that require Laplace domain evaluations on the complex right-half space. We present different implementation strategies, which only differ slightly from the classical realization of CQ methods. Numerical experiments demonstrate the use of the method with a time-dependent acoustic scattering problem, which was discretized by the boundary element method in space.

Paper Structure

This paper contains 18 sections, 4 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. The convolution quadrature method
  4. Standard assembly of the quadrature weights
  5. Parsimonious CQ by piecewise polynomial kernel approximation
  6. Kernel approximation by piecewise polynomials
  7. The domain of analyticity in the half space case
  8. Parsimonious CQ based on the implicit Euler method
  9. A specific choice of mesh for the implicit Euler method
  10. Approximation result for the implicit Euler method
  11. Parsimonious CQ based on higher order methods
  12. The case of BDF2
  13. Higher order methods
  14. The sectorial case
  15. Implementation and numerical experiments
  16. ...and 3 more sections

Key Result

Lemma 3.3

Let $f\,:\, D_\rho(-1,1) \rightarrow \mathbb C$ be holomorphic. The error of the Chebyshev interpolant $I^{[-1,1]}_p f \in {\mathcal{P}}_p$ on the interval $[-1,1]$ is bounded by If $f|_{(-1,1)}$ is real-valued, then $I_p f$ is likewise real-valued on $(-1,1)$.

Figures (5)

  • Figure 1: The convergence of an implicit Euler based convolution quadrature approximation and the corresponding modified method applied to the hyperbolic example \ref{['eq:experiment-scalar']}. Shown is the error on the $y$-axis, which is set in relation to the number of Laplace evaluations on the $x$-axis. On the right-hand side, we observe the error introduced by the interpolation used to compute the modified approximation.
  • Figure 2: Error vs. number of Laplace transform evaluations $L$ for CQ based on different multistep methods applied to the hyperbolic example \ref{['hyperbolic-kernel']}.
  • Figure 3: CQ based on implicit Euler applied to the sectorial example \ref{['sectorial-kernel']} with $\alpha=\frac{2}{3}$, which corresponds to $\gamma=1$ (i.e., a sector with $45^{\circ}$ angle).
  • Figure 4: CQ based on different multistep methods of order $2$ applied to the sectorial example \ref{['sectorial-kernel']} with $\gamma=1$ (which corresponds to a sector with $45^{\circ}$ angle).
  • Figure 5: Temporal convergence of CQ based on implicit Euler applied to the convolution \ref{['eq:experiment-scat']}, which computes the scattered wave solution to the acoustic scattering problem. The grid has been fixed to a coarse grid with $2836$ degrees of freedom. On the right-hand side, we again observe the error introduced by the modification of the method.

Theorems & Definitions (14)

  • Remark 2.1: On the assumptions on $K$
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1: Intervals for the implicit Euler method
  • proof
  • Remark 4.2
  • Theorem 4.3: error of parsimonious CQ
  • proof
  • ...and 4 more