A μ-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the mu-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB (R)/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach

Backward error analysis of polynomial approximations for computing the action of the matrix exponential

We describe how to perform the backward error analysis for the approximation of exp(A)v by p(s 121 A) s v, for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja\u2013Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples

A μ\mu-mode integrator for solving evolution equations in Kronecker form

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem

A μ\mu-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the $\mu$-mode product (also known as tensor-matrix product or mode- $n$ product) and related operations, such as the Tucker operator. Their MathWorks MATLAB/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach

A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation

The numerical approximation of nonsmooth solutions of the semilinear Klein–Gordon equation in the d-dimensional space, with d = 1, 2, 3, is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method (i.e., exponential Euler method), which can significantly improve the accuracy of the numerical solutions under low-regularity conditions compared with other second-order methods. In particular, the proposed time-stepping method can have second-order convergence in the energy space under the regularity condition $(u,{\mathrm{\partial }}_tu)\in {L}^{\mathrm{\infty }}(0,T;{H}^{1+\frac{d}{4}}\times {H}^{\frac{d}{4}})$ . In one dimension, the proposed method is shown to have almost $\frac{4}{3}$ -order convergence in L∞(0, T; H1 × L2) for solutions in the same space, i.e., no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. The numerical experiments show that the proposed time-stepping method is much more accurate than previously proposed methods for approximating the time dynamics of nonsmooth solutions of the semilinear Klein–Gordon equation