论文标题
在指数积分器中利用kronecker结构：通过正交矩阵的$φ$ functions的动作的快速近似
Exploiting Kronecker structure in exponential integrators: fast approximation of the action of $φ$-functions of matrices via quadrature
论文作者
论文摘要
在本文中，我们提出了一种算法，用于近似矩阵对向量的$φ-$函数的作用，这是指数时间积分器中的关键操作。特别是，我们考虑具有Kronecker总结构的矩阵，这是由于承认张量产品表示的问题而引起的。该方法基于$φ-$函数的积分形式的正交近似，结合了缩放和修改的平方方法。由于Kronecker总和表示，只能避免在每个正交节点和完整矩阵的组装时需要1D矩阵指数的动作。此外，我们得出了正交误差的\ emph {a先验}界限，这表明，正如经典理论所预期的那样，我们方法的收敛速率是超测量的。在我们的分析的指导下，我们构建了一种快速，可靠的方法，用于估计最佳缩放系数和正交节点的数量，从而最大程度地减少了规定的误差耐性的总成本。我们通过在2D和3D中求解几个线性和半线性时间依赖性问题来研究算法的性能。结果表明，我们的方法是准确的，比当前的最新时间更快。
In this article, we propose an algorithm for approximating the action of $φ-$functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the $φ-$functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive \emph{a priori} bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.
