Volterra积分算子的快速不经意矩阵压缩算法
A fast and oblivious matrix compression algorithm for Volterra integral operators
摘要：
具有记忆的动力系统的数值解需要以进化的方式对Volterra积分算子进行有效的计算。在适当的离散化之后，基本问题可以表示为具有较低对角但密集填充的矩阵的矩阵向量积。对于典型的应用，如分数扩散或具有延迟的大规模动态系统，用于存储矩阵近似和数据的完整历史的存储器成本对于精确的数值逼近将变得过高。对于卷积类型的Volterra-积分算子，Schädle、Lopez-Fernandez和Lubich的\emph{fast and oblivious convolution quadrature}方法允许以$O(N \log N)$复杂度计算具有$N$时间步长的离散化赋值，并且仅需要$O(\log N)$活动存储器来存储数据的完整历史的压缩版本。我们将表明，该算法可以被解释为一个$\mathcal{H}$矩阵近似的基本积分算子，因此，可以实现进一步的改进，在原则上，通过诉诸$\mathcal{H}^2$矩阵压缩技术。我们制定了一个变种的$\mathcal{H}^2$ -矩阵向量积离散Volterra积分算子，可以在一个渐进的和不经意的方式进行，只需要$O(N)$操作和$O(\log N)$主动记忆。除了加速，更一般的渐近光滑内核可以处理和算法不需要先验知识的时间步长的数量。所提出的方法的效率被证明通过应用到一些典型的测试问题。
英文摘要：
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then would become prohibitive for an accurate numerical approximation. For Volterra-integral operators of convolution type, the \emph{fast and oblivious convolution quadrature} method of Schädle, Lopez-Fernandez, and Lubich allows to compute the discretized valuation with $N$ time steps in $O(N \log N)$ complexity and only requiring $O(\log N)$ active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an $\mathcal{H}$-matrix approximation of the underlying integral operator and, consequently, a further improvement can be achieved, in principle, by resorting to $\mathcal{H}^2$-matrix compression techniques. We formulate a variant of the $\mathcal{H}^2$-matrix vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only $O(N)$ operations and $O(\log N)$ active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a-priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.
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