The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, 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 becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in \(O(N \log N)\) complexity and only requires \(O(\log N)\) 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 \({{\mathscr{H}}}\)-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to \({{\mathscr{H}}}^{2}\)-matrix compression techniques. Following this idea, we formulate a variant of the \({{\mathscr{H}}}^{2}\)-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.

具有记忆的动态系统的数值解需要以进化的方式对 Volterra 积分算子进行有效评估。经过适当的离散化，基本问题可以表示为具有较低对角线但密集矩阵的矩阵向量乘积。对于典型应用，如分数扩散或具有延迟的大规模动态系统，用于存储矩阵近似值和完整数据历史的内存成本对于精确的数值近似值变得过高。对于卷积类型的 Volterra 积分算子，Schädle、Lopez-Fernandez 和 Lubich的快速且不经意的卷积求积方法解决了这个问题，并允许计算具有N 个时间步长的离散化评估\(O(N \log N)\)复杂度并且只需要\(O(\log N)\)活动内存来存储数据完整历史记录的压缩版本。我们将证明该算法可以解释为基础积分运算符的\({{\mathscr{H}}}\) -矩阵近似。因此，原则上可以通过采用\({{\mathscr{H}}}}^{2}\) -矩阵压缩技术来实现进一步的改进。遵循这个想法，我们为离散化的 Volterra 积分算子制定了\({{\mathscr{H}}}^{2}\) -matrix-vector 乘积的变体，它可以以进化和不经意的方式执行，并且只需要O ( N ) 操作和\(O(\log N)\)主动记忆。除了加速之外，还可以处理更一般的渐近平滑内核，并且该算法不需要时间步数的先验知识。通过对一些典型测试问题的应用，证明了所提出方法的有效性。