$\mathcal{H}$-矩阵基于迭代和直接求解器的理论与实现：用于亥姆霍兹和弹性动力学振荡核
Theory and implementation of $\mathcal{H}$-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
AI总结：
本文研究了基于$\mathcal{H}$-矩阵的快速方法在求解三维弹性动力学格林张量离散化所产生密集线性系统中的精度与效率。通过$\mathcal{H}^2$矩阵和方向方法克服了标准$\mathcal{H}$-矩阵在振荡核中的不足，展示了在机械工程中该方法的有效性。
摘要：
本文研究了基于$\mathcal{H}$-矩阵的快速方法在求解三维弹性动力学格林张量离散化所产生密集线性系统中的精度与效率。已知标准$\mathcal{H}$-矩阵方法虽然在渐近光滑核中非常高效，但不适用于振荡核。已提出$\mathcal{H}^2$矩阵和方向方法以克服此问题。然而，此类方法的实现比标准$\mathcal{H}$-矩阵表示更为复杂。本文探讨了两个核心问题：(i) $\mathcal{H}$-矩阵格式在何种频率范围内能高效表示三维弹性动力学问题？(ii) 该方法在机械工程问题建模中能带来哪些预期效果？我们展示了尽管方法并非最优（在更复杂的表示能带来更快算法的意义上），但可以轻松开发出高效的求解器。通过边界元方法的数值示例展示了该方法的能力。
英文摘要：
In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.
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