Fast algorithm for computing nonlocal operators with finite interaction distance
Fast algorithm for computing nonlocal operators with finite interaction distance
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DOI：
10.4310/cms.2019.v17.n6.a7
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发表时间：
2019-05
影响因子：
1
通讯作者：
Xiaochuan Tian;B. Engquist
中科院分区：
文献类型：
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作者：
Xiaochuan Tian;B. Engquist
在过去的几年里，用于建模过程的非局部算子的发展在过去几年中越来越活跃，这些算子传统上是由局部微分算子描述的。一个例子是脆性材料的周期动力学，另一个例子是非标准扩散，包括使用分数导数。这些方法应用的一个主要障碍是非局部算子的数值实现带来的高计算代价。考虑快速多极或分层矩阵类型的快速方法来克服这一挑战是很自然的。不幸的是，相关的核不满足标准的必要条件。本文提出并分析了一类新的快速算法，在某些情况下，将非局部算子应用到与标准局部数值方法的复杂度基本相同的数量级，从而降低了计算复杂度。
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.