Communications in Mathematical Sciences
Volume 17 (2019)
Fast algorithm for computing nonlocal operators with finite interaction distance
Pages: 1653 – 1670
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 in 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.
nonlocal operator, fast algorithm, nonlocal diffusion, peridynamics, heterogeneous material, fast multipole method, finite interaction
2010 Mathematics Subject Classification
45K05, 65F05, 65R20, 82C21
The research of Bjorn Engquist is supported in part by the U.S. NSF grant DMS-1620396. The research of Xiaochuan Tian is supported in part by the U.S. NSF grant DMS-1819233.
Received 23 January 2019
Accepted 1 July 2019
Published 26 December 2019