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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.
We present a new fast algorithm for computing the Boys function using nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.
We present a fast method for evaluating expressions of the form $$ u_j = sum_{i = 1,i ot = j}^n frac{alpha_i}{x_i - x_j}, quad text{for} quad j = 1,ldots,n, $$ where $alpha_i$ are real numbers, and $x_i$ are points in a compact interval of $mathbb{R
In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive comput
Kolmogorov famously proved that multivariate continuous functions can be represented as a superposition of a small number of univariate continuous functions, $$ f(x_1,dots,x_n) = sum_{q=0}^{2n+1} chi^q left( sum_{p=1}^n psi^{pq}(x_p) right).$$ Fridma
The Cadzows algorithm is a signal denoising and recovery method which was designed for signals corresponding to low rank Hankel matrices. In this paper we first introduce a Fast Cadzows algorithm which is developed by incorporating a novel subspace p