"Truncated Fourier Filtering" method for fast and high-order evaluation of integrals and convolutions in general domains

Abstract

This paper introduces and analyzes a novel algorithm---Truncated Fourier Filtering (TFF)---for the fast, high-order accurate evaluation of standard integrals and convolutions involving piecewise-smooth (possibly discontinuous) integrands over general $m$-dimensional domains ($m \ge 1$) employing an $m$-dimensional Cartesian grid. For an $N$-point discretization, the method runs at a computational cost of $\mathcal{O}(N)$ operations for standard integrals and $\mathcal{O}(N \log N)$ operations for convolutions, following, in either case, a one-time $\mathcal{O}(N \log N)$ precomputation step (not required in dimension $m = 1$). The core idea underlying TFF is to approximate the characteristic function of the integration domain by a truncated Fourier expansion of it over a suitably extended periodic domain, and to evaluate the resulting integrals via trapezoidal quadrature on a Cartesian grid with an appropriately chosen discretization size. Despite its conceptual simplicity, TFF attains high-order accuracy even for complex, possibly non-smooth or even non-Lipschitz geometries. A complete theoretical analysis is provided that establishes the superalgebraic convergence (i.e., convergence faster than any negative power of $N$) of the overall approach.