Fast discrete convolution in \(\mathbb{R}^2\) with radial kernels using non-uniform fast Fourier transform with nonequispaced frequencies. (English) Zbl 1431.65252

Let \(z_k\in {\mathbb R}^2\) be given points of the unit disk and \(f_k\in \mathbb C\), \(k=1,\ldots,N\). Further, \(\|x\|\) is the Euclidean norm of \(x\in {\mathbb R}^2\). Let \(g:(0,\infty) \to \mathbb R\) be given. Later \(g(t)\) is chosen as \(\log t\), \(t^2 \log t\), and \(t^{-2}\), \(t>0\). In this paper, the author presents a new algorithm, the so-called efficient Bessel decomposition (EBD), for the fast evaluation of discrete convolutions with the radial kernel \(g(\|x\|)\) of the form \[ q_k = \sum_{\ell=1}^N g(\|z_k -z_{\ell}\|)\,f_{\ell}\,, \quad k = 1,\ldots,N\,. \] Using a trigonometric approximation of \(g(\|x\|)\), the discrete convolution can be rapidly computed by non-uniform fast Fourier transforms, see [D. Potts et al., Numer. Math. 98, No. 2, 329–351 (2004; Zbl 1056.65146)]. The new EBD method approximates \(g(\|x\|)\) by a Bessel series outside the origin by a small number \(P\) of terms \[ g(\|x\|) \approx g(1) + \sum_{n=1}^P \alpha_n\,J_0(\rho_n \|x\|)\,, \quad \delta \le \|x\|\le 1\,, \] where \(J_0\) is the Bessel function of first kind, \(\rho_n\) is the \(n\)th positive zero of \(J_0\), and \(\delta >0\) is a cutoff parameter. Applying the trapezoidal rule to the integral \[ J_0(\rho_n \|x\|) = \frac{1}{2\pi} \int_{\|y\|=1} {\mathrm e}^{{\mathrm i}\rho_n\,x\cdot y}\,{\mathrm d}y\,, \] the author obtains a trigonometric approximation of \(g(\|x\|)\). An important consequence of EBD is the fact that much less terms last for a given accuracy. This allows for a faster evaluation of the discrete convolution at the cost of longer precomputation. A Matlab code of this method is available online. The computational cost and the error of the EBD method are estimated. Numerical results are presented too.


65T50 Numerical methods for discrete and fast Fourier transforms
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
41A55 Approximate quadratures


Zbl 1056.65146


Full Text: DOI


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