We present an efficient algorithm for the evaluation of the Caputo fractional derivative C 0 D α t f (t) of order α ∈ (0, 1), which can be expressed as a convolution of f ′ (t) with the kernel t −α . The algorithm is based on an efficient sum-of-exponentials approximation for the kernel t −1−α on the interval [∆t, T ] with a uniform absolute error ε, where the number of exponentials Nexp needed is ofAs compared with the direct method, the resulting algorithm reduces the storage requirement from O(N T ) to O(Nexp) and the overall computational cost from O(N 2 T ) to O(N T Nexp) with N T the total number of time steps. Furthermore, when the fast evaluation scheme of the Caputo derivative is applied to solve the fractional diffusion equations, the resulting algorithm requires only O(N S Nexp) storage and O(N S N T Nexp) work with N S the total number of points in space; whereas the direct methods require O(N S N T ) storage and O(N S N 2 T ) work. The complexity of both algorithms is nearly optimal since Nexp is of the order O(log N T ) for T ≫ 1 or O(log 2 N T ) for T ≈ 1 for fixed accuracy ε. We also present a detailed stability and error analysis of the new scheme for solving linear fractional diffusion equations. The performance of the new algorithm is illustrated via several numerical examples. Finally, the algorithm can be parallelized in a straightforward manner.

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This study relates to the heteroepitaxy of InP on patterned Si substrates using the defect trapping technique. We carefully investigated the growth mechanism in shallow trench isolation trenches to optimize the nucleation layer. By comparing different recess engineering options: rounded-Ge versus V-grooved, we could show a strong enhancement of the crystalline quality and growth uniformity of the InP semiconductor. The demonstration of III-V heteroepitaxy at scaled dimensions opens the possibility for new applications integrated on Silicon.

Abstract. We present a fast and accurate algorithm for the evaluation of nonlocal (longrange) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel U (x) and a density function ρ(x) = |ψ(x)| 2 , for some complexvalued wave function ψ(x), permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel U (k) has a singularity at the origin k = 0 in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in k which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity inÛ (k) at the origin is canceled, so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function ρ(x) is smooth and decays sufficiently fast as x → ∞. More precisely, the calculation requires O(N log N ) operations, where N is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.Key words. Coulomb interaction, dipole-dipole interaction, interaction energy, nonuniform FFT, nonlocal, Poisson equation. AMS subject classifications. 33C10, 33F05, 44A35, 65R10, 65T50, 81Q401. Introduction. Nonlocal (long-range) interactions are encountered in modeling a variety of problems from quantum physics and chemistry to materials science and biology. A typical example is the Coulomb interaction in the nonlinear Schrödinger equation (or Schrödinger-Poisson system in three dimensions (3D)) as a "mean field limit" for N -electrons, assuming binary Coulomb interactions [10, 11, 23] and the Kohn-Sham equation of density functional theory (DFT) for electronic structure calculations in materials simulation and design [23, 38, 53, 56, 57]. Dipole-dipole interactions arise in quantum chemistry [32, 43], in dipolar Bose-Einstein condensation (BEC) [2-4, 31, 40, 46, 55, 62-64], in dipolar Fermi gases [48], and in dipole-dipole interacting Rydberg molecules [35][36][37].In physical space, the interaction kernel is both long-range and singular at the origin, requiring both accurate quadrature techniques and suitable fast algorithms. When the density function is smooth, however, it is often more convenient to use Fourier methods since the frequency content is well-controlled. Unfortunately, the Fourier transform of the interaction kernel is singular at the origin of Fourier (phase) space as well, resulting in significant numerical burdens and challenges [3, 4, 9, 13, 14, 26, 59, 66].In this paper, we present a fast and accurate algorithm for the numerical evalua-