Shidong Jiang

Shidong Jiang, Jiwei Zhang, Qian Zhang et al.

We present an efficient algorithm for the evaluation of the Caputo fractional derivative $_0^C\!D_t^αf(t)$ of order $α\in (0,1)$, 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 $[Δt, T]$ with a uniform absolute error $\varepsilon$, where the number of exponentials $N_{\text{exp}}$ needed is of the order $O\left(\log\frac{1}{\varepsilon}\left( \log\log\frac{1}{\varepsilon}+\log\frac{T}{Δt}\right) +\log\frac{1}{Δt}\left( \log\log\frac{1}{\varepsilon}+\log\frac{1}{Δt}\right) \right)$. As compared with the direct method, the resulting algorithm reduces the storage requirement from $O(N_T)$ to $O(N_{\text{exp}})$ and the overall computational cost from $O(N_T^2)$ to $O(N_TN_{\text{exp}})$ 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 $O(N_SN_{\text{exp}})$ storage and $O(N_SN_TN_{\text{exp}})$ work with $N_S$ the total number of points in space; whereas the direct methods require $O(N_SN_T$) storage and $O(N_SN_T^2)$ work. The complexity of both algorithms is nearly optimal since $N_{\text{exp}}$ is of the order $O(\log N_T)$ for $T\gg 1$ or $O(\log^2N_T)$ for $T\approx 1$ for fixed accuracy $\varepsilon$. 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.

Jiuyang Liang, Libin Lu, Alex Barnett et al.

The evaluation of long-range Coulomb interactions is a significant cost in molecular dynamics (MD), even when using Particle Mesh Ewald (PME) or Particle-Particle-Particle-Mesh (PPPM) methods, which rely on Ewald splitting and the fast Fourier transform to achieve near-linear scaling. We introduce ESP -- Ewald summation with prolate spheroidal wave functions (PSWFs) -- which leads to a more efficient Fourier representation and a reduction in the required grid size, global communication, and particle-grid operations, without loss of accuracy. We have integrated the ESP method into two widely-used open-source MD packages, LAMMPS and GROMACS, enabling rapid comparison and adoption. Relative to PME/PPPM baselines at error tolerances $10^{-3}$ to $10^{-4}$, ESP gives roughly a $3$-fold acceleration of electrostatic interactions, and a $2.5$-fold speed-up in the MD simulation when using about $10^3$ compute cores. At high accuracy ($10^{-5}$), these increase to $10$-fold for the far-field electrostatics and $5$-fold for MD simulation. Furthermore, we show that the accelerated codes have improved strong scaling with core count, and validate them in realistic long-time biological and material simulations. ESP thus offers a practical, drop-in path to reduce the time-to-solution and energy footprint of MD workflows.

Leslie Greengard, Shidong Jiang, Jun Wang

Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts - a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algorithms. The local part, however, involves complicated singularities in the underlying Green's function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

Jun Lai, Shidong Jiang

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.

Alex H. Barnett, Charles L. Epstein, Leslie Greengard et al.

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step $Δt$ is of the order $O(Δx^2)$, where $Δx$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_dT^{d/2}$ times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient $κ$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $Δt < C/κ^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in $L^\infty$-norm.

Leslie Greengard, Shidong Jiang

We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each individual equation is well-condtioned, and we show that using them in predictor-corrector fashion, combined with spectral deferred correction, leads to high-order accuracy solvers. The fundamental unknowns in the mixed potential representation are densities supported on the boundary of the domain. We refer to one as the vortex source, the other as the pressure source and the coupled system as the combined source integral equation.

Weizhu Bao, Shidong Jiang, Qinglin Tang et al.

We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schrödinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in \cite{JGB} for nonlocal interactions where the singularity of the Fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates. We then extend the method to compute other nonlocal interactions whose Fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform. Many of these interactions do not decay at infinity in the physical space, which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain. The performance of our method against other existing methods is illustrated numerically, with particular attention on the effect of the size of the computational domain in the physical space. Finally, to study the ground state and dynamics of the NLSE, we propose efficient and accurate numerical methods by combining the NUFFT method for potential evaluation with the normalized gradient flow using backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with various nonlocal interactions. Numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency.