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We present a fast direct solver for the simulation of electromagnetic scattering from an arbitrarily shaped, large, empty cavity embedded in an infinite perfectly conducting half-space. The governing Maxwell equations are reformulated as a well-conditioned second kind integral equation, and the resulting linear system is solved in nearly linear time using a hierarchical matrix factorization technique. We illustrate the performance of the scheme with several numerical examples for complex cavity shapes over a wide range of frequencies.

The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N²) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, \"SIAM J. Sci. Comput.\", 14 (1993), pp. 1368-1383]. In this paper, we observe that one of the standard interpolation or \"gridding\" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and three-dimensional settings, saving either$10^{d}N$in storage in d dimensions or a factor of about 5-10 in CPU time (independent of dimension).

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-conditioned, 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 to the coupled system as the combined source integral equation.

Some of the recently developed fast summation methods that have arisen in scientific computing are described. These methods require an amount of work proportional to N or N log N to evaluate all pairwise interactions in an ensemble of N particles. Traditional methods, by contrast, require an amount of work proportional to N$^2$. As a result, large-scale simulations can be carried out using only modest computer resources. In combination with supercomputers, it is possible to address questions that were previously out of reach. Problems from diffusion, gravitation, and wave propagation are considered.

We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson--Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or low-frequency Helmholtz equations, both phases typically have complexity $\\mathcal{O} (N)$ in two dimensions, where $N$ is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are $\\mathcal{O} (N^{3/2})$ and $\\mathcal{O} (N \\log N)$ for precomputation and solution, respectively. Extensive numerical experiments show a speedup of ${\\sim}100$ for the solution phase over modern fast multipole methods; however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with ill-conditioned linear systems or when the same system is to be solved with multiple right-hand sides. Our algorithm is implemented in Fortran and freely available. [PUBLICATION ABSTRACT]

We have developed a mathematical approach to the study of dynamical biological networks, based on combining large-scale numerical simulation with nonlinear \"dimensionality reduction\" methods. Our work was motivated by an interest in the complex organization of the signaling cascade centered on the neuronal phosphoprotein DARPP-32 (dopamine- and cAMP-regulated phosphoprotein of molecular weight 32,000). Our approach has allowed us to detect robust features of the system in the presence of noise. In particular, the global network topology serves to stabilize the net state of DARPP-32 phosphorylation in response to variation of the input levels of the neurotransmitters dopamine and glutamate, despite significant perturbation to the concentrations and levels of activity of a number of intermediate chemical species. Further, our results suggest that the entire topology of the network is needed to impart this stability to one portion of the network at the expense of the rest. This could have significant implications for systems biology, in that large, complex pathways may have properties that are not easily replicated with simple modules.

We present an adaptive fast multipole method for solving the Poisson equation in two dimensions. The algorithm is direct, assumes that the source distribution is discretized using an adaptive quad-tree, and allows for Dirichlet, Neumann, periodic, and free-space conditions to be imposed on the boundary of a square. The amount of work per grid point is comparable to that of classical fast solvers, even for highly nonuniform grids.

We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\\mathbf{x})$ and a density function $\\rho(\\mathbf{x})=|\\psi(\\mathbf{x})| arrow up $ for some complex-valued wave function $\\psi(\\mathbf{x})$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\\widehat{U}(\\mathbf{k})$ has a singularity and/or $\\widehat{\\rho}(\\mathbf{k})\\ne0$ at the origin $\\mathbf{k}={\\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\\mathbf{k}$ 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 in $\\hat{U}(\\mathbf{k})$ 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 $(\\rho\\mathbf{x})$ is smooth and decays sufficiently quickly as $|\\mathbf{x}| \\rightarrow \\infty$. 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.

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen Lp space and adaptively refined in the internal layer.

We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic, or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [F. Vico, L. Greengard, M. Ferrando, J. Comput. Phys., 323 (2016), pp. 191-203] to discretize and apply the relevant integral operators.