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23
Accelerating the nonuniform Fast Fourier Transform
 SIAM REVIEW
, 2004
"... 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 recon ..."
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Cited by 72 (5 self)
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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(N 2) 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 threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).
Fast Evaluation of Nonreflecting Boundary Conditions for the Schrödinger Equation in One Dimension
, 2004
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On the numerical solution of the heat equation I: Fast solvers in free space
 Journal of Computational Physics
"... Abstract We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain. It relies on a recently developed spectral approximation of the freespace heat kernel coupled with the nonuniform fast Fourier transform. Unlik ..."
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Cited by 8 (3 self)
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Abstract We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain. It relies on a recently developed spectral approximation of the freespace heat kernel coupled with the nonuniform fast Fourier transform. Unlike finite difference and finite element techniques, there is no need for artificial boundary conditions on a finite computational domain. The method is explicit, unconditionally stable, and requires an amount of work of the order OðNM log N Þ, where N is the number of discretization points in physical space and M is the number of time steps. We refer to the approach as the fast recursive marching (FRM) method.
HIGH ORDER ACCURATE METHODS FOR THE EVALUATION OF LAYER HEAT POTENTIALS
, 2009
"... We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly d ..."
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Cited by 7 (3 self)
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We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as “geometrically induced stiffness,” but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the “history part” of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal
The Chebyshev Fast Gauss and Nonuniform Fast Fourier Transforms and their application to the evaluation of distributed heat potentials
, 2007
"... We present a method for the fast and accurate computation of distributed (or volume) heat potentials in two dimensions. The distributed source is assumed to be given in terms of piecewise spacetime Chebyshev polynomials. We discretize uniformly in time, whereas in space the polynomials are defined ..."
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Cited by 7 (3 self)
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We present a method for the fast and accurate computation of distributed (or volume) heat potentials in two dimensions. The distributed source is assumed to be given in terms of piecewise spacetime Chebyshev polynomials. We discretize uniformly in time, whereas in space the polynomials are defined on the leaf nodes of a quadtree data structure. The quadtree can vary at each time step. We combine a product integration rule with fast algorithms (fast heat potentials, nonuniform FFT, fast Gauss transform) to obtain a highorder accurate method with optimal complexity. If the input contains q 3 polynomial coefficients at M leaf nodes and N time steps, our method requires O(q 3 MN log M) work to evaluate the heat potential at arbitrary MN spacetime target locations. The overall convergence rate of the method is of order q. We present numerical experiments for q =4, 8, and 16, and we verify the theoretical convergence rate of the method. When the solution is sufficiently smooth, the 16thorder variant results in significant computational savings, even in the case in which we require only a few digits of accuracy.
The fast generalized Gauss transform
 In press) SIAM Journal on Scientific Computing
, 2010
"... Abstract. The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N +M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains ..."
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Cited by 6 (2 self)
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Abstract. The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N +M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust highorder product integration methods require convolution with O(q) distinct Gaussiantype kernels in order to obtain qth order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop planewave representations of these Gaussiantype fields, permitting the “diagonal translation ” version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be avoided. Under these conditions, our implementation has a lower operation count than the fast Fourier transform (FFT). Key words. Gauss transform, fast algorithms, heat potentials, highorder accuracy, tensorproduct grids AMS subject classifications. 35K05, 31A10, 65N38, 65Y20 1. Introduction. The fast Gauss transform (FGT) [9] computes discrete sums of the form N∑ k=1
A HighOrder Solver for the Heat Equation in 1D domains with Moving Boundaries
"... We describe a fast highorder accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of L. Greengard and J. Strain, “A fast algorithm fo ..."
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Cited by 5 (2 self)
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We describe a fast highorder accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of L. Greengard and J. Strain, “A fast algorithm for the evaluation of heat potentials”, Comm. Pure & Applied Math. 1990. Our scheme is based on a timespace Chebyshev pseudospectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with the Green’s function for the heat equation. We present numerical results that exhibit up to eighthorder convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in spacetime requires O(NM log M) work. Thus, our scheme can be characterized as “fast”, that is, it is workoptimal up to a logarithmic factor. Key words. Integral equations, spectral methods, Chebyshev polynomials, moving boundaries, heat equation, quadratures, Nyström’s method, collocation methods, potential theory.
Spectral Lagrangian methods for Collisional Models of Non Equilibrium Statistical States
, 710
"... We propose a new spectral Lagrangian based deterministic solver for the nonlinear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or nonconservative binary interactions. The method is based on symmetries of the Fourier transform of the collision i ..."
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We propose a new spectral Lagrangian based deterministic solver for the nonlinear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or nonconservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computing is reduced to a separate integral over the unit sphere S 2. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space is very versatile and adjusts in a very simple manner, to several cases that involve energy dissipation due to local microreversibility (inelastic interactions) or elastic model of slowing down process. Our simulations are benchmarked with the available exact selfsimilar solutions, exact moment equations and analytical estimates for homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the simulations involves the selection of a time selfsimilar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in [13] and generalized to a wide range of related models in [12]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hardspheres (inelastic collisions under thermal bath), where overpopulated nonGaussian exponential tails have been conjectured in computations by stochastic methods in [49; 26; 46; 35] and rigourously proven in [34] and [15].
An integral equation method for epitaxial stepflow growth simulations
 Journal of Computational Physics
"... In this paper, we describe an integral equation approach for simulating epitaxial stepflow growth. The solutions are represented as moving layer potentials where the unknowns are only defined on the steps between different terraces. The resulting integrodifferential equation (IDE) system is solved ..."
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In this paper, we describe an integral equation approach for simulating epitaxial stepflow growth. The solutions are represented as moving layer potentials where the unknowns are only defined on the steps between different terraces. The resulting integrodifferential equation (IDE) system is solved using spectral deferred correction techniques developed for general differential algebraic equation (DAE) systems, and the time dependent potentials are evaluated efficiently using fast convolution algorithms. This approach can be applied to the accurate and efficient solutions of general moving interface problems arising in science and engineering.
Nyström Discretization of parabolic Boundary Integral Equations
"... A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal ..."
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A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for secondkind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers. 1