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17
Discrete symbol calculus
, 2008
"... This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phasespace, i.e., functions of space x and frequency ξ. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equati ..."
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Cited by 8 (6 self)
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This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phasespace, i.e., functions of space x and frequency ξ. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fastconverging, nonasymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbolbased numerical methods handle operators and not functions, their complexity depends on the desired resolution N very weakly, typically only through log N factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into oneway components and 3) depthstepping in reflection seismology.
A FAST BUTTERFLY ALGORITHM FOR THE COMPUTATION OF FOURIER INTEGRAL OPERATORS
, 2009
"... This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, wherekisafrequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest, for such fundamental computation ..."
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Cited by 6 (2 self)
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This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, wherekisafrequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest, for such fundamental computations are connected with the problem of finding numerical solutions to wave equations and also frequently arise in many applications including reflection seismology, curvilinear tomography, and others. In two dimensions, when the input and output are sampled on N × N Cartesian grids, a direct evaluation requires O(N 4) operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in O(N 2 log N) time, i.e., with nearoptimal computational complexity, and whose overall structure follows that of the butterfly algorithm. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel e2πıΦ(x,k) to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel is approximately lowrank; we propose constructing such lowrank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part, and finally remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.
Sparse Fourier transform via butterfly algorithm
, 2008
"... We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatia ..."
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Cited by 4 (4 self)
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We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their radii are bounded by the maximum frequency. Based on this property, equivalent sources located at Cartesian grids are used to speed up the computation of the interaction between these two regions. The overall structure of our algorithm follows the recentlyintroduced butterfly algorithm. The computation is further accelerated by exploiting the tensorproduct property of the Fourier kernel in two and three dimensions. The proposed algorithm is accurate and has an O(N log N) complexity. Finally, we present numerical results in both two and three dimensions.
V.: “Krylov Subspace Spectral Methods for the TimeDependent Schrödinger Equation with NonSmooth Potentials
 Numer. Alg
"... Abstract—This paper presents modifications of ..."
A butterfly algorithm for synthetic aperture radar imaging
, 2010
"... Abstract. In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from N data points in provable nearlinear time complexity. This paper seeks to close this gap by proposing an algorith ..."
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Cited by 3 (0 self)
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Abstract. In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from N data points in provable nearlinear time complexity. This paper seeks to close this gap by proposing an algorithm which runs in complexity O(N log N log(1/ɛ)) without making the farfield approximation or imposing the beampattern approximation required by timedomain backprojection, with ɛ the desired pixelwise accuracy. It is based on the butterfly scheme, which unlike the FFT works for vastly more general oscillatory integrals than the discrete Fourier transform. A complete error analysis is provided: the rigorous complexity bound has additional powers of log N and log(1/ɛ) that are not observed in practice. Acknowledgment. LD would like to thank Stefan Kunis for early discussions on error propagation analysis
A Spectral TimeDomain Method for Computational Electrodynamics
"... over forty years ago, the finitedifference timedomain (FDTD) method has been a widelyused technique for solving the timedependent Maxwell’s equations. This paper presents an alternative approach to these equations in the case of spatiallyvarying electric permittivity and/or magnetic permeabilit ..."
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Cited by 1 (1 self)
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over forty years ago, the finitedifference timedomain (FDTD) method has been a widelyused technique for solving the timedependent Maxwell’s equations. This paper presents an alternative approach to these equations in the case of spatiallyvarying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variablecoefficient heat equation and wave equation, and have demonstrated highorder accuracy, as well as stability characteristic of implicit timestepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gérard Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell’s equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the implementation of appropriate boundary conditions for simulation on infinite computational domains, and how discontinuous coefficients can be handled.
Fast Wave Computation via Fourier Integral Operators
, 2010
"... This paper presents a numerical method for “time upscaling ” wave equations, i.e., performing time steps not limited by the CFL condition. The proposed method leverages recent work on fast algorithms for pseudodifferential and Fourier integral operators (FIO). This algorithmic approach is not asympt ..."
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Cited by 1 (1 self)
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This paper presents a numerical method for “time upscaling ” wave equations, i.e., performing time steps not limited by the CFL condition. The proposed method leverages recent work on fast algorithms for pseudodifferential and Fourier integral operators (FIO). This algorithmic approach is not asymptotic: it is shown how to construct an exact FIO propagator by 1) solving HamiltonJacobi equations for the phases, and 2) sampling rows and columns of lowrank matrices at random for the amplitudes. In the setting of scalar waves in twodimensional smooth periodic media, it is demonstrated that the algorithmic complexity for solving the wave equation to fixed time T ≃ 1 can be as low as O(N 2 log N), where N is the bandlimit of the initial condition, with controlled accuracy. Numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest.
Wave imaging
, 2010
"... This chapter discusses imaging methods related to wave phenomena, and in particular inverse problems for the wave equation will be considered. The first part of the chapter explains the boundary control method for determining a wave speed of a medium from the response operator which models boundary ..."
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This chapter discusses imaging methods related to wave phenomena, and in particular inverse problems for the wave equation will be considered. The first part of the chapter explains the boundary control method for determining a wave speed of a medium from the response operator which models boundary measurements. The second part discusses the scattering relation and travel times, which are different types of boundary data contained in the response operator. The third part gives a brief introduction to curvelets in wave imaging for media with nonsmooth wave speeds. The focus will be on theoretical results and methods.
Block Krylov Subspace Spectral Methods for VariableCoefficient Elliptic PDE
"... Abstract—Krylov subspace spectral (KSS) methods have been demonstrated to be effective tools for solving timedependent variablecoefficient PDE. They employ techniques developed by Golub and Meurant for computing elements of functions of matrices to approximate each Fourier coefficient of the solut ..."
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Abstract—Krylov subspace spectral (KSS) methods have been demonstrated to be effective tools for solving timedependent variablecoefficient PDE. They employ techniques developed by Golub and Meurant for computing elements of functions of matrices to approximate each Fourier coefficient of the solution using a Gaussian quadrature rule that is tailored to that coefficient. In this paper, we apply this same approach to timeindependent PDE of the form Lu = f, where L is an elliptic differential operator. Numerical results demonstrate the effectiveness of this approach for Poisson’s equation and the Helmholtz equation in two dimensions.