Results 1  10
of
32
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
Abstract

Cited by 83 (13 self)
 Add to MetaCart
The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Penalty Methods For American Options With Stochastic Volatility
, 1998
"... The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. ..."
Abstract

Cited by 62 (18 self)
 Add to MetaCart
The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.
Efficient Kernel Density Estimation using the Fast Gauss Transform with Applications to Color Modeling and Tracking
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... The study of many vision problems is reduced to the estimation of a probability density function from observations. Kernel density estimation techniques are quite general and powerful methods for this problem, but have a significant disadvantage in that they are computationally intensive. In this pa ..."
Abstract

Cited by 39 (0 self)
 Add to MetaCart
The study of many vision problems is reduced to the estimation of a probability density function from observations. Kernel density estimation techniques are quite general and powerful methods for this problem, but have a significant disadvantage in that they are computationally intensive. In this paper we explore the use of kernel density estimation with the fast gauss transform (FGT) for problems in vision. The FGT allows the summation of a mixture of M Gaussians at N evaluation points in O(M + N) timeasopposedtoO(MN)time for a naive evaluation, and can be used to considerably speed up kernel density estimation. We present applications of the technique to problems from image segmentation and tracking, and show that the algorithm allows application of advanced statistical techniques to solve practical vision problems in real time with today’s computers. 1
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 ..."
Abstract

Cited by 36 (2 self)
 Add to MetaCart
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).
Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes
 IMA Journal of Numerical Analysis
, 2003
"... An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models wit ..."
Abstract

Cited by 32 (13 self)
 Add to MetaCart
An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wraparound effects. Numerical tests of convergence for a variety of options are presented.
A framework for discrete integral transformations II – the 2D 31 Radon transform
"... This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopola ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopolar Fourier transform that samples the Fourier transform on the pseudopolar grid, also known as the concentric squares grid. The pseudopolar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudopolar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Fast evaluation of quadrature formulae on the sphere
 Math. Comput
, 2006
"... Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2 ..."
Abstract

Cited by 12 (11 self)
 Add to MetaCart
Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules. We give a formulation in matrixvector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical GaußLegendre and ClenshawCurtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution. 1.
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.
Fast errorbounded surfaces and derivatives computation for volumetric particle data
, 2005
"... Volumetric smooth particle data arise as atomic coordinates with electron density kernels for molecular structures, as well as fluid particle coordinates with a smoothing kernel in hydrodynamic flow simulations. In each case there is the need for efficiently computing approximations of relevant surf ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Volumetric smooth particle data arise as atomic coordinates with electron density kernels for molecular structures, as well as fluid particle coordinates with a smoothing kernel in hydrodynamic flow simulations. In each case there is the need for efficiently computing approximations of relevant surfaces (molecular surfaces, material interfaces, shock waves, etc), along with surface and volume derivatives (normals, curvatures, etc.), from the irregularly spaced smooth particles. Additionally, molecular properties (charge density, polar potentials), as well as field variables from numerical simulations are often evaluated on these computed surfaces. In this paper we show how all the above problems can be reduced to a fast summation of irregularly spaced smooth kernel functions. For a scattered smooth particle system of M smooth kernels in R 3, where the Fourier coefficients have a decay of the type 1/ω 3, we present an O(M + n 3 log n + N) time, Fourier based algorithm to compute N approximate, irregular samples of a level set surface and its derivatives within a relative L2 error norm ǫ, where n is O(M 1/3 ǫ 1/3). Specifically, a truncated Gaussian of the form e −bx2 has the above decay, and n grows as √ b. In the case when the N output points are samples on a uniform grid, the back transform can be done exactly using a Fast Fourier transform algorithm, giving us an algorithm with O(M + n 3 log n + N log N) time complexity, where n is now approximately half its previously estimated value.