WorldCat Identities

Rokhlin, V.

Overview
Works: 41 works in 75 publications in 1 language and 102 library holdings
Roles: Author
Publication Timeline
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Most widely held works by V Rokhlin
A rapid numerical procedure for determining axisymmetric transverse electric electromagnetic fields via boundary integrals by Ira B Bernstein( Book )

3 editions published in 1988 in English and held by 6 WorldCat member libraries worldwide

An important engineering problem is the determination of the electromagnetic fields in microwave systems, for example tapered waveguides, horns, scatterers, close cavities, and open resonators. Consider the case of axisymmetric transverse electric modes. Such problems for monochromatic radiation can be reduced to consideration of an elliptic partial differential equation similar to the Helmholtz equation. Methods have been developed for the direct numerical solution of the partial differential equation. Variational principles have been used to optimally determine approximate values of object of interest like reflection and transmission coefficients. An alternative approach is the reduction of the problem to consideration of an integral equation defined on the metallic walls defining the object (the boundary integral method). These have been solved for the case of scalar fields described by the Helmholtz equation. The boundary integral equation method is feasible when the Greens function is known in a computationaly convenient form, and is very often much more computationaly efficient than its competitors, particularly when the geometry is complex. The theory and effective numerical implementation are described of such a boundary integral equation approach for the case of an axisymmetric transverse electric electromagnetic field. The technique is readily generalizable to arbitrary axisymmetric fields. (JHD)
A fast algorithm for the evaluation of legendre expansions by Bradley K Alpert( Book )

4 editions published in 1989 in English and held by 5 WorldCat member libraries worldwide

An algorithm is presented for a rapid calculation of the values and coefficients of finite Legendre series. Given an n-term Legendre expansion, the algorithm produces its values at n Chebyshev nodes on the interval -1,1 for a cost proportional to n log n. Similarly, given the values of a function f at n Chebyshev nodes, the algorithm produces the n-term Legendre expansion of the polynomial of degree n - 1 that is equal of f at these nodes. The cost of the algorithm is roughly 3 times that of the fast Fourier transform of length n, provided that calculations are performed to single precision accuracy. In double precision, the ratio is approximately 5.5. The method employed admits far-reaching generalizations and is currently being applied to several other problems
On the numerical solution of two-point boundary value problems by Leslie Greengard( Book )

3 editions published in 1989 in English and held by 5 WorldCat member libraries worldwide

Abstract: "In this paper, we present a new numerical method for the solution of linear two-point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N [multiplied by] p[superscript 2]) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end-point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods."
On the inverse scattering problem for the Helmholtz equation in one dimension by Yu Chen( Book )

3 editions published in 1990 in English and held by 5 WorldCat member libraries worldwide

Thus, a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has the asymptotic cost O(n²), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated with several numerical examples."
Rapid solution of integral equations of scattering theory in two dimensions by Yale University( Book )

4 editions published in 1985 in English and held by 4 WorldCat member libraries worldwide

This paper describes an algorithm for rapid solution of boundary value problems for the Helmholtz equation in two dimensions based on iteratively solving integral equations of acoustic scattering theory. CPU time requirements of previously published algorithms of this type are of the order sq n, where n is the number of nodes in the discretization of the boundary of the scatterer. The CPU time requirements of the algorithm of the present paper are n raised, and can be further reduced, making it considerably more practical for large scale problems. Keywords: radiation fields; operators (mathematics). (Author)
End-point corrected trapezoidal quadrature rules for singular functions by Yale University( Book )

4 editions published in 1985 in English and held by 4 WorldCat member libraries worldwide

A group of quadrature formulae for end-point singular functions is presented generalizing classical end-point corrected trapezoidal quadrature rules. The actual values of the end-point corrections are obtained for each singularity as a solution of a system of linear algebraic equations. The algorithm is applicable to a wide class of monotonic singularities and does not require that an analytical expression for the singularity be known; only the knowledge of its first several moments and the ability to evaluate it on the interval of integration are needed. Keywords: convergence; numerical analysis; numerical integration. (Author)
On the efficient implementation of the fast multipole algorithm by Leslie Greengard( Book )

3 editions published in 1988 in English and held by 4 WorldCat member libraries worldwide

The Fast Multipole Method (FMM) is a recently developed algorithm for the evaluation of potential fields in particle systems. In order to evaluate the fields induced by a set of N charges (or masses) on each other, the FMM requires order O(N) work rather than the O(N squared) work required by the direct evaluation of pairwise interactions. The constant of proportionality for the method depends on the cost of applying a translation operator to a multiple or Taylor expansion. In existing implementations, this is O(p squared) in two dimensions and O(p to the 4th power) in three, where p is the degree of the expansion. In this paper we describe a procedure permitting translation operators to be applied to p to the 4th power degree expansions for a cost proportional to p.log p in two dimensions, and p squared. log p in three. The incorporation of this technique into the FMM scheme speeds up the execution of two-dimensional single precision codes by a factor of two or three, and the execution of three-dimensional codes by roughly a factor of eight
Rapid evaluation of potential fields in three dimensions by L Greengard( Book )

3 editions published in 1987 in English and held by 4 WorldCat member libraries worldwide

This paper describes a three dimensional version of the fast multipole algorithm for the rapid evaluation of the potential and force fields in systems of particles whose interactions are Coulombic or gravitational in nature. For a system of N particles, an amount of work of the order O(N-square) has traditionally been required to evaluate all pairwise interactions, unless some approximation or truncation method is used. The algorithm presented here requires an amount of work proportional to N to evaluate all interactions to within roundoff error, making it considerably more practical for large scale problems encountered in plasma physics, fluid dynamics, molecular dynamics and celestial mechanics
On the rapid evaluation of trigonometric series by A Dutt( Book )

3 editions published in 1992 in English and held by 4 WorldCat member libraries worldwide

Abstract: "A group of algorithms is presented generalizing the Fast Fourier Transform to the case of non-integer frequencies and non- equispaced nodes on the interval [-[pi], [pi]]. The schemes of this paper are based on a combination of certain analytical considerations with the classical Fast Fourier Transform, and generalize both the forward and backward FFTs. Each of the algorithms requires O(N x log N + N x log (1/[epsilon])) arithmetic operations, where [epsilon] is the precision of computations and N is the number of nodes. The efficiency of the approach is illustrated by several numerical examples."
Diagonal forms of translation operators for the Helmholtz equation in three dimensions by V Rokhlin( Book )

1 edition published in 1992 in English and held by 3 WorldCat member libraries worldwide

It is an extension to the three-dimensional case of the results of [13], where a similar apparatus is developed in the two-dimensional case."
A fast algorithm for the discrete Laplace transformation by V Rokhlin( Book )

2 editions published in 1987 in English and held by 3 WorldCat member libraries worldwide

An algorithm is presented for the rapid evaluation of expressions of the form sum over j = 1 to m of (a sub j) exp ( - beta sub j)x) at multiple points x1, x2 ..., xn. In order to evaluate the above sum at n points, the algorithm requires order O(n + m) operations, and a simple modification of the scheme provides an order O(n) procedure for the evaluation of an order n polynomial at n arbitrary real points. The algorithm is numerically stable, and its practical usefulness is demonstrated by numerical examples
A fast algorithm for particle simulations by Leslie Greengard( Book )

3 editions published in 1986 in English and held by 3 WorldCat member libraries worldwide

An algorithm is presented for the rapid evaluation of the potential and force fields in systems involving large numbers of particles whose interactions are Coulombic or gravitational in nature. For a system of N particles, an amount of work of the order O(N2) has traditionally been required to evaluate all pairwise interactions, unless some approximation or truncation method is used. The algorithm of this paper requires an amount of work proportional to N to evaluate all interactions to within roundoff error, making it considerably more practical for large-scale problems encountered in plasma physics, fluid dynamics, molecular dynamics and celestial mechanics. Keywords: N-body problem; Molecular dynamics, Plasma physics, Potential theory. (Author)
The fast multipole method for gridless particle simulations by J. J Ambrosiano( Book )

1 edition published in 1987 in English and held by 3 WorldCat member libraries worldwide

On the numerical solution of two-point boundary value problems by P Starr( Book )

3 editions published in 1990 in English and held by 3 WorldCat member libraries worldwide

Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end- point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods. We in addition present a Newton method for solving boundary value problems for nonlinear first order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems."
A new version of the fast multipole method for the Laplace equation in three dimensions by Leslie Greengard( )

2 editions published in 1996 in English and held by 2 WorldCat member libraries worldwide

Abstract: "We introduce a new version of the Fast Multipole Method for the evaluation of potential fields in three dimensions. It is based on a new diagonal form for translation operators and yields high accuracy at a reasonable cost."
An improved fast multipole algorithm for potential fields on one- dimensional structures by Norman Yarvin( )

1 edition published in 1997 in English and held by 2 WorldCat member libraries worldwide

Abstract: "A new version of the Fast Multipole Method for the evaluation of potential fields on one-dimensional structures is introduced. The scheme uses a new representation of potential fields, based on generalized Gaussian quadratures [6,9]; in this representation, most translation operators are diagonal. To incorporate this representation into the FMM, an apparatus is introduced for transforming between different types of expansions; this apparatus is somewhat general, and is based on formulae for the least squares approximation of linear operators. The performance of the method is illustrated with several numerical examples; it is roughly twice as fast as previously published algorithms."
On the Riccati equations for the scattering matrices in two dimensions by Yu Chen( )

2 editions published in 1995 in English and held by 2 WorldCat member libraries worldwide

Abstract: "This is the first of a series of papers addressing the solution of the inverse scattering problem for the Helmholtz equation in two dimensions. Here, we derive a system of differential equations for the scattering matrices which 1. Directly govern the whole behavior of the scattering problem, 2. Can be easily implemented numerically with any prescribed accuracy. In the subsequent papers, we will use this apparatus to design stable inversion algorithms for the acoustical inverse scattering problem. Specifically, in the second paper, we will present a scheme based on the trace formula which is a direct extension to the one employed in [16]. The algorithm is quite satisfactory analytically, but requires excessive amounts of CPU time. Finally, in the third paper, we will present a radically accelerated version of the algorithm."
Fast Fourier transforms for nonequispaced data by A Dutt( Book )

2 editions published in 1993 in English and held by 2 WorldCat member libraries worldwide

Abstract: "A group of algorithms is presented generalizing the Fast Fourier Transform to the case of noninteger frequencies and nonequispaced nodes on the interval [-[pi], [pi]]. The schemes of this paper are based on a combination of the classical Fast Fourier Transform with a version of the Fast Multipole Method, and generalize both the forward and backward FFTs. Each of the algorithms requires O(N [multiplied by] log N + N [multiplied by] log(1/[epsilon])) arithmetic operations, where [epsilon] is the precision of computations and N is the number of nodes. The efficiency of the approach is illustrated by several numerical examples."
Generalized Gaussian quadrature rules for systems of arbitrary functions by Jin Hong Ma( Book )

2 editions published in 1993 in English and held by 2 WorldCat member libraries worldwide

Abstract: "In [6], a far-reaching generalization of the classical Gaussian quadrature rules is introduced, replacing the polynomials with a wide class of functions. While the rules of [6] possess most of the desirable properties of the classical Gaussian integration formulae (positivity of the weights, etc.), it is not clear from [6] how such quadrature rules can be obtained numerically. In this paper, we present a numerical scheme for the generation of such generalized Gaussian quadratures. The algorithm is applicable to a variety of functions, including smooth functions as well as functions with endpoint singularities. The performance of the algorithm is demonstrated with several numerical examples."
Fast algorithms for polynomial interpolation, integration and differentiation by A Dutt( Book )

2 editions published in 1993 in English and held by 2 WorldCat member libraries worldwide

Abstract: "For functions tabulated at Chebyshev nodes on an interval, spectral interpolation, integration and differentiation can be performed stably and efficiently via the fast Fourier transform. In this paper, a group of algorithms is presented for the efficient evaluation of Lagrange polynomial interpolants at multiple points on the line, and for the rapid spectral integration and differentiation of functions tabulated at nodes other than Chebyshev. The interpolation scheme requires O(N [multiplied by] log(1/[epsilon])) arithmetic operations, and O(N [multiplied by] log N + N [multiplied by] log(1/[epsilon])) operations are required for the integration and differentiation schemes, where [epsilon] is the precision of computations and N is the numer of nodes. The algorithms utilize efficient versions of the fast multipole method which have been designed specifically for one-dimensional problems; these are also described in the present paper. Several experiments are included to illustrate the numerical performance of the approach."
 
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