V. Rokhlin’s research while affiliated with University of Colorado Boulder and other places

: 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.

A fast solver is presented for the solution of scattering problems in which the scatterer is a relatively thin, elongated object. The scheme presented here is a version of an algorithm previously published by the authors, and is based on the observation that under certain conditions and with certain modifications, the scheme will retain its O(n) CPU time esti-mate independently of the size of the scatterer in wavelengths. The performance of the scheme is illustrated with numerical examples. Ó 2006 Published by Elsevier Inc.

This paper describes a simple version of the Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. We discuss both the underlying theory and some of the practical aspects of its implementation to allow for stability and high accuracy at all wavelengths.

We propose a “fast” algorithm for the construction of a data-sparse inverse of a generalToeplitz matrix. The computational cost for inverting an N × N Toeplitz matrix equals the cost of four length-N FFTs plus an O(N)-term. This cost should be compared to the O(N log2N) cost of previously published methods. Moreover, while those earlier methods are based on algebraic considerations, the procedure of this paper is analysis-based; as a result, its stability does not depend on the symmetry and positive-definiteness of the matrix being inverted. The performance of the scheme is illustrated with numerical examples.

We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two dimensions. The algorithm is “fast” in the sense that its asymptotic complexity is O(n), where n is the number of nodes in the discretization. Unlike previous fast techniques based on iterative solvers, the present algorithm directly constructs a compressed factorization of the inverse of the matrix; thus it is suitable for problems involving relatively ill-conditioned matrices, and is particularly efficient in situations involving multiple right hand sides. The performance of the scheme is illustrated with several numerical examples.

Polynomials are one of the principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc, it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well developed apparatus of prolate spheroidal wavefunctions to construct quadratures, interpolation and differentiation formulae, etc, for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc, the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.

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 the present 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.

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.