Showing papers by "Leslie Greengard published in 1989"

A fast algorithm for the evaluation of heat potentials

Abstract: A Fast Algorithm for the Evaluation of Heat Potentials (to appear in Communications and Pure and Applied Mathematics) L. Greengard J. Strain May, 1989 Abstract Numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials. When M points are used in the discretization of the boundary and N time steps are computed, an amount of work of the order O(N 2 M 2 ) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(N M ), and we observe speedups of five orders of magnitude for large-scale problems. Thus, the method makes it possible to solve the heat equation by potential theory in practical situations. Keywords: Heat Equation, Potential Theory, Fast Algorithms, Integral Equations Introduction A classical approach to the solution of the heat equation U t = ∆U Q T in a space-time domain Ω T = t=0 Ω(t) (see Fig. 1) is through the use of heat potentials [3, 7]. Given zero initial conditions, one seeks a representation of U as a single layer heat potential Z t Z Sµ(x, t) = K(x, x 0 , t − t 0 )µ(x 0 , t 0 ) dx 0 dt 0 Γ(t 0 ) or a double layer heat potential Dµ(x, t) = Z t Z Γ(t 0 ) ∂K (x, x 0 , t − t 0 )µ(x 0 , t 0 ) dx 0 dt 0 , ∂n 0 where K is a fundamental solution of the heat equation in some region containing Q Ω, n 0 denotes the unit T outward normal to Γ(t ) = ∂Ω(t ) at x , and µ is a surface density defined on Γ T = t=0 ∂Ω(t). ∗ Yale University and Courant Institute of Mathematical Sciences, New York University. The work of this author was supported in part by the Office of Naval Research under Grant N00014-89-J-1527, in part by IBM under grant P00038437 and in part by a NSF Mathematical Sciences Postdoctoral Fellowship. † Courant Institute of Mathematical Sciences, New York University. The work of this author was supported by DARPA/AFOSR Contract No. F-49620-87-C-0065.

On the Numerical Solution of Two-Point Boundary Value Problems

Abstract: : This paper, presents 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. The latter are discretized via a high order Nystrom scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N (dot) p2) 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. Keywords: Two-Point boundary value problems; Integral equations; Chebyshev polynomials; Approximation theory.

A parallel version of the fast multipole method

Abstract: This paper presents a parallel version of the fast multipole method (FMM) The FMM is a recently developed scheme for the evaluation of the potential and force fields in systems of particles whose interactions are Coulombic or gravitational in nature The sequential method requires O(N) operations to obtain the fields due to N charges, rather than the O(N2) operations required by the direct calculation Here, we describe the modifications necessary for implementation of the method on parallel architectures and show that the expected time requirements grow as log N when using N processors Numerical results are given for a shared memory machine (the Encore Multimax 320)