Fast Multipole Method for Multivariable Integrals

We give a fast numerical algorithm to evaluate a class of multivariable integrals. A direct numerical evaluation of these integrals costs $N^m$, where $m$ is the number of variables and $N$ is the number of the quadrature points for each variable. For $m=2$ and $m=3$ and for only one-dimensional variables, we present an algorithm which is able to reduce this cost from $N^m$ to a cost of the order of $O((-\log \epsilon )^{\mu_m} N)$, where $\epsilon$ is the desired accuracy and $\mu_m$ is a constant that depends only on $m$. Then, we make some comments about possible extensions of such algorithms to number of variables $m\geq 4$ and to higher dimensions. This recursive algorithm can be viewed as an extension of ``fast multipole methods" to situations where the interactions between particles are more complex than the standard case of binary interactions. Numerical tests illustrating the efficiency and the limitation of this method are presented.

[1]  V. Rokhlin,et al.  Rapid Evaluation of Potential Fields in Three Dimensions , 1988 .

[2]  W. Kutzelnigg,et al.  Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. III. Second‐order Mo/ller–Plesset (MP2‐R12) calculations on molecules of first row atoms , 1991 .

[3]  Mohammed Lemou,et al.  Fast multipole method for multidimensional integrals , 1998 .

[4]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix. IV. Multipole accelerated formation of the exchange matrix , 1999 .

[5]  Mohammed Lemou,et al.  Multipole expansions for the Fokker-Planck-Landau operator , 1998 .

[6]  Wim Klopper,et al.  Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory , 1991 .

[7]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[8]  Wilson,et al.  Optimized trial wave functions for quantum Monte Carlo calculations. , 1988, Physical review letters.

[9]  Werner Kutzelnigg,et al.  Rates of convergence of the partial‐wave expansions of atomic correlation energies , 1992 .

[10]  J. Noga,et al.  A CCSD(T)-R12 study of the ten-electron systems Ne, F-, HF, H2O, NH3, NH4+ and CH4 , 1997 .

[11]  Benny G. Johnson,et al.  Linear scaling density functional calculations via the continuous fast multipole method , 1996 .

[12]  Stephen W. Taylor,et al.  KWIK: Coulomb Energies in O(N) Work , 1996 .

[13]  Benny G. Johnson,et al.  THE CONTINUOUS FAST MULTIPOLE METHOD , 1994 .

[14]  L. Greengard,et al.  A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy , 1996 .

[15]  S. Ten-no,et al.  New Transcorrelated Method Improving the Feasibility of Explicitly Correlated Calculations , 2002 .

[16]  Eric Darve,et al.  The Fast Multipole Method , 2000 .

[17]  Eric Darve,et al.  The Fast Multipole Method I: Error Analysis and Asymptotic Complexity , 2000, SIAM J. Numer. Anal..

[18]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix , 1997 .

[19]  M. Caffarel,et al.  Zero-Variance Principle for Monte Carlo Algorithms , 1999, cond-mat/9911396.

[20]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[21]  Leslie Greengard,et al.  The numerical solution of the N-body problem , 1990 .

[22]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .