A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions

Greengard, Leslie; Huang, Jingfang

doi:10.1006/jcph.2002.7110

Cited by 118 publications

(108 citation statements)

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“…As the particles only interact with boxes and other particles at the finest level, and information at higher levels is transferred by using a combination of multipole and local expansions as explained in Fig. 3 (11). Preliminary numerical experiments show that the overall break-even point of the new version FMM becomes 600 with 6-digit accuracy, and Ϸ400 for 3-digit accuracy.…”

Section: Discussionmentioning

confidence: 99%

“…For the matrix vector multiplications in each iteration, we use the new version FMM developed for the screened Coulombic interaction by J.H. and his collaborators (11). Compared with the original FMM, the plane wave expansion-based diagonal translation operators dramatically reduce the prefactor in the O(N) new version FMM, especially in three dimensions where a break-even point of Ϸ600 for 6-digit precision is numerically observed.…”

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confidence: 99%

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Order N algorithm for computation of electrostatic interactions in biomolecular systems

Cheng

Huang

et al. 2006

Proc. Natl. Acad. Sci. U.S.A.

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141

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Poisson-Boltzmann electrostatics is a well established model in biophysics; however, its application to large-scale biomolecular processes such as protein-protein encounter is still limited by the efficiency and memory constraints of existing numerical techniques. In this article, we present an efficient and accurate scheme that incorporates recently developed numerical techniques to enhance our computational ability. In particular, a boundary integral equation approach is applied to discretize the linearized Poisson-Boltzmann equation; the resulting integral formulas are well conditioned and are extended to systems with arbitrary numbers of biomolecules. The solution process is accelerated by Krylov subspace methods and a new version of the fast multipole method. In addition to the electrostatic energy, fast calculations of the forces and torques are made possible by using an interpolation procedure. Numerical experiments show that the implemented algorithm is asymptotically optimal O(N) in both CPU time and required memory, and application to the acetylcholinesterasefasciculin complex is illustrated. In recent years, because of the rapid advances in biotechnology, both the temporal and spatial scales of biomolecular studies have been increased significantly: from single molecules to interacting molecular networks in a cell, and from the static molecular structures at different resolutions to the dynamical interactions in biophysical processes. In these studies, the electrostatics modeled by the well established Poisson-Boltzmann (PB) equation has been shown to play an important role under physiological solution conditions. Therefore, its accurate and efficient numerical treatment becomes extremely important, especially in the study of large-scale dynamical processes such as protein-protein association and dissociation in which the PB equation has to be solved repetitively during a simulation.Traditional numerical schemes for PB electrostatics include the finite difference methods, where difference approximations are used on structured grids, and finite element methods in which arbitrarily shaped biomolecules are discretized by using elements and the associated basis functions. The resulting algebraic systems for both are commonly solved by using multigrid or domain decomposition accelerations for optimal efficiency. However, as the grid number (and thus the storage, number of operations, and condition number of the system) increases proportionally to the volume size, finite difference and finite element methods become less efficient and accurate for systems with large spatial sizes, e.g., as encountered in protein association and dissociation. Alternative methods include the boundary element method (BEM) and the boundary integral equation (BIE) method. In these methods, only the surfaces of the molecules are discretized; hence, the number of unknowns is greatly reduced. Unfortunately, in earlier versions of BEM, the matrix is stored explicitly and the resulting dense linear system is solved by using Gauss eli...

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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Cheng

Huang

et al. 2006

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

141

158

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show abstract

“…For these reasons many researchers tried to devise algorithms which were hybrids of tree codes and FMM, in order to combine the high accuracy of FMM methods with the simplicity of tree codes. In addition, the extension of the FMM to more general kernels like the modified Laplacian [13], the Stokes [10] , and the Navier [9,26] operators can be quite cumbersome, due to the need to implement efficient translation operators. In this paper, we only review algorithms that could be used to develop kernel independent methods.…”

Section: Synopsis Of the New Methodmentioning

confidence: 99%

“…We discuss relative performance of our method in greater detail in the 3D case since this is more difficult to implement efficiently. We compare with results from two papers: the single-layer 3D Laplacian results of Cheng, Greengard, and Rokhlin [7] and modified single-layer 3D Laplacian results of Greengard and Huang [13].…”

Section: Figmentioning

confidence: 99%

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Ying

Biros

Zorin

2004

Journal of Computational Physics

449

505

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We present a new fast multipole method for particle simulations. The main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel, and it is based only on kernel evaluations. Instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical FMM tree, we use a continuous distribution of an equivalent density on a surface enclosing the box. To find this equivalent density we match its potential to the potential of the original sources at a surface, in the far field, by solving local Dirichlet-type boundary value problems. The far field evaluations are sparsified with singular value decomposition in 2D or fast Fourier transforms in 3D. We have tested the new method on the single and double layer operators for the Laplacian, the modified Laplacian, the Stokes, the modified Stokes, the Navier, and the modified Navier operators in two and three dimensions. Our numerical results indicate that our method compares very well with the best known implementations of the analytic FMM method for both the Laplacian and modified Laplacian kernels. Its advantage is the (relative) simplicity of the implementation and its immediate extension to more general kernels.

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“…The N-body problem, which simulates the interactions between individual particles in a multi-particle system, is an active research topic in physics. However, direct computation for this problem is computationally expensive and there are two major efficient algorithms: the fast multipole method (FMM, [14] [15] [16] [17]) and treecode [13] [18]. FMM is relatively complicated and the performance is not sufficient for the actual computation.…”

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confidence: 99%

Application of Fast N-Body Algorithm to Option Pricing under CGMY Model

Sakuma¹

2017

JMF

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The N-body problem is an active research topic in physics for which there are two major algorithms for efficient computation, the fast multipole method and treecode, but these algorithms are not popular in financial engineering. In this article, we apply a fast N-body algorithm called the Cartesian treecode to the computation of the integral operator of integro-partial differential equations to compute option prices under the CGMY model, a generalization of a jump-diffusion model. We present numerical examples to illustrate the accuracy and effectiveness of the method and thereby demonstrate its suitability for application in financial engineering.

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A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions

Cited by 118 publications

References 14 publications

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Order N algorithm for computation of electrostatic interactions in biomolecular systems

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Application of Fast N-Body Algorithm to Option Pricing under CGMY Model

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