Jingfang Huang

A wideband fast multipole method for the Helmholtz equation in three dimensions

We describe a wideband version of the Fast Multipole Method for the Helmholtz equation in three dimensions. It unifies previously existing versions of the FMM for high and low frequencies into an algorithm which is accurate and efficient for any frequency, having a CPU time of O(N) if low-frequency computations dominate, or O(NlogN) if high-frequency computations dominate. The performance of the algorithm is illustrated with numerical examples.

Accelerating fast multipole methods for the Helmholtz equation at low frequencies

The authors describe a diagonal form for translating far-field expansions to use in low frequency fast multipole methods. Their approach combines evanescent and propagating plane waves to reduce the computational cost of FMM implementation. More specifically, we present the analytic foundations for a new version of the fast multipole method for the scalar Helmholtz equation in the low frequency regime. The computational cost of existing FMM implementations, is dominated by the expense of translating far field partial wave expansions to local ones, requiring 189p/sup 4/ or 189p/sup 3/ operations per box, where harmonics up to order p/sup 2/ have been retained. By developing a new expansion in plane waves, we can diagonalize these translation operators. The new low frequency FMM (LF-FMM) requires 40p/sup 2/+6p/sup 2/ operations per box.

Order N algorithm for computation of electrostatic interactions in biomolecular systems

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 acetylcholinesterase–fasciculin complex is illustrated.

Accelerating the convergence of spectral deferred correction methods

In the recent paper by Dutt, Greengard and Rokhlin, a variant of deferred or defect correction methods is presented which couples Gaussian quadrature with the Picard integral equation formulation of the initial value ordinary differential equation. The resulting spectral deferred correction (SDC) methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy. In this paper, we show that for linear problems, the iterations in the SDC algorithm are equivalent to constructing a preconditioned Neumann series expansion for the solution of the standard collocation discretization of the ODE. This observation is used to accelerate the convergence of SDC using the GMRES Krylov subspace method. For nonlinear problems, the GMRES acceleration is coupled with a linear implicit approach. Stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy, efficiency, and stability of the original SDC approach. Furthermore, preliminary numerical experiments show that accelerating the convergence of SDC methods can effectively eliminate the order reduction previously observed for stiff ODE systems.

Arbitrary order Krylov deferred correction methods for differential algebraic equations

In this paper, a new framework for the construction of accurate and efficient numerical methods for differential algebraic equation (DAE) initial value problems is presented. The methods are based on applying spectral deferred correction techniques as preconditioners to a Picard integral collocation formulation for the solution. The resulting preconditioned nonlinear system is solved using Newton-Krylov schemes such as the Newton-GMRES method. Least squares based orthogonal polynomial approximations are computed using Gaussian type quadratures, and spectral integration is used to avoid the numerically unstable differentiation operator. The resulting Krylov deferred correction (KDC) methods are of arbitrary order of accuracy and very stable. Preliminary results show that these new methods are very competitive with existing DAE solvers, particularly when high precision is desired.

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Abstract An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each “particle” in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N 2) to O(N), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.

An Adaptive Fast Multipole Boundary Element Method for Poisson-Boltzmann Electrostatics

The numerical solution of the Poisson−Boltzmann (PB) equation is a useful but a computationally demanding tool for studying electrostatic solvation effects in chemical and biomolecular systems. Recently, we have described a boundary integral equation-based PB solver accelerated by a new version of the fast multipole method (FMM). The overall algorithm shows an order N complexity in both the computational cost and memory usage. Here, we present an updated version of the solver by using an adaptive FMM for accelerating the convolution type matrix-vector multiplications. The adaptive algorithm, when compared to our previous nonadaptive one, not only significantly improves the performance of the overall memory usage but also remarkably speeds the calculation because of an improved load balancing between the local- and far-field calculations. We have also implemented a node-patch discretization scheme that leads to a reduction of unknowns by a factor of 2 relative to the constant element method without sacrificing accuracy. As a result of these improvements, the new solver makes the PB calculation truly feasible for large-scale biomolecular systems such as a 30S ribosome molecule even on a typical 2008 desktop computer.

AFMPB: An adaptive fast multipole Poisson Boltzmann solver for calculating electrostatics in biomolecular systems

A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole to local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage.

A Fast Direct Solver for Elliptic Partial Differential Equations on Adaptively Refined Meshes

We present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes. These solvers rely on a combination of standard fast solvers for uniform grids and potential theory. Unlike standard iterative approaches, they have a well-defined operation count. They also preserve the order of accuracy of the uniform grid solver, despite the presence of coarse/fine interfaces.

Improving the robustness of a surface integral formulation for wideband impendance extraction of 3D structures

In order for parasitic extraction of high-speed integrated circuit interconnect to be sufficiently efficient, and fit with model-order reduction techniques, a robust wideband surface integral formulation is essential. One recently developed surface integral formulation has shown promise, but was plagued with numerical difficulties of poorly understood origin. We show that one of that formulations difficulties was related to the inaccuracy in the approach to evaluate integrals over discretization panels, and we present an accurate approach based on an adapted piecewise quadrature scheme. We also show that the condition number of the original system of integral equations can be reduced by differentiating one of the integral equations. Computational results on a ring and a spiral inductor are used to show that the new quadrature scheme and the differentiated integral formulation improve accuracy and accelerate the convergence of iterative solution methods.