Henrik G. Petersen

Error estimates on averages of correlated data

We describe how the true statistical error on an average of correlated data can be obtained with ease and efficiency by a renormalization group method. The method is illustrated with numerical and analytical examples, having finite as well as infinite range correlations.

Accuracy and efficiency of the particle mesh Ewald method

In this article, a recently proposed method called the particle mesh Ewald (PME) method for computing the long ranged Coulomb interactions in for example molecular dynamics simulations is studied. The PME method has a complexity O(Nu2009logu2009N), where N is the total number of charges. This complexity should in particular be compared with the complexity O(N3/2) for the well known Ewald method and O(N) for the rather new (but already famous) fast multipole method (FMM). However, these complexities say nothing about which method is fastest at some finite N. The purpose of this article is thus to study the PME method and compare its efficiency with the Ewald method and the fast multipole method. To enable this, a theoretical estimate for the accuracy of the PME method as function of its truncation parameters is derived. It is shown that this estimate is very precise by comparing it with results obtained from molecular dynamics simulations of a molten NaCl. Based on this estimate and very careful time experiments, ...

The very fast multipole method

The fast multipole method (FMM) has become an important alternative to traditional methods such as the Ewald method for computing the long‐range interactions necessary to simulate charged or dipolar systems. In this paper, we present an improvement of this method, which we shall call the very fast multipole method (VFMM). The VFMM is shown to be a factor of about 1.2 faster than the FMM for two‐dimensional systems and a factor about 2–3 times faster for three‐dimensional systems without losing any accuracy for the worst case error.

Hamilton's Equations for Constrained Dynamical Systems

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.

ERROR ESTIMATES FOR THE FAST MULTIPOLE METHOD. II: THE THREE-DIMENSIONAL CASE

The Greengard-Rokhlin algorithm is a new and interesting method for computing long-range interactions in particle systems. Although the method already has been implemented and claimed to be superior to traditional and other methods, no reliable estimates of the size of the error of the method have been given. We illustrate what the error actually is for the two-dimensional case, and derive an estimate for it. The estimate has a simple analytic form which will allow its use in tuning the algorithm for best efficiency.

Molecular dynamics on transputer arrays. I: Algorithm design, programming issues, timing experiments and scaling projections

We extend earlier work on implementing boxing algorithms for simulating very large systems of particles on massively parallel S.I.M.D. arrays to more modest ones. We report details and preliminary results obtained using an OCCAM 2 program running on a linear array of transputers.

Error Estimates for the Fast Multipole Method

Error estimates for algorithms based on truncations for evaluating electrostatic interactions in molecular dynamics applications are very important for several reasons. For example, the estimates are necessary to establish the validity of the simulations and can be used to estimate various simulation parameters. Very precise estimates have been found for the Ewald method and the related particle mesh Ewald method. However, for the very popular fast multipole method such a precise estimate is not available. In this paper, we illustrate the rather complicated error behavior of the fast multipole method and we use statistical methods to derive an estimate for the root mean square error on the forces. Furthermore, the expected maximum error on the force acting on a single particle is studied. The estimates are tested against errors obtained from simulations and are found to be very precise.

Motion Planning for an Articulated Robot: A Multi-Agent Approach

Traditional motion planning techniques consider the problem of collision-free motion of an articulated robot as a global high level planning problem for one agent with internal degrees of freedom. The consequence is centralized explicit control of all joints. We suggest a simple multi-agent approach to motion planning with local low level collision avoidance. The control is distributed and there is no explicit control of the joints. Each individual link of a robot is a self-contained agent and the motion planning problem is formulated as a constraint satisfaction problem. Equations of motion for the multi-agent system incorporate satisfaction of the equality constraints between joined agents, while artificial forces ensure the satisfaction of inequality constraints preventing collisions. The artificial forces are local to each individual agent and solution of the equations of motion gives an emergent behaviour of the multi-agent system. The presented method has been successfully applied to various problems, including the simulation of a multi-tool robot and a 19-link snakelike robot moving through a maze. Finally, the method is used in an actual application: an industrial robot welding ship sections.

A rigorous comparison of the Ewald method and the fast multipole method in two dimensions

The most efficient and proper standard method for simulating charged or dipolar systems is the Ewald method, which asymptotically scales as N32 where N is the number of charges. However, recently the “fast multipole method” (FMM) which scales linearly with N has been developed. The break-even of the two methods (that is, the value of N below which Ewald is faster and above which FMM is faster) is very sensitive to the way the methods are optimized and implemented and to the required simulation accuracy. n nIn this paper we use theoretical estimates and simulation results for the accuracies to carefully compare the two methods with respect to speed. We have developed and implemented highly efficient algorithms for both methods for a serial computer (a SPARCstation ELC) as well as a parallel computer (a T800 transputer based MEIKO computer). Breakevens in the range between N = 10 000 and N = 30 000 were found for reasonable values of the average accuracies found in our simulations. Furthermore, we illustrate how huge but rare single charge pair errors in the FMM inflate the error for some of the charges.

Estimation of convergence orders in repeated Richardsen extrapolation

LetA(h) be an approximation depending on a single parameterh to a fixed quantityA, and assume thatA−A(h)=c1hk1+c2hk2+.... Given a sequence ofh-valuesh1>h2>...>hn and corresponding computed approximationsA(hi), the orders for repeated Richardson extrapolation are estimated, and the repeated extrapolation is performed. Hence in this case the algorithm described here can do the same work as BrezinskisE-algorithm and at the same time provide a check whether repeated extrapolation is justified.