An adaptive treecode for computing nonbonded potential energy in classical molecular systems

A treecode algorithm is presented for rapid computation of the nonbonded potential energy in classical molecular systems. The algorithm treats a general form of pairwise particle interaction with the Coulomb and London dispersion potentials as special cases. The energy is computed as a sum of group–group interactions using a variant of Appel's recursive strategy. Several adaptive techniques are employed to reduce the execution time. These include an adaptive tree with nonuniform rectangular cells, variable order multipole approximation, and a run-time choice between direct summation and multipole approximation for each group–group interaction. The multipole approximation is derived by Taylor expansion in Cartesian coordinates, and the necessary coefficients are computed using a recurrence relation. An error bound is derived and used to select the order of approximation. Test results are presented for a variety of systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 184–195, 2001Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/34695/1/6_ftp.pd

Treecodes for Potential and Force Approximations

N-body problems encompass a variety of fields such as electrostatics, molecularbiology and astrophysics. If there are N particles in the system, the brute force algorithmfor these problems based on particle-particle interaction takes O(N2), whichis clearly expensive for large values of N. There have been some approximation algorithmslike the Barnes-Hut Method and the Fast Multipole Method (FMM) proposedfor these problems to reduce the complexity. However, the applicability of these algorithmsare limited to operators with analytic multipole expansions or restricted tosimulations involving low accuracy. The shortcoming of N-body treecodes are moreevident for particles in motion where the movement of the particles are not consideredwhen evaluating the potential. If the displacement of the particles are small, thenupdating the multipole coefficients for all the nodes in the tree may not be requiredfor computing the potential to a reasonable accuracy. This study focuses on some ofthe limitations of the existing approximation schemes and presents new algorithmsthat can be used for N-body simulations to efficiently compute potentials and forces.In the case of electrostatics, existing algorithms use Cartesian coordinates to evaluatethe potentials of the form r−, where 1. The use of such coordinates toseparate the variables results in cumbersome expressions and does not exploit the inherent spherical symmetry found in these kernels. For such potentials, we providea new multipole expansion series and construct a method which is asymptoticallysuperior than the current treecodes. The advantage of this expansion series is furtherdemonstrated by an algorithm that can compute the forces to the desired accuracy.For particles in motion, we introduce a new method in which we retain the multipolecoefficients when performing multipole updates (to the parent nodes) at every timestep. This results in considerable savings in time while maintaining the accuracy. Wefurther illustrate the efficiency of our algorithms through numerical experiments

Development and Application of Numerical Methods in Biomolecular Solvation

This work addresses the development of fast summation methods for long range particle interactions and their application to problems in biomolecular solvation, which describes the interaction of proteins or other biomolecules with their solvent environment. At the core of this work are treecodes, tree-based fast summation methods which, for N particles, reduce the cost of computing particle interactions from O(N^2) to O(N log N). Background on fast summation methods and treecodes in particular, as well as several treecode improvements developed in the early stages of this work, are presented. Building on treecodes, dual tree traversal (DTT) methods are another class of tree-based fast summation methods which reduce the cost of computing particle interactions for N particles to O(N). The primary result of this work is the development of an O(N) dual tree traversal fast summation method based on barycentric Lagrange polynomial interpolation (BLDTT). This method is implemented to run across multiple GPU compute nodes in the software package BaryTree. Across different problem sizes, particle distributions, geometries, and interaction kernels, the BLDTT shows consistently better performance than the previously developed barycentric Lagrange treecode (BLTC). The first major biomolecular solvation application of fast summation methods presented is to the Poisson–Boltzmann implicit solvent model, and in particular, the treecode-accelerated boundary integral Poisson–Boltzmann solver (TABI-PB). The work on TABI-PB consists of three primary projects and an application. The first project investigates the impact of various biomolecular surface meshing codes on TABI-PB, and integrated the NanoShaper software into the package, resulting in significantly better performance. Second, a node patch method for discretizing the system of integral equations is introduced to replace the previous centroid collocation scheme, resulting in faster convergence of solvation energies. Third, a new version of TABI-PB with GPU acceleration based on the BLDTT is developed, resulting in even more scalability. An application investigating the binding of biomolecular complexes is undertaken using the previous Taylor treecode-based version of TABI-PB. In addition to these projects, work performed over the course of this thesis integrated TABI-PB into the popular Adaptive Poisson–Boltzmann Solver (APBS) developed at Pacific Northwest National Laboratory. The second major application of fast summation methods is to the 3D reference interaction site model (3D-RISM), a statistical-mechanics based continuum solvation model. This work applies cluster-particle Taylor expansion treecodes to treat long-range asymptotic Coulomb-like potentials in 3D-RISM, and results in significant speedups and improved scalability to the 3D-RISM package implemented in AmberTools. Additionally, preliminary work on specialized GPU-accelerated treecodes based on BaryTree for 3D-RISM long-range asymptotic functions is presented.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168120/1/lwwilson_1.pd