P3M Method Research Articles

Charge Central Interpretation of the Full Nonlinear PB Equation: Implications for Accurate and Scalable Modeling of Solvation Interactions.

Continuum solvation modeling based upon the Poisson-Boltzmann equation (PBE) is widely used in structural and functional analysis of biomolecules. In this work, we propose a charge-central interpretation of the full nonlinear PBE electrostatic interactions. The validity of the charge-central view or simply charge view, as formulated as a vacuum Poisson equation with effective charges, was first demonstrated by reproducing both electrostatic potentials and energies from the original solvated full nonlinear PBE. There are at least two benefits when the charge-central framework is applied. First the convergence analyses show that the use of polarization charges allows a much faster converging numerical procedure for electrostatic energy and forces calculation for the full nonlinear PBE. Second, the formulation of the solvated electrostatic interactions as effective charges in vacuum allows scalable algorithms to be deployed for large biomolecular systems. Here, we exploited the charge-view interpretation and developed a particle-particle particle-mesh (P3M) strategy for the full nonlinear PBE systems. We also studied the accuracy and convergence of solvation forces with the charge-view and the P3M methods. It is interesting to note that the convergence of both the charge-view and the P3M methods is more rapid than the original full nonlinear PBE method. Given the developments and validations documented here, we are working to adapt the P3M treatment of the full nonlinear PBE model to molecular dynamics simulations.

How to Convert SPME to P3M: Influence Functions and Error Estimates.

We demonstrate explicitly how the two seemingly different particle mesh Ewald methods, the smooth particle mesh Ewald (SPME) and the particle-particle particle mesh (P3M), can be mathematically transformed into each other. This allows us in particular to convert the error estimate of the P3M method in the energy-conserving scheme (also known as "P3M with analytic differentiation") into an error estimate for the SPME method, via a simple change of the lattice Green function. Our error estimate is valid for any values of the SPME parameters (mesh size, spline interpolation order, Ewald splitting parameter, real-space cutoff distance), including odd orders of splines. The problem with the self-forces is avoided thanks to an analytical formula that allows to subtract them directly within the particle mesh calculation. Plots of the accuracy of the SPME forces are provided for a wide range of parameter values. The main use of the error estimate is to allow a simulation program to scan quickly the multidimensional parameter space to find the best set of parameters to achieve a target accuracy at the smallest computational cost. As a byproduct, we show how a SPME code can be transformed into a P3M version by changing a few lines of code. We demonstrate also that the P3M lattice Green function can be approximated by a closed form expression, computable on-the-fly, that provides essentially the same accuracy as the full function.

Particle-particle particle-mesh method for dipolar interactions: On error estimates and efficiency of schemes with analytical differentiation and mesh interlacing

The interlaced and non-interlaced versions of the dipolar particle-particle particle-mesh (P(3)M) method implemented using the analytic differentiation scheme (AD-P(3)M) are presented together with their respective error estimates for the calculation of the forces, torques, and energies. Expressions for the optimized lattice Green functions, and for the Madelung self-forces, self-torques and self-energies are given. The applicability of the theoretical error estimates are thoroughly tested and confirmed in several numerical examples. Our results show that the accuracy of the calculations can be improved substantially when the approximate (mesh computed) Madelung self-interactions are subtracted. Furthermore, we show that the interlaced dipolar AD-P(3)M method delivers a significantly higher accuracy (which corresponds approximately to using a twice finer mesh) than the conventional method, allowing thereby to reduce the mesh size with respect to the non-interlaced version for a given accuracy. In addition, we present similar expressions for the dipolar ik-differentiation interlaced scheme, and we perform a comparison with the AD interlaced scheme. Rough tests for the relative speed of the dipolar P(3)M method using ik-differentiation and the interlaced/non-interlaced AD schemes show that when FFT computing time is the bottleneck, usually when working at high precisions, the interlaced AD-scheme can be several times faster than the other two schemes. For calculations with a low accuracy requirement, the interlaced version can perform worse than the ik and the non-interlaced AD schemes.

Parallelization of Particle-Particle, Particle-Mesh Method within N-Body Simulation

N-Body problem has become an intricate part of the computational sciences, and there has been rise to many methods to solve and approximate the problem. The solution potentially requires on the order of calculations each time step, therefore efficient performance of these N-Body algorithms is very significant (5). This work describes the parallelization and optimization of the Particle-Particle, Particle-Mesh (P3M) algorithm within GalaxSeeHPC, an open-source N-Body Simulation code. Upon successful profiling, MPI (Message Passing Interface) routines were implemented into the population of the density grid in the P3M method in GalaxSeeHPC. Each problem size recorded different results, and for a problem set dealing with 10,000 celestial bodies, speedups up to 10x were achieved. However, in accordance to Amdahl's Law, maximum speedups for the code should have been closer to 16x. In order to achieve maximum optimization, additional research is needed and parallelization of the Fourier Transform routines could prove to be rewarding. In conclusion, the GalaxSeeHPC Simulation was successfully parallelized and obtained very respectable results, while further optimization remains possible.

Interlaced P3M algorithm with analytical and ik-differentiation

The interlacing technique of Hockney and Eastwood is extended to the particle-particle, particle-mesh (P3M) algorithm with analytical and ik-differentiation that computes the approximate Coulomb forces between N point particles in a periodic box. Interlacing means that one makes two separate computations of the reciprocal-space Ewald force, using two grids shifted with respect to each other by half of the diagonal of the grid subcell, and then takes the average of the two forces. The resulting algorithms compare favorably against their own noninterlaced versions and against the interlaced smooth particle-mesh Ewald algorithm. In our tests, the accuracy of the interlaced P3M methods was usually more than an order of magnitude higher than that of the other particle-mesh algorithms with the same parameter values. This accuracy gain can be converted into a speedup if the parameters of the algorithm are changed. Interlacing allows one to increase the grid spacing by up to a factor of 2 while keeping the same accuracy. A priori error estimates for the new algorithms are constructed, and the removal of the spurious self-force term is discussed. The success of interlacing is shown to be due to the fact that it suppresses the aliasing effects in the forces. It should be easy to incorporate the interlaced P3M algorithms into an existing simulation package, since this only requires a minor modification of the particle-mesh Ewald part of the code.

Fast and Accurate P 3 M Cosmological Simulations Using a Hardware Accelerator

We implemented a fast and accurate particle-particle particle-mesh (P3M) method using MDGRAPE-2, a hardware accelerator for arbitrary central forces. The results of cosmological N-body simulations using the MDGRAPE-2 P3M method agree with those using other P3M methods. We measured calculation times required for the MDGRAPE-2 P3M method on the system of a PC and one MDGRAPE-2 board. When we reduce memory requirement, the MDGRAPE-2 P3M method is ~10 and ~20 times faster than the adaptive and original P3M methods on the PC, respectively, under heavy particle clustering. When each P3M method is optimized for performance, the MDGRAPE-2 P3M method is as fast as the adaptive P3M method and ~10 times faster than the original P3M method.

Optimal charge-shaping functions for the particle–particle—particle–mesh (P3M) method for computing electrostatic interactions in molecular simulations

The application of the particle–particle—particle–mesh (P3M) method for computing electrostatic interactions in molecular simulations relies on the use of a charge-shaping function to split the potential into two contributions, evaluated in real and reciprocal space, respectively. Although the charge-shaping function is traditionally taken to be a Gaussian, many other choices are possible. In the present study, we investigate the accuracy of the P3M method employing, as charge-shaping functions, polynomials truncated to a finite spacial range (TP functions). We first discuss and test analytical estimates of the P3M root-mean-square force error for both types of shaping functions. These estimates are then used to find the optimal values of the free parameters defining the two types of charge-shaping function (width of the Gaussian or coefficients of the TP function). Finally, we compare the accuracy properties of these optimized functions, using both analytical estimates and numerical results for a model ionic system. It is concluded that the use of specific TP functions instead of the traditional Gaussian function leads to improvements in terms of computational speed, simplicity of use, and accuracy of results.

GRAPE-5: A Special-Purpose Computer for N-Body Simulations

We have developed a special-purpose computer for gravitational many-body simulations, GRAPE-5. GRAPE-5 accelerates the force calculation which dominates the calculation cost of the simulation. All other calculations, such as the time integration of orbits, are performed on a general-purpose computer (host computer) connected to GRAPE-5. A GRAPE-5 board consists of eight custom pipeline chips (G5 chip) and its peak performance is 38.4 Gflops. GRAPE-5 is the successor of GRAPE-3. The differences between GRAPE-5 and GRAPE-3 are: (1) The newly developed G5 chip contains two pipelines operating at 80 MHz, while the GRAPE chip, which was used for GRAPE-3, had one at 20 MHz. The calculation speed of GRAPE-5 is 8-times faster than that of GRAPE-3. (2) The GRAPE-5 board adopted a PCI bus as the interface to the host computer instead of VME of GRAPE-3, resulting in a communication speed one order of magnitude faster. (3) In addition to the pure 1/r potential, the G5 chip can calculate forces with arbitrary cutoff functions, so that it can be applied to the Ewald or P3M methods. (4) The pairwise force calculated on GRAPE-5 is about 10-times more accurate than that on GRAPE-3. On one GRAPE-5 board, one timestep with a direct summation algorithm takes 14 (N/128 k)2 seconds. With the Barnes-Hut tree algorithm (θ = 0.75), one timestep can be done in 15 (N/106) seconds.

Measuring the three-dimensional shear from simulation data, with applications to weak gravitational lensing

We have developed a new three-dimensional algorithm, based on the standard P3M method, for computing deflections resulting from weak gravitational lensing. We compare the results of this method with those of the two-dimensional planar approach, and rigorously outline the conditions under which the two approaches are equivalent. Our new algorithm uses a Fast Fourier Transform convolution method for speed, and has a variable softening feature to provide a realistic interpretation of the large-scale structure in a simulation. The output values of the code are compared with those from the Ewald summation method, which we describe and develop in detail. With an optimal choice of the high-frequency filtering in the Fourier convolution, the maximum errors, when using only a single particle, are about 7 per cent, with an rms error less than 2 per cent. For ensembles of particles, used in typical N-body simulations, the rms errors are typically 0.3 per cent. We describe how the output from the algorithm can be used to generate distributions of magnification, source ellipticity, shear and convergence for large-scale structure.

Softened Lagrangian hydrodynamics for cosmology

view Abstract Citations (95) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Softened Lagrangian Hydrodynamics for Cosmology Gnedin, Nickolay Y. Abstract A new approach to cosmological hydrodynamics is discussed that is based on a moving, quasi-Lagrangian mesh. The softened Lagrangian hydrodynamics (SLH) method utilizes a high-resolution Lagrangian hydrodynamic code combined with a low-resolution Eulerian solver to deal with severe mesh distortions. Most of the volume of a simulation is treated with the Lagrangian code and only in sites where the Lagrangian approach fails, due to mesh distortions, does the Eulerian part of the code step in. This approach utilizes a high-resolution gravity solver without use of TREE or P3M methods; Poisson's equation is solved on the moving baryonic mesh using a simple relaxation technique. The dark matter is included by means of the cloud-in-cell method on the Lagrangian mesh. All three components of the cosmological code--gravity, dark matter, and baryons--are thus treated selft-consistently with exactly the same resolution. The computer code based on the SLH approach is described in detail, and comparison with existing Eulerian and smooth particle hydrodynamics (SPH) codes is presented. For most purposes the SLH approach turns out to be the intermediate between Eulerian and SPH codes, but it outperforms both of these approaches in resolving caustics. Thus, it may turn out to be a valuable tool to study galaxy formation. Publication: The Astrophysical Journal Supplement Series Pub Date: April 1995 DOI: 10.1086/192141 Bibcode: 1995ApJS...97..231G Keywords: Astronomical Models; Computational Grids; Computerized Simulation; Cosmology; Euler-Lagrange Equation; Hydrodynamics; Lagrangian Function; Mathematical Models; Baryons; Dark Matter; Galactic Evolution; Gravitation; Missing Mass (Astrophysics); Poisson Equation; Astrophysics; COSMOLOGY: THEORY; HYDRODYNAMICS; METHODS: NUMERICAL full text sources ADS |