Interval Arithmetic and Computational Science: Performance Considerations (original) (raw)
Related papers
Interval arithmetic and computational science: Rounding and truncation errors in N-body methods
2007
Interval arithmetic is an alternative computational paradigm that enables arithmetic operations to be performed with guarantee error bounds. In this paper interval arithmetic is used to compare the accuracy of various methods for computing the electrostatic energy for a system of point charges. A number of summation approaches that scale as O(N 2 ) are considered, as is an O(N) scaling Fast Multipole Method (FMM). Results are presented for various sizes of water cluster in which each water molecule is described using the popular TIP3P water model. For FMM a subtle balance between the dominance of either rounding or truncation errors is demonstrated Fifth International Conference on Computational Science and Applications 0-7695-2945-3/07 $25.00
The Journal of Chemical Physics, 2020
In molecular dynamics (MD) calculations of the free energies of ions and ionic molecules, we often encounter net charged molecular systems where the electrical neutrality condition is broken. This charge causes a problem in the evaluation of long-range Coulombic interactions under periodic boundary conditions. A standard remedy for this problem is to consider a hypothetical homogeneous background charge density to neutralize the total system. Here, we present a new expression for the evaluation of Coulombic interactions for such systems including background charge using the fast multipole method (FMM). Furthermore, an efficient scheme is developed to evaluate solute–solvent interaction energies using the FMM, reducing the computational burden for the far-field part. We calculate the hydration free energies of Mg2+, Na+, and Cl− ions dissolved in a neutral solvent using the new expression. The calculated free energies show good agreement with the results obtained using the well-estab...
Characterization of the accuracy of the fast multipole method in particle simulations
International Journal for Numerical Methods in Engineering, 2009
The Fast Multipole Method (FMM) is a fast summation algorithm capable of accelerating pairwise interaction calculations, known as N -body problems, from an algorithmic complexity of O(N 2 ) to O(N ) for N particles. The algorithm has brought dramatic increase in the capability of particle simulations in many application areas, such as electrostatics, particle formulations of fluid mechanics, and others. Although the literature on the subject provides theoretical error bounds for the FMM approximation, there are not many reports of the measured errors in a suite of computational experiments. We have performed such an experimental investigation, and summarized the results of about 1000 calculations using the FMM algorithm, to characterize the accuracy of the method in relation with the different parameters available to the user. In addition to the more standard diagnostic of the maximum error, we supply illustrations of the spatial distribution of the errors, offering visual evidence of all the contributing factors to the overall approximation accuracy: multipole expansion, local expansion, hierarchical spatial decomposition (interaction lists, local domain, far domain). This presentation is a contribution to any researcher wishing to incorporate the FMM acceleration to their application code, as it aids in understanding where accuracy is gained or compromised.
Molecular Multipole Potential Energy Functions for Water
The Journal of Physical Chemistry B, 2016
Water is the most common liquid on this planet, with many unique properties that make it essential for life as we know it. These properties must arise from features in the charge distribution of a water molecule, so it is essential to capture these features in potential energy functions for water to reproduce its liquid state properties in computer simulations. Recently, models that utilize a multipole expansion located on a single site in the water molecule, or "molecular multipole models", have been shown to rival and even surpass site models with up to five sites in reproducing both the electrostatic potential around a molecule and a variety of liquid state properties in simulations. However, despite decades of work using multipoles, confusion still remains about how to truncate the multipole expansions efficiently and accurately. This is particularly important when using molecular multipole expansions to describe water molecules in the liquid state, where the short-range interactions must be accurate, because the higher order multipoles of a water molecule are large. Here, truncation schemes designed for a recent efficient algorithm for multipoles in molecular dynamics simulations are assessed for how well they reproduce results for a simple three-site model of water when the multipole moments and Lennard-Jones parameters of that model are used. In addition, the multipole analysis indicates that site models that do not account for out-of-plane electron density overestimate the stability of a non-hydrogen-bonded conformation, leading to serious consequences for the simulated liquid.
Multilevel Summation Method for Electrostatic Force Evaluation
Biophysical Journal, 2015
The multilevel summation method (MSM) offers an efficient algorithm utilizing convolution for evaluating long-range forces arising in molecular dynamics simulations. Shifting the balance of computation and communication, MSM provides key advantages over the ubiquitous particle−mesh Ewald (PME) method, offering better scaling on parallel computers and permitting more modeling flexibility, with support for periodic systems as does PME but also for semiperiodic and nonperiodic systems. The version of MSM available in the simulation program NAMD is described, and its performance and accuracy are compared with the PME method. The accuracy feasible for MSM in practical applications reproduces PME results for water property calculations of density, diffusion constant, dielectric constant, surface tension, radial distribution function, and distancedependent Kirkwood factor, even though the numerical accuracy of PME is higher than that of MSM. Excellent agreement between MSM and PME is found also for interface potentials of air−water and membrane−water interfaces, where long-range Coulombic interactions are crucial. Applications demonstrate also the suitability of MSM for systems with semiperiodic and nonperiodic boundaries. For this purpose, simulations have been performed with periodic boundaries along directions parallel to a membrane surface but not along the surface normal, yielding membrane pore formation induced by an imbalance of charge across the membrane. Using a similar semiperiodic boundary condition, ion conduction through a graphene nanopore driven by an ion gradient has been simulated. Furthermore, proteins have been simulated inside a single spherical water droplet. Finally, parallel scalability results show the ability of MSM to outperform PME when scaling a system of modest size (less than 100 K atoms) to over a thousand processors, demonstrating the suitability of MSM for large-scale parallel simulation.
Point Charges Optimally Placed to Represent the Multipole Expansion of Charge Distributions
PLoS ONE, 2013
We propose an approach for approximating electrostatic charge distributions with a small number of point charges to optimally represent the original charge distribution. By construction, the proposed optimal point charge approximation (OPCA) retains many of the useful properties of point multipole expansion, including the same far-field asymptotic behavior of the approximate potential. A general framework for numerically computing OPCA, for any given number of approximating charges, is described. We then derive a 2-charge practical point charge approximation, PPCA, which approximates the 2-charge OPCA via closed form analytical expressions, and test the PPCA on a set of charge distributions relevant to biomolecular modeling. We measure the accuracy of the new approximations as the RMS error in the electrostatic potential relative to that produced by the original charge distribution, at a distance2 2| the extent of the charge distribution-the mid-field. The error for the 2-charge PPCA is found to be on average 23% smaller than that of optimally placed point dipole approximation, and comparable to that of the point quadrupole approximation. The standard deviation in RMS error for the 2-charge PPCA is 53% lower than that of the optimal point dipole approximation, and comparable to that of the point quadrupole approximation. We also calculate the 3-charge OPCA for representing the gas phase quantum mechanical charge distribution of a water molecule. The electrostatic potential calculated by the 3-charge OPCA for water, in the mid-field (2.8 Å from the oxygen atom), is on average 33.3% more accurate than the potential due to the point multipole expansion up to the octupole order. Compared to a 3 point charge approximation in which the charges are placed on the atom centers, the 3-charge OPCA is seven times more accurate, by RMS error. The maximum error at the oxygen-Na distance (2.23 Å) is half that of the point multipole expansion up to the octupole order.
A rigorous comparison of the Ewald method and the fast multipole method in two dimensions
Computer Physics Communications, 1995
The most efficient and proper standard method for simulating charged or dipolar systems is the Ewald method, which asymptotically scales as N 3/2 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.
The Journal of Chemical Physics, 2014
Improved parameterization of interatomic potentials for rare gas dimers with density-based energy decomposition analysis Efficient particle-mesh Ewald based approach to fixed and induced dipolar interactions Next-generation molecular force fields deliver accurate descriptions of non-covalent interactions by employing more elaborate functional forms than their predecessors. Much work has been dedicated to improving the description of the electrostatic potential (ESP) generated by these force fields. A common approach to improving the ESP is by augmenting the point charges on each center with higher-order multipole moments. The resulting anisotropy greatly improves the directionality of the non-covalent bonding, with a concomitant increase in computational cost. In this work, we develop an efficient strategy for enumerating multipole interactions, by casting an efficient spherical harmonic based approach within a particle mesh Ewald (PME) framework. Although the derivation involves lengthy algebra, the final expressions are relatively compact, yielding an approach that can efficiently handle both finite and periodic systems without imposing any approximations beyond PME. Forces and torques are readily obtained, making our method well suited to modern molecular dynamics simulations.
Scoring Multipole Electrostatics in Condensed-Phase Atomistic Simulations
2013
Permanent multipoles (MTP) embody a natural extension to common point-charge (PC) representations in atomistic simulations. In this work, we propose an alternative to the computationallyexpensive MTP molecular dynamics simulations by running a simple PC simulation and later reevaluate-"score"-all energies using the more detailed MTP force field. The method, which relies on the assumption that the PC and MTP force fields generate closely related phase spaces, is accomplished by enforcing identical sets of monopoles between the two force fields-effectively highlighting the higher MTP terms as a correction to the PC approximation. We first detail our consistent parametrization of the electrostatics and van der Waals interactions for the two force fields. We then validate the method by comparing the accuracy of protein-ligand binding free energies from both PC and MTP-scored representations with experimentally-determined binding constants obtained by us. Specifically, we study the binding of several arylsulfonamide ligands to human carbonic anhydrase II. We find that both representations yield an accuracy of 1 kcal/mol with respect to experiment. Finally, we apply the method to rank the energetic contributions of individual atomic MTP coefficients for molecules solvated in water. All in all, MTP scoring is a computationally appealing method that can provide insight in the multipolar electrostatic interactions of condensed-phase systems.
Multipole electrostatics in hydration free energy calculations
Journal of Computational Chemistry, 2011
Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here, we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio method. We have compared two approaches to derive the atomic multipoles from quantum mechanical calculations: one directly from the new distributed multipole analysis and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-31111G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-cc-pVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-31111G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12Å , and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the HFE calculation. q 2010 Wiley Periodicals, Inc. J Comput Chem 32: 967-977, 2011