An Optimizing Symbolic Algebra Approach for Generating Fast Multipole Method Operators

We have developed a symbolic algebra approach to automatically produce, verify, and optimize computer code for the Fast Multipole Method (FMM) operators. This approach allows for flexibility in choosing a basis set and kernel, and can generate computer code for any expansion order in multiple languages. The procedure is implemented in the publicly available Python program Mosaic. Optimizations performed at the symbolic level through algebraic manipulations significantly reduce the number of mathematical operations compared with a straightforward implementation of the equations. We find that the optimizer is able to eliminate 20-80% of the floating-point operations and for the expansion orders $p \le 10$ it changes the observed scaling properties. We present our approach using three variants of the operators with the Cartesian basis set for the harmonic potential kernel $1/r$, including the use of totally symmetric and traceless multipole tensors.Comment: Updated to final version submitted to Computer Physics Communications. Accepted on 20 November 201

Quantum drude oscillators for accurate many-body intermolecular forces

One of the important early applications of Quantum Mechanics was to explain the Van-der-Waal’s 1/R6 potential that is observed experimentally between two neutral species, such as noble gas atoms, in terms of correlated uncertainty between interacting dipoles, an effect that does not occur in the classical limit [London-Eisenschitz,1930]. When many-body correlations and higher-multipole interactions are taken into account they yield additional many-body and higher-multipole dispersion terms. Dispersion energies are closely related to electrostatic interactions and polarisation [Hirschfelder-Curtiss-Bird,1954]. Hydrogen bonding, the dominant force in water, is an example of an electrostatic effect, which is also strongly modified by polarisation effects. The behaviour of ions is also strongly influenced by polarisation. Where hydrogen bonding is disrupted, dispersion tends to act as a more constant cohesive force. It is the only attractive force that exists between hydrophobes, for example. Thus all three are important for understanding the detailed behaviour of water, and effects that happen in water, such as the solvation of ions, hydrophobic de-wetting, and thus biological nano-structures. Current molecular simulation methods rarely go beyond pair-wise potentials, and so lose the rich detail of many-body polarisation and dispersion that would permit a force field to be transferable between different environments. Empirical force-fields fitted in the gas phase, which is dominated by two-body interactions, generally do not perform well in the condensed (many-body) phases. The leading omitted dispersion term is the Axilrod-Teller-Muto 3-body potential, which does not feature in standard biophysical force-fields. Polarization is also usually ommitted, but it is sometimes included in next-generation force-fields following seminal work by Cochran [1971]. In practice, many-body forces are approximated using two-body potentials fitted to reflect bulk behaviour, but these are not transferable because they do not reproduce detailed behaviour well, resulting in spurious results near inhomogeneities, such as solvated hydrophobes and ions, surfaces and interfaces. The Quantum Drude Oscillator model (QDO) unifies many-body, multipole polarisation and dispersion, intrinsically treating them on an equal footing, potentially leading to simpler, more accurate, and more transferable force fields when it is applied in molecular simulations. The Drude Oscillator is simply a model atom wherein a single pseudoelectron is bound harmonically to a single pseudonucleus, that interacts via damped coulomb interactions [Drude,1900]. Path Integral [Feynman-Hibbs,1965] Molecular Dynamics (PIMD) can, in principle, provide an exact treatment for moving molecules at finite temperature on the Born- Oppenheimer surface due to their pseudo-electrons. PIMD can be applied to large systems, as it scales like N log(N), with multiplicative prefactor P that can be effectively parallelized away on modern supercomputers. There are other ways to treat dispersion, but all are computationally intensive and cannot be applied to large systems. These include, for example, Density Functional Theory provides an existence proof that a functional exists to include dispersion, but we dont know the functional. We outline the existing methods, and then present new density matrices to improve the discretisation of the path integral. Diffusion Monte Carlo (DMC), first proposed by Fermi, allows the fast computation of high-accuracy energies for static nuclear configurations, making it a useful method for model development, such as fitting repulsion potentials, but there is no straightforward way to generate forces. We derived new methods and trial wavefunctions for DMC, allowing the computation of energies for much larger systems to high accuracy. A Quantum Drude model of Xenon, fit in the gas-phase, was simulated in the condensed-phase using both DMC and PIMD. The new DMC methods allowed for calculation of the bulk modulus and lattice constant of FCC-solid Xenon. Both were in excellent agreement with experiment even though this model was fitted in the gasphase, demonstrating the power of Quantum Drudes to build transferable models by capturing many-body effects. We also used the Xenon model to test the new PIMD methods. Finally, we present the outline of a new QDO model of water, including QDO parameters fitted to the polarisabilities and dispersion coefficients of water

Fast Numerical Methods for Non-local Operators

[no abstract available

Development of low-scaling electronic structure methods using rank factorizations and an attenuated Coulomb metric

Novel low-scaling techniques for molecular electronic structure and property calculations are introduced. Through the use of rank-revealing matrix factorizations, overheads compared to canonical molecular orbital-based formulations are virtually eliminated. Asymptotic computational complexity is linear or sub-linear (depending on the property) through the use of sparsity-preserving transformations throughout. For electron correlation energy calculations within the random phase approximation, these techniques are combined with an attenuated Coulomb metric in the resolution-of-the-identity to improve the accuracy over existing low-scaling methods and to reduce the scaling compared to existing canonical methods. For the resolution-of-the-identity itself, a novel method for the compression of auxiliary bases is introduced, powered by removal of the particle-hole-interaction nullspace through projection. Furthermore, efficient schemes for the calculation of molecular response properties at the Hartree–Fock and density functional theory levels are introduced: For the linear scaling calculation of vibrational frequencies, the exact cancellation of different long-range operator derivatives is employed in combination with Laplace-transformed and Cholesky-decomposed coupled-perturbed self-consistent field theories. Using related techniques, calculations of indirect nuclear spin-spin coupling constants with asymptotically constant time complexity are demonstrated and used to explore the dependence of spin-spin couplings in a peptide on the size of a surrounding solvent environment

Self-force and radiation reaction in general relativity

[Abridged] This review surveys the theory of gravitational self-force in curved spacetime and its application to the gravitational two-body problem in the extreme-mass-ratio regime. We first lay the relevant formal foundation, describing the rigorous derivation of the equation of self-forced motion using matched asymptotic expansions and other ideas. We then review the progress that has been achieved in numerically calculating the self-force and its physical effects in the astrophysical scenario of a compact object inspiralling into a (rotating) massive black hole. We highlight the way in which, nowadays, self-force calculations make a fruitful contact with other approaches to the two-body problem and help inform an accurate universal model of binary black hole inspirals, valid across all mass ratios. We conclude with a summary of the state of the art, open problems and prospects. Our review is aimed at non-specialist readers and is for the most part self-contained and non-technical; only elementary-level acquaintance with General Relativity is assumed. Where useful, we draw on analogies with familiar concepts from Newtonian gravity or classical electrodynamics.Comment: 79 pages, 11 figures; invited by Reports on Progress in Physics. v2 contains minor corrections and it is the published versio

A generalized approach to planar induction heating magnetics

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 85-90).This thesis describes an efficient numerical simulation technique of magnetoquasistatic electromagnetic fields for planar induction heating applications. The technique is based on a volume-element discretization, integral formulation of Maxwell's equations, and uses the multilayer Green's function to avoid volumetric meshing of the heated material. The technique demonstrates two orders of magnitude of computational advantage compared to existing FEM techniques. Single-objective and multiobjective optimization of a domestic induction heating coil are performed using the new technique, using more advanced algorithms than those previously used due to the increase in speed. Both optimization algorithms produced novel, three-dimensional induction coil designs.by Richard Yi Zhang.S.M