Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

The mechanical properties of ordinary materials degrade substantially with reduced density because their structural elements bend under applied load. We report a class of microarchitected materials that maintain a nearly constant stiffness per unit mass density, even at ultralow density. This performance derives from a network of nearly isotropic microscale unit cells with high structural connectivity and nanoscale features, whose structural members are designed to carry loads in tension or compression. Production of these microlattices, with polymers, metals, or ceramics as constituent materials, is made possible by projection microstereolithography (an additive micromanufacturing technique) combined with nanoscale coating and postprocessing. We found that these materials exhibit ultrastiff properties across more than three orders of magnitude in density, regardless of the constituent material.

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Ultralight, ultrastiff mechanical metamaterials

Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.

Physics-informed machine learning

DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations

In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form—hence the term equation-free. We selectively discuss basic algorithms and underlyin...

Equation-Free Multiscale Computation: Algorithms and Applications

An important class of problems exhibits smooth behavior in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multiscale problems, an ``...

The gap-tooth scheme for homogenization problems

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the rightmost, stability-determining roots of the characteristic equation which can subsequently be used as starting values in a Newton procedure. For periodic solutions, approximations to the Floquet multipliers are computed. The manual describes the structure of the package, its routines, and its data and method parameter structures.

DDE-BIFTOOL Manual - Bifurcation analysis of delay differential equations

An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an ``equation-free framework'' has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with non-linear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.

Giovanni Samaey

Papers

DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations

Equation-Free Multiscale Computation: Algorithms and Applications

The gap-tooth scheme for homogenization problems

DDE-BIFTOOL Manual - Bifurcation analysis of delay differential equations

The gap-tooth scheme for homogenization problems