3 research outputs found
Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice
International audienceWe are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or substracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles
GNN-Assisted Phase Space Integration with Application to Atomistics
Overcoming the time scale limitations of atomistics can be achieved by
switching from the state-space representation of Molecular Dynamics (MD) to a
statistical-mechanics-based representation in phase space, where approximations
such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic
ensemble in a time-coarsened fashion. In practice, this requires the
computation of expensive high-dimensional integrals over all of phase space of
an atomistic ensemble. This, in turn, is commonly accomplished efficiently by
low-order numerical quadrature. We show that numerical quadrature in this
context, unfortunately, comes with a set of inherent problems, which corrupt
the accuracy of simulations -- especially when dealing with crystal lattices
with imperfections. As a remedy, we demonstrate that Graph Neural Networks,
trained on Monte-Carlo data, can serve as a replacement for commonly used
numerical quadrature rules, overcoming their deficiencies and significantly
improving the accuracy. This is showcased by three benchmarks: the thermal
expansion of copper, the martensitic phase transition of iron, and the energy
of grain boundaries. We illustrate the benefits of the proposed technique over
classically used third- and fifth-order Gaussian quadrature, we highlight the
impact on time-coarsened atomistic predictions, and we discuss the
computational efficiency. The latter is of general importance when performing
frequent evaluation of phase space or other high-dimensional integrals, which
is why the proposed framework promises applications beyond the scope of
atomistics
A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient
In this work, a Reduced Basis (RB) approach is used to solve a large number of boundary value problems parametrized by a stochastic input – expressed as a Karhunen–Loève expansion – in order to compute outputs that are smooth functionals of the random solution fields. The RB method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the RB approach developed previously for deterministic systems. However, the stochastic framework requires the development of new a posteriori estimates for “statistical” outputs – such as the first two moments of integrals of the random solution fields; these error bounds, in turn, permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many-query context.United States. Air Force Office of Scientific Research (Grant FA9550-07-1-0425)Singapore-MIT Alliance for Research and TechnologyChaire d’excellence AC