Optimizing the accuracy of lattice Monte Carlo algorithms for simulating diffusion

Chubynsky, Mykyta V.; Slater, Gary W.

doi:10.1103/physreve.85.016709

Cited by 13 publications

(6 citation statements)

References 50 publications

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“…The choice of p IIs is in principle arbitrary. While p IIs = 0 seems the most efficient choice at first glance, since the particle would move at every step of the algorithm, this, as mentioned, can introduce artifacts and, in fact, we have recently shown [50] that p IIs = 2/3 is optimal from the point of view of accuracy of the algorithm and may even be more efficient than p IIs = 0, since a coarser mesh can be used to achieve the same accuracy. For this reason, we choose p IIs = 2/3 in this work.…”

Section: B MC Ii: Fixed Ps Variable Jump Lengthmentioning

confidence: 99%

“…The parameter η I 0 should not be larger than the smaller of the two viscosities η L and η R to ensure that both p L Is and p R Is are non-negative. In fact, even the situation where η I 0 is equal to or only slightly smaller than either η L or η R should be avoided, since some simulation artifacts, such as spurious oscillations both in time and in space, are possible when the waiting time is zero [50], or, more generally, the variance of the time between successful steps is small. On the other hand, from Eqs.…”

Section: The Basic Algorithmmentioning

confidence: 99%

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Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich Dilemma"

de Haan,

Chubynsky,

Slater

2012

Preprint

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The possibility of different interpretations of the stochastic term (or calculi) in the overdamped Langevin equation for the motion of a particle in an inhomogeneous medium is often referred to as the "Ito-Stratonovich dilemma," although there is, in fact, a continuum of choices. We introduce two Monte Carlo (MC) simulation approaches for studying such systems, with both approaches giving the choice between different interpretations (in particular, Ito, Stratonovich, and "isothermal"). To demonstrate these approaches, we study the diffusion on a 1D interval of a particle released at an interface (in the middle of the system) between two media where this particle has different diffusivities (for example, two fluids with different viscosities). We consider the properties of the particle distribution for reflecting boundary conditions at the ends of the 1D interval. A discontinuity at the interface in the stationary-state particle distribution is found, except for the isothermal case, as expected. We also study the first-passage problem using absorbing boundary conditions. Good agreement is found when comparing the MC approaches against theoretical predictions as well as Brownian and Langevin dynamics simulations. Additionally, while this problem was chosen primarily to verify the algorithms, the results themselves turn out to be interesting -particularly when comparing across interpretations. For instance, we report that: 1) for some calculi, there can be more particles on the low-viscosity side at earlier times and then more particles on the highviscosity side at later times; 2) there is no preference to end up on a particular wall for the Ito variant, but a bias towards the wall on the low-viscosity side in all other cases; 3) the mean first-passage time to the wall on the low-viscosity side grows as the viscosity on the high-viscosity side is increased, except for the isothermal case where it approaches a constant; 4) when the viscosity ratio is high, the first-passage-time distribution for the wall on the lower-viscosity side is much broader than for the other wall, with a power-law dependence of the former in a certain time interval whose exponent depends on the calculus; 5) the average portion of time the particle spends on a particular side can be very different from the probability to reach the wall on that side and depends significantly on how the averaging is done.

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Section: B MC Ii: Fixed Ps Variable Jump Lengthmentioning

confidence: 99%

Section: The Basic Algorithmmentioning

confidence: 99%

Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich Dilemma"

de Haan,

Chubynsky,

Slater

2012

Preprint

Self Cite

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show abstract

“…Particularly, explicit FPT distributions are important for lattice Monte Carlo simulation of normal or anomalous diffusion [12][13][14][15]. The focus of this study is on the FPT distributions of anomalous diffusion governed by the Galilei variant fractional diffusion-advection equation (FDAE) model [5] subject to an absorbing boundary condition.…”

Section: Introductionmentioning

confidence: 99%

First passage time distributions of anomalous biased diffusion with double absorbing barriers

Guo¹,

Qiu²

2014

Physica A: Statistical Mechanics and its Applications

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“…Based on boundary-conforming finite elements, these schemes have definite advantages over finite differences in terms of describing complex boundaries. This advantage is shared with Monte Carlo methods, which are very simple to code but require careful optimization in order to run efficiently [38,39].…”

Section: Introductionmentioning

confidence: 99%

Lattice Boltzmann method for simulation of diffusion magnetic resonance imaging physics in multiphase tissue models

2020

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We report the first implementation of the Lattice Boltzmann method (LBM) to integrate the Bloch-Torrey equation, which describes the evolution of the transverse magnetization vector and the fate of the signal of diffusion magnetic resonance imaging (dMRI). Motivated by the need to interpret dMRI experiments on skeletal muscle, and to offset the small time step limitation of classical LBM, a hybrid LBM scheme is introduced and implemented to solve the Bloch-Torrey equation in a tissue model consisting of cylindrical inclusions (representing bundles of parallel myocytes), delineated by thin permeable membranes (sarcolemma) and surrounded by a extracellular domain (endomysium). As implemented, the hybrid LBM scheme accommodates piece-wise uniform transport, dMRI parameters, periodic outer boundary conditions, and finite membrane permeabilities on non-boundary-conforming inner boundaries. The geometry is discretized in uniform 2D or 3D lattices by locating the curved cylindrical boundaries halfway between lattice nodes. By comparing with analytical solutions of limiting cases, we demonstrate that the hybrid LBM scheme is more accurate that the classical LBM scheme. The proposed explicit LBM scheme maintains second-order spatial accuracy, stability, and first-order temporal accuracy for a wide range of parameters. The parallel implementation of the hybrid LBM code in a multi-CPU computer system with MPI is straightforward. For a domain decomposition algorithm on one million lattice nodes, the speedup is linear up to 168 MPI processes. While offering certain advantages over finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a sui generis scheme that can by easily adapted to model more complex interfacial conditions and physics in heterogeneous multiphase tissue models, and to accommodate sophisticated dMRI sequences.

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Optimizing the accuracy of lattice Monte Carlo algorithms for simulating diffusion

Cited by 13 publications

References 50 publications

Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich Dilemma"

Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich Dilemma"

First passage time distributions of anomalous biased diffusion with double absorbing barriers

Lattice Boltzmann method for simulation of diffusion magnetic resonance imaging physics in multiphase tissue models

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