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Domain decomposition methods for parallel laser-tissue models with Monte Carlo transport
Achieving parallelism in simulations that use Monte Carlo transport methods presents interesting challenges. For problems that require domain decomposition, load balance can be harder to achieve. The Monte Carlo transport package may have to operate with other packages that have different optimal domain decompositions for a given problem. To examine some of these issues, we have developed a code that simulates the interaction of a laser with biological tissue; it uses a Monte Carlo method to simulate the laser and a finite element model to simulate the conduction of the temperature field in the tissue. We will present speedup and load balance results obtained for a suite of problems decomposed using a few domain decomposition algorithms we have developed
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
Comparison of Different Parallel Implementations of the 2+1-Dimensional KPZ Model and the 3-Dimensional KMC Model
We show that efficient simulations of the Kardar-Parisi-Zhang interface
growth in 2 + 1 dimensions and of the 3-dimensional Kinetic Monte Carlo of
thermally activated diffusion can be realized both on GPUs and modern CPUs. In
this article we present results of different implementations on GPUs using CUDA
and OpenCL and also on CPUs using OpenCL and MPI. We investigate the runtime
and scaling behavior on different architectures to find optimal solutions for
solving current simulation problems in the field of statistical physics and
materials science.Comment: 14 pages, 8 figures, to be published in a forthcoming EPJST special
issue on "Computer simulations on GPU
Thimble regularization at work besides toy models: from Random Matrix Theory to Gauge Theories
Thimble regularization as a solution to the sign problem has been
successfully put at work for a few toy models. Given the non trivial nature of
the method (also from the algorithmic point of view) it is compelling to
provide evidence that it works for realistic models. A Chiral Random Matrix
theory has been studied in detail. The known analytical solution shows that the
model is non-trivial as for the sign problem (in particular, phase quenched
results can be very far away from the exact solution). This study gave us the
chance to address a couple of key issues: how many thimbles contribute to the
solution of a realistic problem? Can one devise algorithms which are robust as
for staying on the correct manifold? The obvious step forward consists of
applications to gauge theories.Comment: 7 pages, 1 figure. Talk given at the Lattice2015 Conferenc
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Unified Gas-kinetic Wave-Particle Methods III: Multiscale Photon Transport
In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method
to the multiscale photon transport. In this method, the photon free streaming
and scattering processes are treated in an un-splitting way. The duality
descriptions, namely the simulation particle and distribution function, are
utilized to describe the photon. By accurately recovering the governing
equations of the unified gas-kinetic scheme (UGKS), the UGKWP preserves the
multiscale dynamics of photon transport from optically thin to optically thick
regime. In the optically thin regime, the UGKWP becomes a Monte Carlo type
particle tracking method, while in the optically thick regime, the UGKWP
becomes a diffusion equation solver. The local photon dynamics of the UGKWP, as
well as the proportion of wave-described and particle-described photons are
automatically adapted according to the numerical resolution and transport
regime. Compared to the -type UGKS, the UGKWP requires less memory cost
and does not suffer ray effect. Compared to the implicit Monte Carlo (IMC)
method, the statistical noise of UGKWP is greatly reduced and computational
efficiency is significantly improved in the optically thick regime. Several
numerical examples covering all transport regimes from the optically thin to
optically thick are computed to validate the accuracy and efficiency of the
UGKWP method. In comparison to the -type UGKS and IMC method, the UGKWP
method may have several-order-of-magnitude reduction in computational cost and
memory requirement in solving some multsicale transport problems.Comment: 27 pages, 15 figures. arXiv admin note: text overlap with
arXiv:1810.0598
Stochastic turbulence modeling in RANS simulations via Multilevel Monte Carlo
A multilevel Monte Carlo (MLMC) method for quantifying model-form
uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS)
simulations is presented. Two, high-dimensional, stochastic extensions of the
RANS equations are considered to demonstrate the applicability of the MLMC
method. The first approach is based on global perturbation of the baseline eddy
viscosity field using a lognormal random field. A more general second extension
is considered based on the work of [Xiao et al.(2017)], where the entire
Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For
two fundamental flows, we show that the MLMC method based on a hierarchy of
meshes is asymptotically faster than plain Monte Carlo. Additionally, we
demonstrate that for some flows an optimal multilevel estimator can be obtained
for which the cost scales with the same order as a single CFD solve on the
finest grid level.Comment: 40 page
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