1,924 research outputs found
A probabilistic linear solver based on a multilevel Monte Carlo Method
We describe a new Monte Carlo method based on a multilevel method for computing the action of the resolvent matrix over a vector. The method is based on the numerical evaluation of the Laplace transform of the matrix exponential, which is computed efficiently using a multilevel Monte Carlo method. Essentially, it requires generating suitable random paths which evolve through the indices of the matrix according to the probability law of a continuous-time Markov chain governed by the associated Laplacian matrix. The convergence of the proposed multilevel method has been discussed, and several numerical examples were run to test the performance of the algorithm. These examples concern the computation of some metrics of interest in the analysis of complex networks, and the numerical solution of a boundary-value problem for an elliptic partial differential equation. In addition, the algorithm was conveniently parallelized, and the scalability analyzed and compared with the results of other existing Monte Carlo method for solving linear algebra systems.info:eu-repo/semantics/acceptedVersio
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
Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method
Computational tools for characterizing electromagnetic scattering from
objects with uncertain shapes are needed in various applications ranging from
remote sensing at microwave frequencies to Raman spectroscopy at optical
frequencies. Often, such computational tools use the Monte Carlo (MC) method to
sample a parametric space describing geometric uncertainties. For each sample,
which corresponds to a realization of the geometry, a deterministic
electromagnetic solver computes the scattered fields. However, for an accurate
statistical characterization the number of MC samples has to be large. In this
work, to address this challenge, the continuation multilevel Monte Carlo
(CMLMC) method is used together with a surface integral equation solver. The
CMLMC method optimally balances statistical errors due to sampling of the
parametric space, and numerical errors due to the discretization of the
geometry using a hierarchy of discretizations, from coarse to fine. The number
of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison
to the standard MC scheme.Comment: 25 pages, 10 Figure
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
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
We provide a framework for the sparse approximation of multilinear problems
and show that several problems in uncertainty quantification fit within this
framework. In these problems, the value of a multilinear map has to be
approximated using approximations of different accuracy and computational work
of the arguments of this map. We propose and analyze a generalized version of
Smolyak's algorithm, which provides sparse approximation formulas with
convergence rates that mitigate the curse of dimension that appears in
multilinear approximation problems with a large number of arguments. We apply
the general framework to response surface approximation and optimization under
uncertainty for parametric partial differential equations using kernel-based
approximation. The theoretical results are supplemented by numerical
experiments
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
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