14 research outputs found
Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly when they are combined with a fast multigrid solver, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. We optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. In a series of numerical tests, the influence on the overall performance of the “scalability window” of the multigrid solver (i.e., the range of processor numbers over which good parallel efficiency can be maintained) is illustrated. Different homogeneous and heterogeneous scheduling strategies are proposed and discussed. Finally, large 3D scaling experiments are carried out, including adaptivity
Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly when they are combined with a fast multigrid solver, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. We optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. In a series of numerical tests, the influence on the overall performance of the “scalability window” of the multigrid solver (i.e., the range of processor numbers over which good parallel efficiency can be maintained) is illustrated. Different homogeneous and heterogeneous scheduling strategies are proposed and discussed. Finally, large 3D scaling experiments are carried out, including adaptivity
Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes
When solving stochastic partial differential equations (SPDEs) driven by
additive spatial white noise, the efficient sampling of white noise
realizations can be challenging. Here, we present a new sampling technique that
can be used to efficiently compute white noise samples in a finite element
method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit
the finite element matrix assembly procedure and factorize each local mass
matrix independently, hence avoiding the factorization of a large matrix.
Moreover, in a MLMC framework, the white noise samples must be coupled between
subsequent levels. We show how our technique can be used to enforce this
coupling even in the case of non-nested mesh hierarchies. We demonstrate the
efficacy of our method with numerical experiments. We observe optimal
convergence rates for the finite element solution of the elliptic SPDEs of
interest in 2D and 3D and we show convergence of the sampled field covariances.
In a MLMC setting, a good coupling is enforced and the telescoping sum is
respected.Comment: 28 pages, 10 figure
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo
In this work we develop a new hierarchical multilevel approach to generate
Gaussian random field realizations in an algorithmically scalable manner that
is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC)
algorithms. This approach builds off of other partial differential equation
(PDE) approaches for generating Gaussian random field realizations; in
particular, a single field realization may be formed by solving a
reaction-diffusion PDE with a spatial white noise source function as the
righthand side. While these approaches have been explored to accelerate forward
uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous
constructions are not directly applicable to multilevel MCMC frameworks which
build fine scale random fields in a hierarchical fashion from coarse scale
random fields. Our new hierarchical multilevel method relies on a hierarchical
decomposition of the white noise source function in which allows us to
form Gaussian random field realizations across multiple levels of
discretization in a way that fits into multilevel MCMC algorithmic frameworks.
After presenting our main theoretical results and numerical scaling results to
showcase the utility of this new hierarchical PDE method for generating
Gaussian random field realizations, this method is tested on a four-level MCMC
algorithm to explore its feasibility
A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system
A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented