Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies

Detommaso, Gianluca; Dodwell, Tim; Scheichl, Robert

doi:10.1137/18m1172259

Cited by 20 publications

(22 citation statements)

References 22 publications

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“…• As a first idea, we could assume that the correlation coefficients ρ τ in (8) are all equal to 1. This allows the computation of the optimal number of samples N on each level as a function of V and C , similar to (5). However, we find numerically that the variance of the estimator, obtained in this way, is a huge overestimation.…”

Section: Obtaining An Estimate For the Variancementioning

confidence: 85%

See 1 more Smart Citation

Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

Robbe¹,

Nuyens²,

Vandewalle³

2019

SIAM J. Sci. Comput.

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The Multilevel Monte Carlo method is an efficient variance reduction technique. It uses a sequence of coarse approximations to reduce the computational cost in uncertainty quantification applications. The method is nowadays often considered to be the method of choice for solving PDEs with random coefficients when many uncertainties are involved. When using Full Multigrid to solve the deterministic problem, coarse solutions obtained by the solver can be recycled as samples in the Multilevel Monte Carlo method, as was pointed out by Kumar, Oosterlee and Dwight [Int. J. Uncertain. Quantif., 7 (2017), pp. 57-81]. In this article, an alternative approach is considered, using Quasi-Monte Carlo points, to speed up convergence. Additionally, our method comes with an improved variance estimate which is also valid in case of the Monte Carlo based approach. The new method is illustrated on the example of an elliptic PDE with lognormal diffusion coefficient. Numerical results for a variety of random fields with different smoothness parameters in the Matérn covariance function show that sample recycling is more efficient when the input random field is nonsmooth.

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Section: Obtaining An Estimate For the Variancementioning

confidence: 85%

“…A variance estimate can be obtained using the debiasing technique from Rhee and Glynn, see [29]. See also [5] for details on how this can be applied in the usual MLMC setting.…”

Section: Obtaining An Estimate For the Variancementioning

confidence: 99%

Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

Robbe¹,

Nuyens²,

Vandewalle³

2019

SIAM J. Sci. Comput.

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“…MLMC estimators are used in this work for the accurate and efficient estimation of statistics of various QoI. MLMC estimators have shown significant performance improvements over standard Monte Carlo algorithms [11,15,17,20] when tuned properly. There exist different algorithms to calibrate MLMC estimators for simple expectations [8,16].…”

Section: Multi-level Monte Carlo Methodsmentioning

confidence: 99%

“…Some notable extensions of the notion of levels are mentioned here for completenessfor example, the notion of fidelity [23] that leads to MFMC algorithms and the notion of continuous levels [11]. MIMC algorithms are also another notable extension of levels to the case of multiple discretization parameters [17].…”

Section: Mse(m Mlmcmentioning

confidence: 99%

D5.3 Report on theoretical work to allow the use of MLMC with adaptive mesh refinement

Ayoul-Guilmard¹,

Ganesh²,

Núñez³

et al. 2021

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This documents describes several studies undertaken to assess the applicability of MultiLevel Monte Carlo (MLMC) methods to problems of interest; namely in turbulent fluid flow over civil engineering structures. Several numerical experiments are presented wherein the convergence of quantities of interest with mesh parameters are studied at different Reynolds’ numbers and geometries. It was found that MLMC methods could be used successfully for low Reynolds’ number flows when combined with appropriate Adaptive Mesh Refinement (AMR) strategies. However, the hypotheses for optimal MLMC performance were found to not be satisfied at higher turbulent Reynolds’ numbers despite the use of AMR strategies. Recommendations are made for future research directions based on these studies. A tentative outline for an MLMC algorithm with adapted meshes is made, as well as recommendations for alternatives to MLMC methods for cases where the underlying assumptions for optimal MLMC performance are not satisfied.

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“…Our unbiased multi‐index Monte Carlo estimator, inspired by the work by Rhee and Glynn 8 and Detomasso et al, 18 does allow for sample reuse. The estimator can be written as

𝒬_{\infty, N} = \frac{1}{N} \sum_{n = 1}^{N} \sum_{0 ⩽ ℓ ⩽ L^{(n)}} \frac{1}{p_{ℓ}} Δ Q_{ℓ}^{(n)},

where for every sample n , we draw a sample L ( n ) of a d ‐variate discrete random variable L according to some nonzero probability mass function

π_{L} : N_{0}^{d} \mapsto [0, 1]

with π L (ℓ)≥0, componentwise, and

\sum_{ℓ \in N_{0}^{d}} π_{L} (ℓ) = 1

, independent from the random samples of Δ Q ℓ .…”

Section: Multi‐index Monte Carlomentioning

confidence: 99%

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

Robbe

Nuyens

Vandewalle

2020

Numerical Linear Algebra App

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In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncertain permeability of the material is modelled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [15,16,17], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method, that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC) [6] to Multi-Index Monte Carlo (MIMC) [9]. We present an unbiased MIMC method that reuses the MSG coarse solutions, similar to the work in [11]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and unifies the previous work on adaptive MIMC [19] and unbiased estimation [18]. We analyse the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

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Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies

Cited by 20 publications

References 22 publications

Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method

D5.3 Report on theoretical work to allow the use of MLMC with adaptive mesh refinement

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

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