The Sparse Grids Matlab kit -- a Matlab implementation of sparse grids for high-dimensional function approximation and uncertainty quantification

Piazzola, Chiara; Tamellini, Lorenzo

doi:10.48550/arxiv.2203.09314

Cited by 3 publications

(1 citation statement)

References 35 publications

(57 reference statements)

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“…The goal in those studies was to perform pathwise adaptivity for functional approximation for the expected value of the solution and not to compute a quantity of interest as we do here. The more recent work [55] is focused on adaptivity with multilevel stochastic collocation for solving elliptic PDEs with random data by combining adaptive (anisotropic) sparse Smolyak grid approximations in the stochastic space, using the Sparse grid Matlab kit (version 17-5) [80], for each collocation point by applying the dual weighted residual method [13] for spatial goal-oriented adaptive mesh refinements.…”

Section: Introductionmentioning

confidence: 99%

Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

Beck¹,

Liu²,

Schwerin³

et al. 2022

Preprint

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In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of R d . Our AMLMC algorithm is built on the results of the weak convergence rates in the work [Moon et al., BIT Numer. Math., 46 (2006), pp. 367-407] for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation.From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem

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Section: Introductionmentioning

confidence: 99%

Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

Beck¹,

Liu²,

Schwerin³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution II: computing moments and sparse grid acceleration

et al. 2022

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Sensitivity-based Parameter Calibration of Single- and Dual-continuum Coreflooding Simulation Models

et al. 2022

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Our study is keyed to the development of a viable framework for the stochastic characterization of coreflooding simulation models under two- and three-phase flow conditions taking place within a core sample in the presence of preferential flow of the kind that can be associated with the presence of a system of fractures. We do so considering various modeling strategies based on (spatially homogeneous or heterogeneous) single- and dual-continuum formulations of black-oil computational models and relying on a global sensitivity-driven stochastic parameter calibration. The latter is constrained through a set of data collected under a water alternating gas scenario implemented in laboratory-scale coreflooding experiments. We set up a collection of Monte Carlo (MC) numerical simulations while considering uncertainty encompassing (a) rock attributes (i.e., porosity and absolute permeability), as well as (b) fluid–fluid/ fluid–solid interactions, as reflected through characteristic parameters of relative permeability and capillary pressure formulations. Modern moment-based global sensitivity indices are evaluated on the basis of the MC model responses, with the aim of (i) quantifying sensitivity of the coreflooding simulation results to variations of the input uncertain model parameters and (ii) assessing the possibility of reducing the dimensionality of model parameter spaces. We then rest on a stochastic inverse modeling approach grounded on the acceptance–rejection sampling (ARS) algorithm to obtain probability distributions of the key model parameters (as identified through our global sensitivity analyses) conditional to the available experimental observations. The relative skill of the various candidate models to represent the system behavior is quantified upon relying on the deviance information criterion. Our findings reveal that amongst all tested models, a dual-continuum formulation provides the best performance considering the experimental observations available. Only a few of the parameters embedded in the dual-continuum formulation are identified as major elements significantly affecting the prediction (and associated uncertainty) of model outputs, petrophysical attributes and relative permeability model parameters having a stronger effect than parameters related to capillary pressure.

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The Sparse Grids Matlab kit -- a Matlab implementation of sparse grids for high-dimensional function approximation and uncertainty quantification

Cited by 3 publications

References 35 publications

Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution II: computing moments and sparse grid acceleration

Sensitivity-based Parameter Calibration of Single- and Dual-continuum Coreflooding Simulation Models

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