467 research outputs found
Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario
A variety of methods is available to quantify uncertainties arising with\-in
the modeling of flow and transport in carbon dioxide storage, but there is a
lack of thorough comparisons. Usually, raw data from such storage sites can
hardly be described by theoretical statistical distributions since only very
limited data is available. Hence, exact information on distribution shapes for
all uncertain parameters is very rare in realistic applications. We discuss and
compare four different methods tested for data-driven uncertainty
quantification based on a benchmark scenario of carbon dioxide storage. In the
benchmark, for which we provide data and code, carbon dioxide is injected into
a saline aquifer modeled by the nonlinear capillarity-free fractional flow
formulation for two incompressible fluid phases, namely carbon dioxide and
brine. To cover different aspects of uncertainty quantification, we incorporate
various sources of uncertainty such as uncertainty of boundary conditions, of
conceptual model definitions and of material properties. We consider recent
versions of the following non-intrusive and intrusive uncertainty
quantification methods: arbitary polynomial chaos, spatially adaptive sparse
grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The
performance of each approach is demonstrated assessing expectation value and
standard deviation of the carbon dioxide saturation against a reference
statistic based on Monte Carlo sampling. We compare the convergence of all
methods reporting on accuracy with respect to the number of model runs and
resolution. Finally we offer suggestions about the methods' advantages and
disadvantages that can guide the modeler for uncertainty quantification in
carbon dioxide storage and beyond
On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion
This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings. This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings
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