Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Pettersson, Mass Per; Iaccarino, Gianluca; Nordström, Jan

doi:10.1007/978-3-319-10714-1

Cited by 68 publications

(33 citation statements)

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“…This problem is nonlinear hyperbolic and will exhibit discontinuities that tend to emerge around the same time as the discontinuities of the realizations would emerge, were we to use a sampling-based method instead. For more details on the regularity of the statistics and gPC approximations, we refer to Pettersson et al (2015).…”

Section: Numerical Resultsmentioning

confidence: 99%

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Pettersson

2015

Transp Porous Med

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A stochastic Galerkin formulation for the transport of CO 2 in a tilted aquifer with uncertain heterogeneous properties is presented. We consider a simplified physics model assuming capillary pressure to be negligible compared to hydrostatic and viscous pressure. The flow is dominated by buoyancy and capillary trapping. We assume a stochastic permeability field and a stochastic model for the uncertain relative permeabilities. We prove that the proposed stochastic Galerkin formulation results in a hyperbolic system of equations, and we devise a numerical method that captures the expected solution discontinuities. The shockcapturing solver for the flux function is combined with an adaptive quadrature method for discontinuous isosurfaces that is used to compute the discontinuous stochastic accumulation coefficient. The stochastic solver is validated against Monte Carlo sampling of an analytical solution for the deterministic problem. The sharp features of the statistics of the solution are accurately captured by the numerical solver. The polynomial chaos framework admits low-cost post-processing of the output to obtain statistics of interest. By construction of an accurate polynomial chaos surrogate model of the output, fast sampling admits calculation of risk for leakage and failure probabilities.

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Section: Numerical Resultsmentioning

confidence: 99%

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Pettersson

2015

Transp Porous Med

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“…For the purpose of simplifying the presentation, we assume that the polynomial bases are normalized in the sense that E{φ k (ξ)φ k (ξ)} = δ kk . Interested readers are referred to [46,52,55,62,84] for detailed discussions on representing random variables of different types using appropriate orthogonal polynomials.…”

Section: Generalized Polynomial Chaos Approximationmentioning

confidence: 99%

“…The field of uncertainty quantification has experienced tremendous growth in the past decade, with many efficient general-purpose computational algorithms developed and some specific theoretical issues mathematically understood; see, for instance, [8,17,18,21,27,28,32,33,34,39,46,47,48,49,58,59,62,73,78,83,86] and references therein for some recent developments in the field. In this work, we investigate uncertainty quantification issues in image reconstruction problems in quantitative photoacoustic tomography (PAT), one of the recent hybrid imaging modality that combines the advantages of the classical ultrasound imaging and optical tomography [16,81,82].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Impacts of Model Uncertainties in Quantitative Photoacoustics

Ren¹,

Vallélian²

2020

SIAM/ASA J. Uncertainty Quantification

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This work is concerned with uncertainty quantification problems for image reconstructions in quantitative photoacoustic imaging (PAT), a recent hybrid imaging modality that utilizes the photoacoustic effect to achieve high-resolution imaging of optical properties of tissue-like heterogeneous media. We quantify mathematically and computationally the impact of uncertainties in various model parameters of PAT on the accuracy of reconstructed optical properties. We derive, via sensitivity analysis, analytical bounds on error in image reconstructions in some simplified settings, and develop a computational procedure, based on the method of polynomial chaos expansion, for such error characterization in more general settings. Numerical simulations based on synthetic data are presented to illustrate the main ideas.

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“…Table 1 shows the bivariate Hermite polynomial basis constructed from univariate basis {1, X 1 , X 2 1 − 1} and {1, X 2 , X 2 2 − 1} and the link between the main index j and the multi-index (i 1 , i 2 ). For completness, the same Table also provides bivariate Legendre polynomial basis constructed from univariate basis {1, X 1 , [38]. Here we have used the first one which consists in premultiplying (2.13) by Ψ j (X) and by taking the expectation of the resulting product.…”

Section: Polynomial Chaos Expansionmentioning

confidence: 99%

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

Bachi¹,

Chauvière

Djellout

et al. 2019

Communications in Statistics - Simulation and Computation

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In this paper, we develop a numerical approach based on Chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modelling the propagation of uncertainty arising from those parameters. Furthermore, Polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reliable information on the relative importance of each uncertain entry parameters. Numerical results show the benefit of using Polynomial Chaos over standard Monte-Carlo simulations, when considering statistical moments and Sobol' indices as output quantities.

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Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Cited by 68 publications

References 0 publications

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Characterizing Impacts of Model Uncertainties in Quantitative Photoacoustics

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

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