In this session we concentrate on the latest research insights for uncertainty quantification in transport problems and high-dimensional systems under structural uncertainties, with focus on kinetic and hyperbolic PDEs and multiscale interacting particle systems.
08:30
Particle based stochastic Galerkin methods for kinetic equations with random inputs
Lorenzo Pareschi | University of Ferrara | Italy
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Lorenzo Pareschi | University of Ferrara | Italy
In this talk we will present some recent results on the use of stochastic Galerkin methods directly at the particle level in kinetic equations with random inputs. The method permits to use standard particle simulation techniques in the phase space in combination with stochastic Galerkin methods in the random space and is capable to mitigate the curse of dimensionality of such problems.
Different realizations of the general idea are presented for mean-field type problems and for the challenging case of the Boltzmann equation.
09:00
Sensitivity equation method for the Euler equations applied to uncertainty quantification
Camilla Fiorini | Sorbonne Université | France
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Camilla Fiorini | Sorbonne Université | France
Sensitivity analysis (SA) is the study of how the output of a mathematical model is affected by changes in the inputs.The classical SA techniques, in particular the continuous sensitivity equation (CSE) method, cannot be used if the mathematical model is a system of hyperbolic partial differential equations (PDEs) with discontinuous solutions, because these methods are based on the differentiation of the state system. We aim at defining a correction term to be added to the sensitivity equations starting from the Rankine-Hugoniot conditions, which govern the state across a shock. We detail this procedure in the case of the Euler equations. Numerical results show that the standard Godunov and Roe schemes fail in producing good numerical results because of the underlying numerical diffusion. An anti-diffusive numerical method is then successfully proposed and the corrected sensitivity can be used to estimate the variance of the output of the model.
09:30
A new MC scheme for the resolution of intrusive-gPC based reduced models for the uncertain linear Boltzmann equation
Gaël Poëtte | Atomic Energy and Alternative Energies Commission | Italy
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Gaël Poëtte | Atomic Energy and Alternative Energies Commission | Italy
We first present the method described in [G. Poëtte, J. Comput. Phys., 385:135-162, 2019], used to solve a sensitivity analysis problem for the uncertain linear Boltzmann equation. The intrusive method only needs simple and localized modifications of an already existing MC simulation code. It furthermore allows important gains (by avoiding the tensorisation) but remains slightly sensitive to the dimension (as it is still gPC based). The solver now allows performing several uncertainty propagation, sensitivity analysis allowing to tackle robust optimisation problems on the uncertain linear Boltzman equation. The convergence of the MC scheme will be demonstrated for a fixed P.
Second, we present the recent results of [G. Poëtte. Spectral convergence of the generalized Polynomial Chaos reduced model obtained from the uncertain linear Boltzmann equation, paper submitted]: the spectral convergence of the gPC based reduced model (i.e. convergence with respect to P) holds under mild regularity conditions.
The talk will also present several benchmarks in different stochastic dimensions and for various statistical quantities (mean, variance, Sobol indices for sensitivity analysis).
10:00
Structure preserving gPC schemes for kinetic equations
Mattia Zanella | Politecnico di Torino | Italy
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Mattia Zanella | Politecnico di Torino | Italy
We introduce and discuss numerical schemes for the approximation of kinetic equations that incorporate uncertain quantities. In contrast to a direct application of stochastic Galerkin methods, we will develop a class of schemes that preserve the main structural properties of expected quantities. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve spectral accuracy in the random space.
Several tests on kinetic models for collective phenomena validate the proposed methods both in the homogeneous and inhomogeneous setting.