An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems

Pareschi, Lorenzo

doi:10.1007/978-3-030-67104-4_5

Cited by 20 publications

(18 citation statements)

References 59 publications

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“…When introducing the same definition of flux (89) for the additional compartments, 𝐽 𝐸 and 𝐽 𝐴 , integrating system (93) in 𝑣, we get the set of equations for the macroscopic densities [18]. Moreover, defining also 𝐷 𝐸 = 1 2 𝜆 2 𝐸 𝜏 𝐸 and 𝐷 𝐴 = 1 2 𝜆 2 𝐴 𝜏 𝐴 and considering the same procedure discussed in Section 4.2.2, we recover SEIAR system in the diffusive regime for the commuting individuals [18] coupled with (94) for the non-commuting counterpart.…”

Section: Extension To Multi-compartmental Modellingmentioning

confidence: 99%

“…Let us underline that the discretization of the resulting multiscale systems of PDEs is not trivial and therefore requires the construction of a specific numerical method able to correctly describe the transition from a convective to a diffusive regime in realistic geometries. For this purpose, we adopt an asymptotic-preserving IMEX Runge-Kutta method on unstructured grids coupled with a stochastic Collocation method which ensures spectral accuracy in the stochastic space [70,71,94]. At each collocation node, the numerical scheme combines a discrete ordinate method in velocity with the even and odd parity formulation [42,72] and achieves asymptotic preservation in time using suitable IMEX Runge-Kutta schemes [22], namely, to obtain a scheme which consistently captures the diffusion limit and for which the choice of the time discretization step is not related to the smallness of the scaling parameters 𝜏.…”

Section: Emilia-romagna and Lombardy Regionmentioning

confidence: 99%

“…In addition, available experimental data are often affected by large uncertainty, which must therefore be considered as part of the process of modeling the infectious disease and simulating the potential epidemic scenarios and control strategies [29,34,105]. A large amount of research in this direction has been recently carried out in the field of hyperbolic and kinetic equations and it is therefore natural to rely on this scientific background to design new models and numerical methods able to deal efficiently with the presence of uncertain data [19,70,71,94,100,121]. We also mention some other related research based on modeling the diffusion of COVID-19 using PDEs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Albi¹,

Bertaglia²,

Boscheri³

et al. 2021

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In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

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Section: Extension To Multi-compartmental Modellingmentioning

confidence: 99%

Section: Emilia-romagna and Lombardy Regionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Albi¹,

Bertaglia²,

Boscheri³

et al. 2021

Preprint

Self Cite

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“…Intrusive approaches can be very complex and hard to implement, since a parametrization of the uncertainty of the inputs is substituted into the model to derive new governing equations. Examples of these methods are the perturbation method, the momentum equation approach and the stochastic Galerkin method [11,32,36,41,53]. In particular, stochastic Galerkin methods, based on generalized polynomial chaos (gPC) expansions, are very attractive thanks to the spectral convergence property with respect to the random input [29].…”

Section: Numerical Approximationmentioning

confidence: 99%

Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model

Bertaglia,

Caleffi,

Pareschi

et al. 2020

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This work aims at identifying and quantifying uncertainties related to elastic and viscoelastic parameters, which characterize the arterial wall behavior, in one-dimensional modeling of the human arterial hemodynamics. The chosen uncertain parameters are modeled as random Gaussiandistributed variables, making stochastic the system of governing equations. The proposed methodology is initially discussed for a model equation, presenting a thorough convergence study which confirms the spectral accuracy of the stochastic collocation method and the second-order of accuracy of the IMEX finite volume scheme chosen to solve the mathematical model. Then, univariate and multivariate uncertain quantification analyses are applied to the a-FSI blood flow model, concerning baseline and patient-specific single-artery test cases. A different sensitivity is depicted when comparing the variability of flow rate and velocity waveforms to the variability of pressure and area, the latter ones resulting much more sensitive to the parametric uncertainties underlying the mechanical characterization of vessel walls. Simulations performed considering both the simple elastic and the more realistic viscoelastic constitutive law show that including viscoelasticity in the FSI model consistently improves the reliability of pressure waveforms prediction. Results of the patient-specific tests suggest that the proposed methodology could be a valuable tool for improving cardiovascular diagnostics and the treatment of diseases.

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“…Several techniques can be adopted for the approximation of the quantities of interest. Here, following [4] we adopt stochastic Galerkin methods that allow to reduce the problem to a set of deterministic equations for the numerical evaluation of the solution in presence of uncertainties [17,44,52].…”

Section: Introductionmentioning

confidence: 99%

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty

Albi

Pareschi

Zanella

2020

Preprint

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After an initial phase characterized by the introduction of timely and drastic containment measures aimed at stopping the epidemic contagion from SARS-CoV2, many governments are preparing to relax such measures in the face of a severe economic crisis caused by lockdowns. Assessing the impact of such openings in relation to the risk of a resumption of the spread of the disease is an extremely difficult problem due to the many unknowns concerning the actual number of people infected, the actual reproduction number and infection fatality rate of the disease. In this work, starting from a compartmental model with a social structure, we derive models with multiple feedback controls depending on the social activities that allow to assess the impact of a selective relaxation of the containment measures in the presence of uncertain

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An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems

Cited by 20 publications

References 59 publications

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty

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