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Nonlinear Fokker--Planck equations play a ma jor role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker--Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin pro jection in the parameter space of the unknowns of the underlying Fokker--Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones. [ABSTRACT FROM AUTHOR]

Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models. The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model. In particular, we will consider the class of multiscale transport models recently introduced in [ 13 , 7 ] as the high-fidelity reference and use simple two-velocity discrete models for low-fidelity evaluations. Both models share the same diffusive behavior and are solved with ad-hoc asymptotic-preserving numerical discretizations. A series of numerical experiments confirm the validity of the approach. Correction: This article is copyrighted in 2022. We apologize for any inconvenience this may cause. [ABSTRACT FROM AUTHOR]

The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources such as the amount of infectious individuals. In this paper, we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale. Through a diffusive rescaling, the model allows us to recover classical diffusion equations related to commuting dynamics. The numerical solution of the resulting multiscale hyperbolic system with uncertainty is then tackled using a stochastic collocation approach in combination with a finite volume Implicit–Explicit (IMEX) method. The ability of the model to correctly describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is confirmed with a study of the outbreak of COVID-19 in Italy and its spread in the Lombardy Region. [ABSTRACT FROM AUTHOR]

The design of particle simulation methods for collisional plasma physics has always represented a challenge due to the unbounded total collisional cross section, which prevents a natural extension of the classical Direct Simulation Monte Carlo (DSMC) method devised for the Boltzmann equation. One way to overcome this problem is to consider the design of Monte Carlo algorithms that are robust in the so-called grazing collision limit. In the first part of this manuscript, we will focus on the construction of collision algorithms for the Landau-Fokker-Planck equation based on the grazing collision asymptotics and which avoids the use of iterative solvers. Subsequently, we discuss problems involving uncertainties and show how to develop a stochastic Galerkin projection of the particle dynamics which permits to recover spectral accuracy for smooth solutions in the random space. Several classical numerical tests are reported to validate the present approach. • We construct collision algorithms for the Landau-Fokker-Planck equation which avoids the use of iterative solvers. • We extend the method to the equation involving uncertainties developing a new class of stochastic Galerkin particle methods. • We study the accuracy of the method in the space of the random parameters for several scattering kernels. • We test the method considering classical benchmarks such as BKW, Trubnikov's solution, and the bump-on-tail problem. [ABSTRACT FROM AUTHOR]

The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable of considerably accelerating the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates. We show that the additional degrees of freedom can be used to further improve the variance reduction properties of multiscale control variate methods. [ABSTRACT FROM AUTHOR]

• Uncertainty quantification for the Boltzmann equation. • Variance reduction of standard Monte Carlo sampling methods. • Control variate methods based on reduced models obtained in a multi-scale setting. • Asymptotic behavior of the control variate methods and fluid-dynamic limit. • Error estimates show a strong acceleration of the convergence with respect to standard Monte Carlo. Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. In this paper we consider the construction of novel multi-scale methods for such problems which, thanks to a control variate approach, are capable to reduce the variance of standard Monte Carlo techniques. [ABSTRACT FROM AUTHOR]

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller--Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge--Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multiscale problems. The sAP property will be shown asymptotically and verified numerically in several tests. Other numerical tests are conducted to explore the effect of the randomness in the kinetic system, with the goal of providing more intuition for the theoretic study of the chemotaxis models. [ABSTRACT FROM AUTHOR]

For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based asymptotic-preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales. Compared with previous methods for these problems, our new method uses the implicit-explicit time discretization to gain higher order accuracy, and by using a modified diffusion operator based penalty method, a more relaxed stability condition-a hyperbolic, rather than parabolic, CFL stabil- ity condition-is achieved in the case of a small mean free path in the diffusive regime. The stochastic asymptotic-preserving property of these methods will be shown asymptotically and demonstrated nu- merically, along with a computational cost comparison with previous methods. [ABSTRACT FROM AUTHOR]

The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle method for collisional kinetic models of plasmas under the effect of uncertainties. This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles' position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG projection and to obtain the corresponding system for the gPC coefficients. We show that the sG particle method preserves the main physical properties of the problem, such as conservations and positivity of the solution, while achieving spectral accuracy for smooth solutions in the random space. Furthermore, in the fluid limit the sG particle solver is designed to possess the asymptotic-preserving property necessary to obtain a sG particle scheme for the limiting Euler-Poisson system, thus avoiding the loss of hyperbolicity typical of conventional sG methods based on finite differences or finite volumes. We tested the schemes considering the classical Landau damping problem in the presence of both small and large initial uncertain perturbations, the two stream instability and the Sod shock tube problems under uncertainties. The results show that the proposed method is able to capture the correct behavior of the system in all test cases, even when the relaxation time scale is very small. • New class of stochastic Galerkin particle methods for kinetic models of plasmas under uncertainties. • The particle method preserves the main physical properties of the problem and spectral accuracy in the random space. • Asymptotic-preserving property to obtain Euler-Poisson hydrodynamic limit. • The method avoids loss of hyperbolicity of conventional stochastic Galerkin methods based on finite differences/volumes. • We test the method considering classical Landau damping, two stream instability and Sod shock tube problem. [ABSTRACT FROM AUTHOR]

• Develop a bi-fidelity method for linear transport equations with random parameters. • Employ the Goldstein-Taylor as low-fidelity model to accelerate convergence of the scheme. • The first work to apply multi-fidelity approach to kinetic equations under the diffusive scaling. In this paper, we consider the development of efficient numerical methods for linear transport equations with random parameters and under the diffusive scaling. We extend to the present case the bi-fidelity stochastic collocation method introduced in [33,50,51]. For the high-fidelity transport model, the asymptotic-preserving scheme [29] is used for each stochastic sample. We employ the simple two-velocity Goldstein-Taylor equation as low-fidelity model to accelerate the convergence of the uncertainty quantification process. The choice is motivated by the fact that both models, high fidelity and low fidelity, share the same diffusion limit. Speed-up is achieved by proper selection of the collocation points and reasonable approximation of the high-fidelity solution. Extensive numerical experiments are conducted to show the efficiency and accuracy of the proposed method, even in non diffusive regimes, with empirical error bound estimations as studied in [16]. [ABSTRACT FROM AUTHOR]

• Effects of uncertainties of elastic and viscoelastic parameters of FSI in arteries. • Innovative asymptotic preserving IMEX finite volume stochastic collocation method. • Less sensitivity of flow rate and velocity relative to pressure and area. • Great uncertainty of the viscosity parameter plays a major role in pressure outputs. • The systolic phase is only slightly affected by the viscosity of the wall. 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 Gaussian-distributed variables, making stochastic the system of governing equations. The proposed methodology is initially validated on a model equation, presenting a thorough convergence study which confirms the spectral accuracy of the stochastic collocation method and the second-order 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 the great uncertainty of the viscosity parameter plays a major role in the prediction of pressure waveforms, enlarging the confidence interval of this variable. In-vivo recorded patient-specific pressure data falls within the confidence interval of the output obtained with the proposed methodology and expectations of the computed pressures are comparable to the recorded waveforms. [ABSTRACT FROM AUTHOR]