99 research outputs found
A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
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 multi-scale problems. The stochastic Asymptotic-Preserving property
will be shown asymptotically and verified numerically in several tests. Many
other numerical tests are conducted to explore the effect of the randomness in
the kinetic system, in the aim of providing more intuitions for the theoretic
study of the chemotaxis models
Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model
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
An introduction to uncertainty quantification for kinetic equations and related problems
We overview some recent results in the field of uncertainty quantification
for kinetic equations and related problems with random inputs. Uncertainties
may be due to various reasons, such as lack of knowledge on the microscopic
interaction details or incomplete information at the boundaries or on the
initial data. These uncertainties contribute to the curse of dimensionality and
the development of efficient numerical methods is a challenge. After a brief
introduction on the main numerical techniques for uncertainty quantification in
partial differential equations, we focus our survey on some of the recent
progress on multi-fidelity methods and stochastic Galerkin methods for kinetic
equations
Multi-scale control variate methods for uncertainty quantification in kinetic equations
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
Kinetic Modelling of Epidemic Dynamics: Social Contacts, Control with Uncertain Data, and Multiscale Spatial Dynamics
In this survey we report some recent results in the mathematical modelling 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 modelling 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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