Showing papers by "Lorenzo Pareschi published in 2013"
Posted Content•
TL;DR: In this article, a step-by-step introduction to the mathematical modelling based on a mesoscopic description and the construction of efficient simulation algorithms by Monte Carlo methods is provided, which can shed light on significant problems of the natural sciences as well as our daily lives.
Abstract: The description of emerging collective phenomena and self-organization in systems composed of large numbers of individuals has gained increasing interest from various research communities in biology, ecology, robotics and control theory, as well as sociology and economics. Applied mathematics is concerned with the construction, analysis and interpretation of mathematical models that can shed light on significant problems of the natural sciences as well as our daily lives. To this set of problems belongs the description of the collective behaviours of complex systems composed by a large enough number of individuals. Examples of such systems are interacting agents in a financial market, potential voters during political elections, or groups of animals with a tendency to flock or herd. Among other possible approaches, this book provides a step-by-step introduction to the mathematical modelling based on a mesoscopic description and the construction of efficient simulation algorithms by Monte Carlo methods. The arguments of the book cover various applications, from the analysis of wealth distributions, the formation of opinions and choices, the price dynamics in a financial market, to the description of cell mutations and the swarming of birds and fishes. By means of methods inspired by the kinetic theory of rarefied gases, a robust approach to mathematical modelling and numerical simulation of multi-agent systems is presented in detail. The content is a useful reference text for applied mathematicians, physicists, biologists and economists who want to learn about modelling and approximation of such challenging phenomena.
173 citations
TL;DR: An approach is presented that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly.
Abstract: We consider implicit-explicit (IMEX) Runge--Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such a regime the system relaxes towards a convection-diffusion equation. The first objective of this paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of this paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by a novel reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis of such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, unde...
168 citations
TL;DR: Implicit-Explicit Runge Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type are discussed which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator.
Abstract: We discuss implicit-explicit (IMEX) Runge--Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits us to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally, we show some numerical results which confirm the theoretical analysis.
100 citations
TL;DR: In this paper, the authors proposed a stochastic method for computing approximate solutions as functions of a small scaling parameter at a reduced complexity of $O(N)$ operations.
Abstract: Microscopic models of flocking and swarming take into account large numbers of interacting individuals. Numerical resolution of large flocks implies huge computational costs. Typically for $N$ interacting individuals we have a cost of $O(N^2)$. We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits us to compute approximate solutions as functions of a small scaling parameter $\varepsilon$ at a reduced complexity of $O(N)$ operations. Several numerical results show the efficiency of the algorithms proposed.
73 citations
TL;DR: This work derives the kinetic model through a mean-field limit and the macroscopic system through a suitable hydrodynamic limit of the self-organized systems as flock of birds, school of fishes or herd of sheeps.
59 citations
TL;DR: Order conditions and symplecticity properties of a class of IMEX Runge--Kutta methods in the context of optimal control problems are discussed and suitable transformations of the adjoint equation are used.
Abstract: Implicit-explicit (IMEX) Runge--Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and nonstiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge--Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable transformations of the adjoint equation, order conditions up to order three are proven, and the relation between adjoint schemes obtained through different transformations is investigated as well. Conditions for the IMEX Runge--Kutta methods to be symplectic are also derived. A numerical example illustrating the theoretical properties is presented.
27 citations
TL;DR: In this article, the Boltzmann collision operator is computed using discrete-velocity approximations, and the cost is O(n 2d+1 )w hered, where w is the dimension of the velocity space.
Abstract: Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N 2d+1 )w hered is the dimension of the velocity space. In this paper, following the ideas introduced
23 citations
TL;DR: A class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation with high order time and space discretization schemes which do not suer from the usual parabolic stiness in the diusive limit.
Abstract: In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.
14 citations
Posted Content•
TL;DR: In this article, the authors developed high-order asymptotic-preserving methods for the spatially inhomogeneous quantum Boltzmann equation and showed that the proposed schemes work with highorder accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen numbers ranging from kinetic regime to fluid regime.
Abstract: In this paper we develop high-order asymptotic-preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi, where asymptotic preserving exponential Runge-Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian steady states characterizing the quantum kinetic behavior. We show that the proposed schemes work with high-order accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen numbers ranging from kinetic regime to fluid regime. Computational results are presented for both Bose gas and Fermi gas.
11 citations
Posted Content•
TL;DR: The relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure are investigated for the class of stiffly accurate implicit–explicit Runge–Kutta methods and the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation.
Abstract: We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior.
3 citations