scispace - formally typeset
Search or ask a question
Author

Lorenzo Pareschi

Bio: Lorenzo Pareschi is an academic researcher from University of Ferrara. The author has contributed to research in topics: Boltzmann equation & Monte Carlo method. The author has an hindex of 45, co-authored 236 publications receiving 7402 citations. Previous affiliations of Lorenzo Pareschi include University of Wisconsin-Madison & Union des Industries Ferroviaires Européennes.


Papers
More filters
Journal ArticleDOI
TL;DR: New implicit–explicit (IMEX) Runge–Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms are considered, with high accuracy in space and several applications are presented.
Abstract: We consider new implicit---explicit (IMEX) Runge---Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge---Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented

505 citations

Journal ArticleDOI
TL;DR: This survey considers the development and mathematical analysis of numerical methods for kinetic partial differential equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods, and an overview of the current state of the art.
Abstract: In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

339 citations

Posted Content
TL;DR: In this article, new implicit-explicit (IMEX) Runge-Kutta methods were proposed for hyperbolic systems of conservation laws with stiff relaxation terms. But the implicit part is treated by a strong-stability-preserving (SSP) scheme, and the explicit part is represented by an L-stable diagonally implicit Runge Kutta method (DIRK).
Abstract: We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta methods (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.

292 citations

Book
28 Jan 2014
TL;DR: A robust approach to mathematical modelling and numerical simulation of multi-agent systems is presented in detail and is a useful reference text for applied mathematicians, physicists, biologists and economists who want to learn about modelling and approximation of such challenging phenomena.
Abstract: PART I: KINETIC MODELLING AND SIMULATION 1. A short introduction to kinetic equations 2. Mathematical tools 3. Monte Carlo strategies 4. Monte Carlo methods for kinetic equations PART II: MULTIAGENT KINETIC EQUATIONS 5. Models for wealth distribution 6. Opinion modelling and consensus formation 7. A further insight into economy and social sciences 8. Modelling in life sciences Appendix A: Basic arguments on Fourier transforms Appendix B: Important probability distributions

274 citations

Journal ArticleDOI
TL;DR: In this article, a simple kinetic model of economy involving both exchanges between agents and speculative trading is considered and a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals.
Abstract: In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.

247 citations


Cited by
More filters
Christopher M. Bishop1
01 Jan 2006
TL;DR: Probability distributions of linear models for regression and classification are given in this article, along with a discussion of combining models and combining models in the context of machine learning and classification.
Abstract: Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

10,141 citations

Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

01 Mar 1987
TL;DR: The variable-order Adams method (SIVA/DIVA) package as discussed by the authors is a collection of subroutines for solution of non-stiff ODEs.
Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

1,955 citations

Book ChapterDOI
15 Feb 2011

1,876 citations

Journal ArticleDOI
TL;DR: In this paper, the authors presented an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques, and showed Monte Carlo to be very robust but also slow.
Abstract: Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

1,708 citations