tag:blogger.com,1999:blog-19419195305626374022021-04-13T21:30:18.882+02:00Lorenzo Pareschi Home PageDepartment of Mathematics and Computer Science, University of Ferrara
Dipartimento di Matematica e Informatica, Università di Ferrara
Numerical Analysis, Kinetic equations, Boltzmann equation, Hyperbolic Relaxation Systems, Implicit-Explicit (IMEX) methods, Boltzmann equation, Monte Carlo methods, Spectral methods, Asymptotic Preserving schemes, Multiscale methods, Hybrid methods, Conservation laws, Machine Learning, Particle Swarm Optimization, Consensus Based OptimizationLorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comBlogger81125tag:blogger.com,1999:blog-1941919530562637402.post-67928826772774938602021-04-01T21:21:00.007+02:002021-04-13T21:26:35.370+02:00Anisotropic Diffusion in Consensus-based Optimization on the Sphere<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-bJXQoTYrg3s/YHXv7w6D34I/AAAAAAAACyg/DjASMUu5j3cZ7dxG8IOouUY-52PmQd6iwCLcBGAsYHQ/s1684/Schermata%2B2021-04-13%2Balle%2B21.24.00.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="970" data-original-width="1684" src="https://1.bp.blogspot.com/-bJXQoTYrg3s/YHXv7w6D34I/AAAAAAAACyg/DjASMUu5j3cZ7dxG8IOouUY-52PmQd6iwCLcBGAsYHQ/s320/Schermata%2B2021-04-13%2Balle%2B21.24.00.png" width="320" /></a></div><br />Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen (1/4/2021 preprint <a href="https://arxiv.org/abs/2104.00420">arXiv:2104.00420</a>)<p></p><p style="text-align: justify;">In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration. <span></span></p><a name='more'></a>The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof of convergence combines a mean-field limit result with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. The main innovation with respect to previous work is the introduction of an anisotropic stochastic term, which allows us to ensure the independence of the parameters of the algorithm from the dimension and to scale the method to work in very high dimension. We present several numerical experiments, which show that the algorithm proposed in the present paper is extremely versatile and outperforms previous formulations with isotropic stochastic noise.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-37085387062635312142021-02-04T12:45:00.003+01:002021-02-05T12:50:37.921+01:00Mean-field control variate methods for kinetic equations with uncertainties and applications to socio-economic sciences<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-jGf9OI1XmVw/YB0wTPDMbcI/AAAAAAAAClY/Y7nWX_veu7w1DxQo8lQh85XrvANov_wDACLcBGAsYHQ/s2776/Schermata%2B2021-02-05%2Balle%2B12.47.04.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1070" data-original-width="2776" src="https://1.bp.blogspot.com/-jGf9OI1XmVw/YB0wTPDMbcI/AAAAAAAAClY/Y7nWX_veu7w1DxQo8lQh85XrvANov_wDACLcBGAsYHQ/s320/Schermata%2B2021-02-05%2Balle%2B12.47.04.png" width="320" /></a></div>Lorenzo Pareschi, Torsten Trimborn, Mattia Zanella (4/2/2021 preprint <a href="https://arxiv.org/abs/2102.02589">arXiv:2102.02589</a>) <p></p><p style="text-align: justify;">In this paper, we extend a recently introduced multi-fidelity control variate for the uncertainty quantification of the Boltzmann equation to the case of kinetic models arising in the study of multiagent systems. For these phenomena, where the effect of uncertainties is particularly evident, several models have been developed whose equilibrium states are typically unknown. In particular, we aim to develop efficient numerical methods based on solving the kinetic equations in the phase space by Direct Simulation Monte Carlo (DSMC) coupled to a Monte Carlo sampling in the random space. <span></span></p><a name='more'></a>To this end, exploiting the knowledge of the corresponding mean-field approximation we develop novel mean-field Control Variate (MFCV) methods that are able to strongly reduce the variance of the standard Monte Carlo sampling method in the random space. We verify these observations with several numerical examples based on classical models , including wealth exchanges and opinion formation model for collective phenomena.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-25827334242711874142020-12-18T17:48:00.009+01:002021-03-06T21:50:10.733+01:00Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-5Q9bmu9PzTk/X-TIMU7-voI/AAAAAAAACbU/vNnO1qM-CDkanAskZFAG9lsbwOd9Bv7xwCLcBGAsYHQ/s617/Schermata%2B2020-12-24%2Balle%2B17.51.39.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="617" data-original-width="487" height="320" src="https://1.bp.blogspot.com/-5Q9bmu9PzTk/X-TIMU7-voI/AAAAAAAACbU/vNnO1qM-CDkanAskZFAG9lsbwOd9Bv7xwCLcBGAsYHQ/s320/Schermata%2B2020-12-24%2Balle%2B17.51.39.png" /></a></div><br />Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (18/12/2020, preprint <a href="https://arxiv.org/pdf/2012.10101">arXiv:2012.10101</a>, to appear in <i>Math. Mod. Meth. App. Scie.</i>)<p></p><p></p><div style="text-align: justify;">In this work we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration dynamics. A construction based on the transport formalism of kinetic theory allows to give a clear model interpretation to the interactions between infected and susceptible in compartmental space-dependent models.</div><span><a name='more'></a></span><div style="text-align: justify;">In addition, in a suitable scaling limit, our approach permits to couple the two populations through a consistent diffusion model acting at the urban scale. A discretization of the system based on finite volumes on unstructured grids, combined with an asymptotic preserving method in time, shows that the model is able to describe correctly the main features of the spatial expansion of an epidemic. An application to the initial spread of COVID-19 is finally presented.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-38321063622386078062020-12-10T09:02:00.004+01:002021-03-19T15:48:32.852+01:00From particle swarm optimization to consensus based optimization: stochastic modeling and mean-field limit<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vMVdn0z8_KQ/X9MoJoQmgJI/AAAAAAAACZg/ViAYrfZUJ_UYRtHDvtzAc2CLcpCI4JoJwCLcBGAsYHQ/s2048/Schermata%2B2020-12-11%2Balle%2B09.04.28.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1905" data-original-width="2048" src="https://1.bp.blogspot.com/-vMVdn0z8_KQ/X9MoJoQmgJI/AAAAAAAACZg/ViAYrfZUJ_UYRtHDvtzAc2CLcpCI4JoJwCLcBGAsYHQ/s320/Schermata%2B2020-12-11%2Balle%2B09.04.28.png" width="320" /></a></div>Sara Grassi, Lorenzo Pareschi (10/12/2020 preprint <a href="https://arxiv.org/abs/2012.05613">arXiv:2012.05613</a>), Math. Mod. Meth. App. Scie. to appear<p></p><p style="text-align: justify;">In this paper we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov-Fokker-Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best. <span></span></p><a name='more'></a>A regularization process for the global best permits to formally derive the respective mean-field description. Subsequently, in the small inertia limit, we compute the related macroscopic hydrodynamic equations that clarify the link with the recently introduced consensus based optimization (CBO) methods. Several numerical examples illustrate the mean field process, the small inertia limit and the potential of this general class of global optimization methods.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-67116894841117753952020-11-11T15:46:00.005+01:002021-03-04T23:44:24.526+01:00On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation<p> Lorenzo Pareschi, Thomas Rey (11/11/2020 preprint <a href="https://arxiv.org/pdf/2011.05811.pdf">arXiv:2011.05811</a>, to appear in <i>Applied Math. Letters</i>)</p><p>Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.</p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-55405721461069470412020-08-07T09:53:00.005+02:002021-03-04T23:43:34.436+01:00High order pressure-based semi-implicit IMEX schemes for the 3D Navier-Stokes equations at all Mach numbers<br /><div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-LPmUh1YpHf0/Xy0IRWx-bTI/AAAAAAAACHU/ZO1jTRuEnSgseCo1bPqUGWlDuWUiWDp-gCLcBGAsYHQ/s724/Schermata%2B2020-08-07%2Balle%2B09.52.08.png" style="clear: left; display: block; float: left; margin-bottom: 1em; margin-right: 1em; padding: 1em 0px;"><img border="0" data-original-height="645" data-original-width="724" height="284" src="https://1.bp.blogspot.com/-LPmUh1YpHf0/Xy0IRWx-bTI/AAAAAAAACHU/ZO1jTRuEnSgseCo1bPqUGWlDuWUiWDp-gCLcBGAsYHQ/w410-h365/Schermata%2B2020-08-07%2Balle%2B09.52.08.png" width="320" /></a></div>Walter Boscheri, Lorenzo Pareschi (4/8/2020 preprint <a href="https://arxiv.org/pdf/2008.01789.pdf">arXiv:2008.01789</a>, to appear in <i>J. Comp. Phys.</i>)<br /><div><br /></div><div style="text-align: justify;">This article aims at developing a high order pressure-based solver for the solution of the 3D compressible Navier-Stokes system at all Mach numbers. We propose a cell-centered discretization of the governing equations that splits the fluxes into a fast and a slow scale part, that are treated implicitly and explicitly, respectively. A novel semi-implicit discretization is proposed for the kinetic energy as well as the enthalpy fluxes in the energy equation, hence avoiding any need of iterative solvers. The implicit discretization yields an elliptic equation on the pressure that can be solved for both ideal gas and general equation of state (EOS). <br /><a name='more'></a>A nested Newton method is used to solve the mildly nonlinear system for the pressure in case of nonlinear EOS. High order in time is granted by implicit-explicit (IMEX) time stepping, whereas a novel CWENO technique efficiently implemented in a dimension-by-dimension manner is developed for achieving high order in space for the discretization of explicit convective and viscous fluxes. A quadrature-free finite volume solver is then derived for the high order approximation of numerical fluxes. Central schemes with no dissipation of suitable order of accuracy are finally employed for the numerical approximation of the implicit terms. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the sound speed, so that the novel schemes work uniformly for all Mach numbers. Convergence and robustness of the proposed method are assessed through a wide set of benchmark problems involving low and high Mach number regimes, as well as inviscid and viscous flows.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-37734014016477347012020-07-08T14:07:00.002+02:002020-12-04T12:46:12.610+01:00Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods<a href="https://1.bp.blogspot.com/-2pDHEJ6N0Jg/XwhZ04QhV1I/AAAAAAAACFY/5rzdrqh6VAANCpnTcZcxcAC-Anzu-P8FwCLcBGAsYHQ/s1600/Schermata%2B2020-07-10%2Balle%2B14.06.21.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1324" data-original-width="1600" height="264" src="https://1.bp.blogspot.com/-2pDHEJ6N0Jg/XwhZ04QhV1I/AAAAAAAACFY/5rzdrqh6VAANCpnTcZcxcAC-Anzu-P8FwCLcBGAsYHQ/s320/Schermata%2B2020-07-10%2Balle%2B14.06.21.png" width="320" /></a>Giulia Bertaglia, Lorenzo Pareschi (8/7/2020, <i>ESAIM Mathematical Modelling and Numerical Analysis</i> to appear, <a href="https://arxiv.org/abs/2007.04019">arXiv:2007.04019</a>)<br /><br /><div style="text-align: justify;">We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. <br /><a name='more'></a>The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-64199546869813865192020-07-03T11:18:00.002+02:002020-12-25T09:44:08.686+01:00Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model<a href="https://1.bp.blogspot.com/-rFrhTpVfbaA/XwQ9V1haElI/AAAAAAAACE4/iFurizq0nEkY1Lx9S3Yz4ulNbf52C8yaQCLcBGAsYHQ/s1600/Schermata%2B2020-07-07%2Balle%2B11.15.41.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><br /></a><a href="https://1.bp.blogspot.com/-rFrhTpVfbaA/XwQ9V1haElI/AAAAAAAACE4/iFurizq0nEkY1Lx9S3Yz4ulNbf52C8yaQCLcBGAsYHQ/s1600/Schermata%2B2020-07-07%2Balle%2B11.15.41.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="1351" data-original-width="1600" height="268" src="https://1.bp.blogspot.com/-rFrhTpVfbaA/XwQ9V1haElI/AAAAAAAACE4/iFurizq0nEkY1Lx9S3Yz4ulNbf52C8yaQCLcBGAsYHQ/s320/Schermata%2B2020-07-07%2Balle%2B11.15.41.png" width="320" /></a><br /><br />Giulia Bertaglia, Valerio Caleffi, Lorenzo Pareschi, Alessandro Valiani (3/7/2020 to appear in <i>J. Comp. Phys.</i> <a href="https://arxiv.org/abs/2007.01907">arXiv:2007.01907</a>)<br /><br /><div style="text-align: justify;">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 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. </div><a name='more'></a><div style="text-align: justify;">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.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-rFrhTpVfbaA/XwQ9V1haElI/AAAAAAAACE4/iFurizq0nEkY1Lx9S3Yz4ulNbf52C8yaQCLcBGAsYHQ/s1600/Schermata%2B2020-07-07%2Balle%2B11.15.41.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><br /></a></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-43221385687612839032020-05-13T12:45:00.000+02:002020-07-07T11:21:54.820+02:00Relaxing lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty<a href="https://1.bp.blogspot.com/-dkDBUXMZ5n0/XsEVb4iKqKI/AAAAAAAACBw/WZLCCwZ4URIg8Twqu35tuZfFfBz8sTbwQCLcBGAsYHQ/s1600/Schermata%2B2020-05-17%2Balle%2B12.42.59.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="1600" data-original-width="1383" height="320" src="https://1.bp.blogspot.com/-dkDBUXMZ5n0/XsEVb4iKqKI/AAAAAAAACBw/WZLCCwZ4URIg8Twqu35tuZfFfBz8sTbwQCLcBGAsYHQ/s320/Schermata%2B2020-05-17%2Balle%2B12.42.59.png" width="276" /></a>Giacomo Albi, Lorenzo Pareschi, Mattia Zanella (13/5/2020 medRxiv preprint doi: <a href="https://doi.org/10.1101/2020.05.12.20099721">10.1101/2020.05.12.20099721</a>)<br /><span style="text-align: justify;"><br /></span><span style="text-align: justify;">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 data.</span><br /><a name='more'></a>Specific contact patterns in the home, work, school and other locations for all countries considered have been used. Results from different scenarios in some of the major countries where the epidemic is ongoing, including Germany, France, Italy, Spain, the United Kingdom and the United States, are presented and discussed.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-26544981720252078932020-04-29T08:48:00.001+02:002020-09-02T13:16:58.870+02:00Wealth distribution under the spread of infectious diseases<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Th7jUxBvGXU/Xqki-__Z_9I/AAAAAAAACBA/G1PgCQ8JRj4nHSbB6ptbZ77J8WbdillfwCEwYBhgL/s1600/Schermata%2B2020-04-29%2Balle%2B08.47.00.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1425" data-original-width="1600" height="284" src="https://1.bp.blogspot.com/-Th7jUxBvGXU/Xqki-__Z_9I/AAAAAAAACBA/G1PgCQ8JRj4nHSbB6ptbZ77J8WbdillfwCEwYBhgL/s320/Schermata%2B2020-04-29%2Balle%2B08.47.00.png" width="320" /></a></div>G. Dimarco, L. Pareschi, G. Toscani, M. Zanella (28/4/2020 <i>Phys. Rev. E</i> 102, 022303, <a href="https://arxiv.org/abs/2004.13620">arXiv:2004.13620</a>)<br /><br /><div style="text-align: justify;">We develop a mathematical framework to study the economic impact of infectious diseases by integrating epidemiological dynamics with a kinetic model of wealth exchange. The multi-agent description leads to study the evolution over time of a system of kinetic equations for the wealth densities of susceptible, infectious and recovered individuals, whose proportions are driven by a classical compartmental model in epidemiology. <br /><a name='more'></a>Explicit calculations show that the spread of the disease seriously affects the distribution of wealth, which, unlike the situation in the absence of epidemics, can converge towards a stationary state with a bimodal form. Furthermore, simulations confirm the ability of the model to describe different phenomena characteristics of economic trends in situations compromised by the rapid spread of an epidemic, such as the unequal impact on the various wealth classes and the risk of a shrinking middle class.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-77394169570042240222020-04-29T08:32:00.000+02:002020-04-29T08:50:11.338+02:00Control with uncertain data of socially structured compartmental epidemic models<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-4vCzK6s6TDs/XqkfN7glMeI/AAAAAAAACA0/DSvwe4aZyU8S7_Nn9imJSvjrDhhTDaLqgCLcBGAsYHQ/s1600/Schermata%2B2020-04-29%2Balle%2B08.30.57.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1243" data-original-width="1600" height="248" src="https://1.bp.blogspot.com/-4vCzK6s6TDs/XqkfN7glMeI/AAAAAAAACA0/DSvwe4aZyU8S7_Nn9imJSvjrDhhTDaLqgCLcBGAsYHQ/s320/Schermata%2B2020-04-29%2Balle%2B08.30.57.png" width="320" /></a></div>G. Albi, L. Pareschi, M. Zanella (27/4/2020 preprint <a href="https://arxiv.org/abs/2004.13067">arXiv:2004.13067</a>)<br /><br /><div style="text-align: justify;">The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. In addition, data are often incomplete and heterogeneous, so a high degree of uncertainty must naturally be incorporated into the models. <br /><a name='more'></a>In this work, we address both these aspects, through an optimal control formulation of the epidemiological model in the presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback-controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The importance of the timing and intensity of interventions is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the recent COVID-19 outbreak in Italy are presented and discussed.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-51297953479162589552020-04-17T19:52:00.004+02:002021-03-02T22:39:08.974+01:00Uncertainty quantification for the BGK model of the Boltzmann equation using multilevel variance reduced Monte Carlo methods<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-6eJdrfYFDZg/XpnstOP-4HI/AAAAAAAAB_4/rHfYxLI4UZs7RSDp9nknQqV9HwJ1cbrIQCLcBGAsYHQ/s1600/Schermata%2B2020-04-17%2Balle%2B19.51.27.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="889" data-original-width="1600" height="176" src="https://1.bp.blogspot.com/-6eJdrfYFDZg/XpnstOP-4HI/AAAAAAAAB_4/rHfYxLI4UZs7RSDp9nknQqV9HwJ1cbrIQCLcBGAsYHQ/s320/Schermata%2B2020-04-17%2Balle%2B19.51.27.png" width="320" /></a></div>Jingwei Hu, Lorenzo Pareschi, Yubo Wang (17/4/2020 to appear in <i>SIAM/ASA Journal on Uncertainty Quantification</i>, preprint <a href="https://arxiv.org/pdf/2004.07638.pdf">arXiv:2004.07638</a>)<br /><br /><div style="text-align: justify;">We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. </div><a name='more'></a>Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical multilevel Monte Carlo method especially for problems with discontinuities.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-31578399516854611942020-04-13T08:34:00.004+02:002020-07-07T11:27:16.633+02:00An introduction to uncertainty quantification for kinetic equations and related problems<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-5LH5phN2Hho/XpQHs0sF58I/AAAAAAAAB_M/vUddYtUW8TEZYLMa9JzOvA5TVRhlNz_JACLcBGAsYHQ/s1600/Schermata%2B2020-04-13%2Balle%2B08.32.59.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="700" data-original-width="1600" height="137" src="https://1.bp.blogspot.com/-5LH5phN2Hho/XpQHs0sF58I/AAAAAAAAB_M/vUddYtUW8TEZYLMa9JzOvA5TVRhlNz_JACLcBGAsYHQ/s320/Schermata%2B2020-04-13%2Balle%2B08.32.59.png" width="320" /></a></div>Lorenzo Pareschi (13/4/2020 preprint <a href="https://arxiv.org/pdf/2004.05072.pdf" target="_blank">arXiv:2004.05072</a> to appear on SEMA-SIMAI Springer series "<i>Trails in kinetic theory: Foundational aspects and numerical methods</i>")<br /><br /><div style="text-align: justify;">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. </div><a name='more'></a>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.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-56153485752302335522020-03-14T18:07:00.002+01:002020-12-25T10:03:05.678+01:00Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-d0E1OWMNpvY/XnuPdaK4yQI/AAAAAAAAB-E/wKPVGOGtNKEbx8XZOGjy-X_rfWRozCp8wCLcBGAsYHQ/s1600/Schermata%2B2020-03-25%2Balle%2B18.05.17.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1325" data-original-width="1600" height="165" src="https://1.bp.blogspot.com/-d0E1OWMNpvY/XnuPdaK4yQI/AAAAAAAAB-E/wKPVGOGtNKEbx8XZOGjy-X_rfWRozCp8wCLcBGAsYHQ/s200/Schermata%2B2020-03-25%2Balle%2B18.05.17.png" width="200" /></a></div>Lorenzo Pareschi, Mattia Zanella (14/3/2020, <i>J. Comp. Phys</i> 423, (2020), 109822, <a href="https://arxiv.org/pdf/2003.06716.pdf">arXiv:2003.06716</a>)<br /><br /><div><div style="text-align: justify;">In this paper we propose a novel numerical approach for the Boltzmann equation with uncertainties. The method combines the efficiency of classical direct simulation Monte Carlo (DSMC) schemes in the phase space together with the accuracy of stochastic Galerkin (sG) methods in the random space. This hybrid formulation makes it possible to construct methods that preserve the main physical properties of the solution along with spectral accuracy in the random space.</div><a name='more'></a><div style="text-align: justify;">The schemes are developed and analyzed in the case of space homogeneous problems as these contain the main numerical difficulties. Several test cases are reported, both in the Maxwell and in the variable hard sphere (VHS) framework, and confirm the properties and performance of the new methods.</div></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-80345807118079499522020-02-03T17:40:00.005+01:002021-03-19T15:50:45.871+01:00Consensus-based Optimization on the Sphere: Convergence to Global Minimizers and Machine Learning<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-4tDJUyvoNZ4/XjhMwXzqOeI/AAAAAAAAB6g/atqdIgXHiJwGVR_8bNC4kC1VEj3P1U8HACLcBGAsYHQ/s1600/Schermata%2B2020-02-03%2Balle%2B17.39.11.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="773" data-original-width="1082" height="227" src="https://1.bp.blogspot.com/-4tDJUyvoNZ4/XjhMwXzqOeI/AAAAAAAAB6g/atqdIgXHiJwGVR_8bNC4kC1VEj3P1U8HACLcBGAsYHQ/s320/Schermata%2B2020-02-03%2Balle%2B17.39.11.png" width="320" /></a></div>Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen (3/2/2020 <a href="https://arxiv.org/abs/2001.11988" target="_blank">arXiv:2001.11988</a>)<br /><br /><div style="text-align: justify;">We present the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. </div><a name='more'></a>The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof combines previous results of mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the algorithm proposed in the present paper scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-74416302184373684992020-01-31T17:35:00.001+01:002020-11-17T15:51:32.254+01:00Consensus-Based Optimization on Hypersurfaces: Well-Posedness and Mean-Field Limit <a href="https://1.bp.blogspot.com/-l1vdU4CBgD4/XjhLmr4R5gI/AAAAAAAAB6U/2upXzwnCDnE_7u-AkGj8EY6TbJPgftVkgCLcBGAsYHQ/s1600/Schermata%2B2020-02-03%2Balle%2B17.34.14.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="655" data-original-width="1102" height="188" src="https://1.bp.blogspot.com/-l1vdU4CBgD4/XjhLmr4R5gI/AAAAAAAAB6U/2upXzwnCDnE_7u-AkGj8EY6TbJPgftVkgCLcBGAsYHQ/s320/Schermata%2B2020-02-03%2Balle%2B17.34.14.png" width="320" /></a>Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen (31/1/2020, <i>Math. Mod. Meth. Appl. Sciences</i> to appear, <a href="https://arxiv.org/abs/2001.11994" target="_blank">arXiv:2001.11994</a>)<br /><br /><div style="text-align: justify;">We introduce a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace's principle. </div><div style="text-align: justify;"></div><a name='more'></a>The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-73826366576412042072020-01-12T09:18:00.002+01:002020-11-23T12:23:33.645+01:00High order semi-implicit multistep methods for time dependent partial differential equations<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ziXsgKRgJqI/Xh126cdNwHI/AAAAAAAAB5A/816FcXJrTj8k6U_fBa334vEYoOMXCRLGACLcBGAsYHQ/s1600/Schermata%2B2020-01-14%2Balle%2B09.07.52.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="549" data-original-width="1029" height="170" src="https://1.bp.blogspot.com/-ziXsgKRgJqI/Xh126cdNwHI/AAAAAAAAB5A/816FcXJrTj8k6U_fBa334vEYoOMXCRLGACLcBGAsYHQ/s320/Schermata%2B2020-01-14%2Balle%2B09.07.52.png" width="320" /></a></div>Giacomo Albi, Lorenzo Pareschi (12/1/2020 <i>Communications on Applied Mathematics and Computation</i> to appear, <a href="https://arxiv.org/abs/2001.03974" target="_blank">arXiv:2001.03974</a>)<br /><br /><div style="text-align: justify;">We consider the construction of semi-implicit linear multistep methods which can be applied to time dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. <br /><a name='more'></a>As shown in Boscarino, Filbet and Russo (2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allows, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-51069768779411163532019-12-25T09:25:00.000+01:002020-09-02T13:19:00.219+02:00Mathematical models and methods for crowd dynamics control<a href="https://1.bp.blogspot.com/-nJefVlYTU4U/XhmGTCXRcyI/AAAAAAAAB4k/vq9IOsTJuag6LJ7HkDe7YcCxTuUCM4OnQCLcBGAsYHQ/s1600/Schermata%2B2020-01-11%2Balle%2B09.24.25.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="587" data-original-width="780" height="150" src="https://1.bp.blogspot.com/-nJefVlYTU4U/XhmGTCXRcyI/AAAAAAAAB4k/vq9IOsTJuag6LJ7HkDe7YcCxTuUCM4OnQCLcBGAsYHQ/s200/Schermata%2B2020-01-11%2Balle%2B09.24.25.png" width="200" /></a>G. Albi, E. Cristiani, L. Pareschi, D. Peri (25/12/2019<i> </i><i>preprint </i><a href="https://arxiv.org/abs/1912.11628" target="_blank">arXiv:1912.11628</a> to appear in <i>"Crowd Dynamics Volume 2 - Theory, Models, and Applications"</i>, Birkhäuser-Springer)<br /><br /><div style="text-align: justify;">In this survey we consider mathematical models and methods recently developed to control crowd dynamics, with particular emphasis on egressing pedestrians. We focus on two control strategies: The first one consists in using special agents, called leaders, to steer the crowd towards the desired direction. Leaders can be either hidden in the crowd or recognizable as such. <br /><a name='more'></a>This strategy heavily relies on the power of the social influence (herding effect), namely the natural tendency of people to follow group mates in situations of emergency or doubt. The second one consists in modify the surrounding environment by adding in the walking area multiple obstacles optimally placed and shaped. The aim of the obstacles is to naturally force people to behave as desired. Both control strategies discussed in this paper aim at reducing as much as possible the intervention on the crowd. Ideally the natural behavior of people is kept, and people do not even realize they are being led by an external intelligence. Mathematical models are discussed at different scales of observation, showing how macroscopic (fluid-dynamic) models can be derived by mesoscopic (kinetic) models which, in turn, can be derived by microscopic (agent-based) models.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-55008642698540964512019-07-10T23:35:00.002+02:002020-12-25T10:02:16.389+01:00Mean field models for large data-clustering problems<a href="https://1.bp.blogspot.com/-EYjHLsa46Uo/XSZaF4hnmuI/AAAAAAAABxo/9W_JDb0xx7kCdpOlY4Q_Ndpq4ASEuecdACLcBGAs/s1600/segmentation.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="646" data-original-width="973" height="130" src="https://1.bp.blogspot.com/-EYjHLsa46Uo/XSZaF4hnmuI/AAAAAAAABxo/9W_JDb0xx7kCdpOlY4Q_Ndpq4ASEuecdACLcBGAs/s200/segmentation.png" width="200" /></a>Michael Herty, Lorenzo Pareschi and Giuseppe Visconti (8/9/2019, <i>Networks and Heterogeneous Media, </i>15(3): 463-487, (2020), <a href="https://arxiv.org/abs/1907.03585" target="_blank">arXiv:1907.03585</a>)<br /><br /><div style="text-align: justify;">We consider mean-field models for data--clustering problems starting from a generalization of the bounded confidence model for opinion dynamics. The microscopic model includes information on the position as well as on additional features of the particles in order to develop specific clustering effects. <br /><a name='more'></a>The corresponding mean--field limit is derived and properties of the model are investigated analytically. In particular, the mean--field formulation allows the use of a random subsets algorithm for efficient computations of the clusters. Applications to shape detection and image segmentation on standard test images are presented and discussed. </div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-1212936763286955782019-04-20T09:48:00.000+02:002020-01-11T10:29:20.014+01:00Vehicular traffic, crowds, and swarms. From kinetic theory and multiscale methods to applications and research perspectives<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-FmVmSTR5E_E/XhmN6chFUTI/AAAAAAAAB40/NbFS2lJewiQ6TTJwksgTVvvRhvTl0KrAwCEwYBhgL/s1600/Schermata%2B2020-01-11%2Balle%2B09.56.55.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="395" data-original-width="887" height="142" src="https://1.bp.blogspot.com/-FmVmSTR5E_E/XhmN6chFUTI/AAAAAAAAB40/NbFS2lJewiQ6TTJwksgTVvvRhvTl0KrAwCEwYBhgL/s320/Schermata%2B2020-01-11%2Balle%2B09.56.55.png" width="320" /></a></div>G Albi, N Bellomo, L Fermo, S-Y Ha, J Kim, L Pareschi, D Poyato, J Soler (<a href="https://www.worldscientific.com/doi/abs/10.1142/S0218202519500374" target="_blank"><i>Mathematical Models and Methods in Applied Sciences</i> 29, No. 10, pp. 1901-2005, 2019</a>)<br /><br /><div style="text-align: justify;">This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. </div><a name='more'></a>It contains a survey of the kinetic models developed in the last ten years on the aforementioned topics so that overlapping with previous reviews can be avoided. Although the main focus of this paper lies on the mesoscopic models for collective dynamics, we provide a brief overview on the corresponding micro and macroscopic models, and discuss intermediate role of mesoscopic model between them. Moreover, we provide a number of selected challenging research perspectives for readers' attention.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-13696399204775784582019-04-10T11:32:00.004+02:002020-07-07T15:36:55.086+02:00Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-z0eHfdxsjlE/XK23x-7LurI/AAAAAAAABuA/NEvmGcb6MoA1twn3abFuJpbSBtNzcw5oACLcBGAs/s1600/Schermata%2B2019-04-10%2Balle%2B11.30.24.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="509" data-original-width="572" height="177" src="https://4.bp.blogspot.com/-z0eHfdxsjlE/XK23x-7LurI/AAAAAAAABuA/NEvmGcb6MoA1twn3abFuJpbSBtNzcw5oACLcBGAs/s200/Schermata%2B2019-04-10%2Balle%2B11.30.24.png" width="200" /></a></div><div style="text-align: justify;">Giacomo Albi, Giacomo Dimarco, Lorenzo Pareschi</div><div style="text-align: justify;">(8/4/2019 <i>preprint</i> <a href="https://arxiv.org/abs/1904.03865" target="_blank">arXiv:1904.03865</a> to appear in <i>SIAM J.Sci. Comp.</i>)</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. <br /><a name='more'></a>From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-2737764259698170802018-12-24T12:14:00.000+01:002020-04-13T10:13:34.814+02:00Hydrodynamic models of preference formation in multi-agent societies<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-VOwDrTlUd7E/XDSGCJnYJ_I/AAAAAAAABok/AEYKutQaBAoFbg3DLmTrebhFLzq6JRlegCLcBGAs/s1600/Schermata%2B2019-01-08%2Balle%2B12.13.33.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="725" data-original-width="856" height="169" src="https://4.bp.blogspot.com/-VOwDrTlUd7E/XDSGCJnYJ_I/AAAAAAAABok/AEYKutQaBAoFbg3DLmTrebhFLzq6JRlegCLcBGAs/s200/Schermata%2B2019-01-08%2Balle%2B12.13.33.png" width="200" /></a></div>Lorenzo Pareschi, Giuseppe Toscani, Andrea Tosin, Mattia Zanella<br />(24/12/2018, <i>J. Nonlin. Science,29 (2019), no. 6, 2761-2796.</i> <a href="https://arxiv.org/abs/1901.00486" target="_blank">arXiv:1901.00486</a>)<br /><br /><div style="text-align: justify;">In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections. </div><a name='more'></a>Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is essentially based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies.Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-28477773345805441382018-12-14T09:42:00.002+01:002020-04-13T10:14:06.210+02:00Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties<a href="https://2.bp.blogspot.com/-GntPIBssTxk/XBNtg68CPWI/AAAAAAAABno/quk53MYobZwaIKK0IXy2VdAv9UE_vmdVgCLcBGAs/s1600/Schermata%2B2018-12-14%2Balle%2B09.44.32.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="539" data-original-width="692" height="155" src="https://2.bp.blogspot.com/-GntPIBssTxk/XBNtg68CPWI/AAAAAAAABno/quk53MYobZwaIKK0IXy2VdAv9UE_vmdVgCLcBGAs/s200/Schermata%2B2018-12-14%2Balle%2B09.44.32.png" width="200" /></a>Giacomo Dimarco, Lorenzo Pareschi<br />(12/12/2018, <i>preprint</i> <a href="https://arxiv.org/abs/1812.05485" target="_blank">arXiv:1812.05485</a>, <i>Multiscale Modeling & Simulation, </i>18(1), 351–382, 2020)<br /><br /><div style="text-align: justify;">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 to accelerate considerably 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. <br /><a name='more'></a>We show that the additional degrees of freedom can be used to improve further the variance reduction properties of multiscale control variate methods.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-53236609099361788842018-10-28T10:14:00.002+01:002020-04-13T10:14:49.251+02:00Multi-scale control variate methods for uncertainty quantification in kinetic equations<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-9dpoigdhmcY/W9V9vzeokfI/AAAAAAAABlo/yFcvBbwUwuknWg8snri4Zip01seGmtNZgCLcBGAs/s1600/Schermata%2B2018-10-28%2Balle%2B10.12.29.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="679" data-original-width="770" height="176" src="https://3.bp.blogspot.com/-9dpoigdhmcY/W9V9vzeokfI/AAAAAAAABlo/yFcvBbwUwuknWg8snri4Zip01seGmtNZgCLcBGAs/s200/Schermata%2B2018-10-28%2Balle%2B10.12.29.png" width="200" /></a></div>Giacomo Dimarco, Lorenzo Pareschi<br /><div>(25/10/2018. <i>J. Comp. Phys</i> 388 (2019), 63--89. <a href="https://arxiv.org/abs/1810.10844">arXiv:1810.10844</a>)<br /><br /><div style="text-align: justify;">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. <br /><a name='more'></a>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.</div></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-34776238309616792922018-08-03T12:26:00.000+02:002018-08-03T12:43:18.196+02:00Uncertainty Quantification for Hyperbolic and Kinetic Equations<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-iQQM5_JJ-xg/W2QvOxT8jcI/AAAAAAAABjA/obpRRaw3YOwmDDlJRC--x1Z2ub0m1zAQACLcBGAs/s1600/sema-simai.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="751" data-original-width="543" height="200" src="https://4.bp.blogspot.com/-iQQM5_JJ-xg/W2QvOxT8jcI/AAAAAAAABjA/obpRRaw3YOwmDDlJRC--x1Z2ub0m1zAQACLcBGAs/s200/sema-simai.png" width="144" /></a></div><i style="text-align: justify;">Shi Jin, Lorenzo Pareschi</i><span style="text-align: justify;"> (Eds.)</span><br /><div style="text-align: justify;"><span style="text-align: start;"><a href="https://www.springer.com/la/book/9783319671093" target="_blank">SEMA SIMAI Springer Series</a></span><br />277 pages, 2018<br /><br />This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.<br /><a name='more'></a><br /><ul><li><span style="background-color: white;">The first-ever book on kinetic equations</span></li><li><span style="background-color: white;">Presents several different approaches by top authors in the field</span></li><li><span style="background-color: white;">Offers an up-to-date survey of current applications, including examples in the social sciences </span></li></ul><div><b>Links</b></div><div><a href="https://www.springer.com/la/book/9783319671093" target="_blank">Web site of the book at Springer</a></div><div><a href="http://www.math.wisc.edu/~jin/" target="_blank">Shi Jin web page</a></div></div><br />Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.com