tag:blogger.com,1999:blog-19419195305626374022021-09-11T22:20:25.213+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.comBlogger88125tag:blogger.com,1999:blog-1941919530562637402.post-2309808095338073402021-08-04T18:10:00.006+02:002021-08-27T15:16:20.909+02:00Mean-field particle swarm optimization<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-NoylOMJafr4/YQq7ncKiuwI/AAAAAAAADFU/pRJhQXnADt8kMwrvI4QYR6akB0r1WALlwCLcBGAsYHQ/s1832/Schermata%2B2021-08-04%2Balle%2B18.08.31.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="748" data-original-width="1832" height="131" src="https://1.bp.blogspot.com/-NoylOMJafr4/YQq7ncKiuwI/AAAAAAAADFU/pRJhQXnADt8kMwrvI4QYR6akB0r1WALlwCLcBGAsYHQ/w320-h131/Schermata%2B2021-08-04%2Balle%2B18.08.31.png" width="320" /></a></div>Sara Grassi, Hui Huang, Lorenzo Pareschi, Jinniao Qiu (3/8/2021 to appear in <i>Modeling and Simulation for Collective Dynamics</i>, IMS Lecture Note Series, World Scientific, preprint <a href="https://arxiv.org/abs/2108.00393" target="_blank">arXiv:2108.00393</a>)<p></p><p style="text-align: justify;">In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods based on particle swarm optimization (PSO), we introduce a continuous formulation via second-order systems of stochastic differential equations that generalize PSO methods and provide the basis for their theoretical analysis. Subsequently, we will show how through the use of mean-field techniques it is possible to derive in the limit of large particles number the corresponding mean-field PSO description based on Vlasov-Fokker-Planck type equations. <span></span></p><a name='more'></a>Finally, in the zero inertia limit, we will analyze the corresponding macroscopic hydrodynamic equations, showing that they generalize the recently introduced consensus-based optimization (CBO) methods by including memory effects. Rigorous results concerning the mean-field limit, the zero-inertia limit, and the convergence of the mean-field PSO method towards the global minimum are provided along with a suite of numerical examples.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-14214444051947655482021-07-20T12:52:00.001+02:002021-07-21T12:55:48.342+02:00A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-qje_vXwa3dY/YPf8_dwbsPI/AAAAAAAADDo/_nGS9qC-nB0axq9WNAZ-INgr2piLbVyGQCLcBGAsYHQ/s2048/Schermata%2B2021-07-21%2Balle%2B12.54.28.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1587" data-original-width="2048" src="https://1.bp.blogspot.com/-qje_vXwa3dY/YPf8_dwbsPI/AAAAAAAADDo/_nGS9qC-nB0axq9WNAZ-INgr2piLbVyGQCLcBGAsYHQ/s320/Schermata%2B2021-07-21%2Balle%2B12.54.28.png" width="320" /></a></div>Liu Liu, Lorenzo Pareschi, Xueyu Zhu (20/07/2021 preprint <a href="https://arxiv.org/abs/2107.09250" target="_blank">arXiv:2107.09250</a>)<p></p><p style="text-align: justify;">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. <span></span></p><a name='more'></a>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 relative 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].<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-31905035451934552242021-07-16T08:13:00.008+02:002021-07-16T08:14:39.955+02:00On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fTdAN5cYaA4/YPEjZxdx_ZI/AAAAAAAADCQ/Mr1aFEkf058vCznGmcGFpsWRm19I5GY4gCLcBGAsYHQ/s2048/Schermata%2B2021-07-16%2Balle%2B08.12.18.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1752" data-original-width="2048" src="https://1.bp.blogspot.com/-fTdAN5cYaA4/YPEjZxdx_ZI/AAAAAAAADCQ/Mr1aFEkf058vCznGmcGFpsWRm19I5GY4gCLcBGAsYHQ/s320/Schermata%2B2021-07-16%2Balle%2B08.12.18.png" width="320" /></a></div><br />Walter Boscheri, Maurizio Tavelli, Lorenzo Pareschi (14/07/2021 preprint <a href="https://arxiv.org/abs/2107.06956" target="_blank">arXiv:2107.06956</a>)<p></p><p style="text-align: justify;">This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. <span></span></p><a name='more'></a>A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy-Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-34531837848806135922021-06-15T11:49:00.014+02:002021-08-27T15:18:31.593+02:00Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-BJAXXTxfnZ0/YMh3bZfRb4I/AAAAAAAAC6c/1aCxiYXWoUsFVwijXdONOtduZ9BIbmpDQCLcBGAsYHQ/s1466/Schermata%2B2021-06-15%2Balle%2B11.47.57.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1418" data-original-width="1466" src="https://1.bp.blogspot.com/-BJAXXTxfnZ0/YMh3bZfRb4I/AAAAAAAAC6c/1aCxiYXWoUsFVwijXdONOtduZ9BIbmpDQCLcBGAsYHQ/s320/Schermata%2B2021-06-15%2Balle%2B11.47.57.png" width="320" /></a></div>Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (15/6/2021, <a href="https://www.aimspress.com/article/doi/10.3934/mbe.2021350" target="_blank"><i>Math. Biosci. Engin. </i>18(5): 7028-7059, 2021</a>)<p></p><p style="text-align: justify;">In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamic of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commuters at the urban scale. <span></span></p><a name='more'></a>Thanks to a suitable scaling limit, the kinetic transport model used to describe the dynamics of commuters, within a given urban area coincides with the diffusion equations that characterize the movement of non-commuting individuals. <span style="text-align: left;">Because of the high uncertainty in the data reported in the early phase of the epidemic, the presence of random inputs in both the initial data and the epidemic parameters is included in the model. A robust numerical method is designed to deal with the presence of multiple scales and the uncertainty quantification process. In our simulations, we considered a realistic geographical domain, describing the Lombardy region, in which the size of the cities, the number of infected individuals, the average number of daily commuters moving from one city to another, and the epidemic aspects are taken into account through a calibration of the model parameters based on the actual available data. The results show that the model is able to describe correctly the main features of the spatial expansion of the first wave of COVID-19 in northern Italy.</span><p></p><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-43221385687612839032021-06-10T12:45:00.003+02:002021-08-27T15:18:59.073+02:00Modelling 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" 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 (10/6/2021, <a href="https://www.aimspress.com/article/doi/10.3934/mbe.2021355" target="_blank"><i>Math. Biosci. Engineer.</i> 18(6):7161-7190, 2021</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-35396917772432470772021-05-31T09:06:00.007+02:002021-08-04T18:20:27.555+02:00Hyperbolic compartmental models for epidemic spread on networks with uncertain data: application to the emergence of Covid-19 in Italy<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-qUkghIC9hoI/YLXdEMdqvcI/AAAAAAAAC5A/9RqCW5T9Iv8ZfW7oX_12c4n5_fwMHNX8ACLcBGAsYHQ/s1290/Schermata%2B2021-06-01%2Balle%2B09.08.41.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1078" data-original-width="1290" src="https://1.bp.blogspot.com/-qUkghIC9hoI/YLXdEMdqvcI/AAAAAAAAC5A/9RqCW5T9Iv8ZfW7oX_12c4n5_fwMHNX8ACLcBGAsYHQ/s320/Schermata%2B2021-06-01%2Balle%2B09.08.41.png" width="320" /></a></div>Giulia Bertaglia, Lorenzo Pareschi (31/5/2021, to appear in <i>Math. Mod. Meth. Appl. Scie.</i>, preprint <a href="https://arxiv.org/abs/2105.14258" target="_blank">arXiv:2105.14258</a>)<p></p><p style="text-align: justify;">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. <span></span></p><a name='more'></a>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 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.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-54726935112076844942021-05-28T12:32:00.000+02:002021-05-28T12:32:02.677+02:00Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tjCeOMPSVOQ/YLDGO18iQyI/AAAAAAAAC4c/SaykTSVMNpsklQaBElxQHzO1MwYcaIvIQCLcBGAsYHQ/s2048/Schermata%2B2021-05-28%2Balle%2B12.29.52.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1618" data-original-width="2048" src="https://1.bp.blogspot.com/-tjCeOMPSVOQ/YLDGO18iQyI/AAAAAAAAC4c/SaykTSVMNpsklQaBElxQHzO1MwYcaIvIQCLcBGAsYHQ/s320/Schermata%2B2021-05-28%2Balle%2B12.29.52.png" width="320" /></a></div> Lorenzo Pareschi, Thomas Rey (28/5/2021 preprint arXiv:2105.13158)<p></p><p><br /></p><div style="text-align: justify;">Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms.</div><span><a name='more'></a></span><div style="text-align: justify;">The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-62946147269921172072021-05-07T09:53:00.004+02:002021-05-07T09:55:26.502+02:00Binary interaction methods for high dimensional global optimization and machine learning<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-b0QI0JmBPG4/YJTxxp5-j0I/AAAAAAAAC1A/szsXsCKc82oVAqTEXwDcraGq2kwLu9T7ACLcBGAsYHQ/s1898/Schermata%2B2021-05-07%2Balle%2B09.52.18.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1546" data-original-width="1898" src="https://1.bp.blogspot.com/-b0QI0JmBPG4/YJTxxp5-j0I/AAAAAAAAC1A/szsXsCKc82oVAqTEXwDcraGq2kwLu9T7ACLcBGAsYHQ/s320/Schermata%2B2021-05-07%2Balle%2B09.52.18.png" width="320" /></a></div><br />Alessandro Benfenati, Giacomo Borghi, Lorenzo Pareschi (7/5/2021 preprint <a href="https://www.blogger.com/#">arXiv:2105.02695</a>)<div><br /></div><div><div style="text-align: justify;">In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. For the latter methods, convergence to the global minimizer can be shown for a large class of functions.</div><span><a name='more'></a></span><div style="text-align: justify;">Algorithmic implementations inspired by the well-known direct simulation Monte Carlo methods in kinetic theory are derived and discussed. Several examples on prototype test functions for global optimization are reported including applications to machine learning.</div><p></p></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag: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-37085387062635312142021-02-04T12:45:00.004+01:002021-06-01T08:47:15.164+02:00Mean-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 <i>International Journal for Uncertainty Quantification</i> to appear, <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-25827334242711874142020-12-18T17:48:00.010+01:002021-08-27T15:27:25.036+02:00Modeling 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, <i>Math. Mod. Meth. App. Scie. </i>31(6):1059-1097, 2021, <a href="https://arxiv.org/pdf/2012.10101">arXiv:2012.10101</a>)<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-38321063622386078062020-12-10T09:02:00.005+01:002021-09-11T18:15:53.111+02:00From 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. Vol. 31, No. 8 (2021) 1625–1657<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-67116894841117753952020-11-11T15:46:00.006+01:002021-08-27T15:34:00.482+02:00On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation<p> Lorenzo Pareschi, Thomas Rey (11/11/2020, <i>Applied Math. Letters, </i>120:107187, 2021, <a href="https://arxiv.org/pdf/2011.05811.pdf">arXiv:2011.05811</a>)</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-55405721461069470412020-08-07T09:53:00.006+02:002021-08-27T15:32:00.775+02:00High 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, <i>J. Comp. Phys. 434: 110206, 2021, </i><a href="https://arxiv.org/pdf/2008.01789.pdf">arXiv:2008.01789</a>)<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-37734014016477347012020-07-08T14:07:00.004+02:002021-08-27T15:36:35.203+02:00Hyperbolic 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>55:381-407, 2021, <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-64199546869813865192020-07-03T11:18:00.003+02:002021-08-27T16:27:24.070+02:00Uncertainty 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, <i>J. Comp. Phys. </i>430: 110102<i>, </i>2021, <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-26544981720252078932020-04-29T08:48:00.002+02:002021-08-27T16:28:33.188+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" 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, 2020, <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-77394169570042240222020-04-29T08:32:00.003+02:002021-08-27T16:30:08.080+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" 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, <i>J. Math. Biol. </i>82(7): 63, 2021, <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-51297953479162589552020-04-17T19:52:00.006+02:002021-08-27T16:32:29.372+02:00Uncertainty 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, <i>SIAM/ASA J. Uncertainty Quantification </i>9(2):650-680, 2021, <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-31578399516854611942020-04-13T08:34:00.006+02:002021-08-27T16:36:27.706+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" 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, <i>Trails in kinetic theory: Foundational aspects and numerical methods</i>, SEMA-SIMAI Springer series 25:141-181, 2021, <a href="https://arxiv.org/pdf/2004.05072.pdf" target="_blank">arXiv:2004.05072</a>)<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><div style="text-align: justify;">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.</div>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-80345807118079499522020-02-03T17:40:00.006+01:002021-07-22T10:34:03.974+02:00Consensus-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" 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> to appear in <i>Journal of Machine Learning Research</i>)<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-74416302184373684992020-01-31T17:35:00.002+01:002021-08-27T16:38:38.039+02:00Consensus-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" 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> 30(14):2725-2751, 2020, <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-73826366576412042072020-01-12T09:18:00.003+01:002021-08-04T19:08:50.788+02:00High 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> (2021) <a href="https://doi.org/10.1007/s42967-020-00110-5">https://doi.org/10.1007/s42967-020-00110-5</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-51069768779411163532019-12-25T09:25:00.001+01:002021-08-27T16:44:12.855+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" 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> <a href="https://link.springer.com/chapter/10.1007/978-3-030-50450-2_8" target="_blank"><i>Crowd Dynamics</i>, Volume 2</a>. Modeling and Simulation in Science, Engineering and Technology, 159-197, 2020, <a href="https://arxiv.org/abs/1912.11628" target="_blank">arXiv:1912.11628</a>)<div><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></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.com