tag:blogger.com,1999:blog-19419195305626374022022-12-03T13:34:19.037+01:00Lorenzo PareschiProfessor of Numerical Analysis - University of FerraraLorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comBlogger104125tag:blogger.com,1999:blog-1941919530562637402.post-80894085566298562902022-11-30T09:16:00.001+01:002022-12-02T09:21:28.550+01:00The kinetic theory of mutation rates<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi26yj-Dq7kXmzj7-7t49_qGMg1QXkTwRF_ECDGTBDJl6SRe1WNQ_b_EM6jEb1TdkP3KFMDpGjIZMOvwqjqJGD-BtLxx7B2lkHZctj8x16s4cFzsIczmRLKPS3ppTJxNGMnOoTx91-1cn2eBHjTAbikfJAvzEdVMINxhk1A5hp7CnCSCY2AV9W5aWHA/s2282/Screenshot%202022-12-02%20alle%2009.19.21.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2126" data-original-width="2282" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi26yj-Dq7kXmzj7-7t49_qGMg1QXkTwRF_ECDGTBDJl6SRe1WNQ_b_EM6jEb1TdkP3KFMDpGjIZMOvwqjqJGD-BtLxx7B2lkHZctj8x16s4cFzsIczmRLKPS3ppTJxNGMnOoTx91-1cn2eBHjTAbikfJAvzEdVMINxhk1A5hp7CnCSCY2AV9W5aWHA/s320/Screenshot%202022-12-02%20alle%2009.19.21.png" width="320" /></a></div>Lorenzo Pareschi, Giuseppe Toscani (preprint <a href="https://arxiv.org/abs/2212.00146">arXiv:2212.00146</a>, 2022)<p></p><p style="text-align: justify;">The Luria-Delbrück mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools, in particular by modeling the phenomenon resorting to methodologies borrowed from classical kinetic theory of rarefied gases. The aim is to construct a linear kinetic model that can reproduce the Luria-Delbrück distribution starting from the elementary interactions that qualitatively and quantitatively describe the variation of mutated cells. <span></span></p><a name='more'></a>The kinetic description is easily adaptable to different situations and makes it possible to clearly identify the differences between the elementary variations leading to the formulations of Luria--Delbrück, Lea--Coulson, and Kendall, respectively. The kinetic approach additionally emphasizes basic principles which not only help to unify existing results but also allow for useful extensions.<p></p><p><br /></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-73350897809021032222022-10-31T08:54:00.018+01:002022-11-10T17:32:34.360+01:00Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equations<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDFZoVMkGROYYE5v5MX0u8NV3QHT8YZeKmQX5qDLTmZnca44_OCZ9UvPd0n_6FlGLtxvFOXEkdje_jcKJ8ltTnJyuTN9HyJfdnGWIelGKfQnLnpSTtK_wLZOmR4zzoa6felhgnqhB0hyv5b_bIwNR8_dznRPsj7QOBdE896ht2dFf_q6M5KFVIxKZP/s1166/Schermata%202022-11-02%20alle%2008.58.25.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="986" data-original-width="1166" height="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDFZoVMkGROYYE5v5MX0u8NV3QHT8YZeKmQX5qDLTmZnca44_OCZ9UvPd0n_6FlGLtxvFOXEkdje_jcKJ8ltTnJyuTN9HyJfdnGWIelGKfQnLnpSTtK_wLZOmR4zzoa6felhgnqhB0hyv5b_bIwNR8_dznRPsj7QOBdE896ht2dFf_q6M5KFVIxKZP/s320/Schermata%202022-11-02%20alle%2008.58.25.png" width="320" /></a></div>Manuel R. Izquierdo, Lorenzo Pareschi, Borja Miñano, Joan Massó, Carlos Palenzuela (preprint <a href="https://arxiv.org/abs/2211.00027" target="_blank">arXiv:2211.00027</a>)<p></p><p style="text-align: justify;"></p><div style="text-align: justify;">Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of 3+1 equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem.</div><span><a name='more'></a></span><div style="text-align: justify;">Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta (IMEX) methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit RK commonly employed for the MHD and Einstein equations. Finally, we analyze their performance in several numerical tests.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-50771693176632100792022-08-02T19:38:00.009+02:002022-10-10T16:14:35.316+02:00An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEichjAbxxuJiGf5uCb72WpwjlO6nHwdlkJypZI0Wogu-z8XSqGfZvfew_TfnBeLkqs7q6SWE87Yw9u1y0Z1LvKh5LfUEHX17ZBYoNYx2Zlb_-S-YQ2ejMb_gv5VrV3nkMLzBrkddlW_fAV3_tlKg-OZy5IVMu9Y3OBYS1ngG2GpQCJR00dLpKIwOO2T/s970/Schermata%202022-08-03%20alle%2019.44.03.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="942" data-original-width="970" height="311" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEichjAbxxuJiGf5uCb72WpwjlO6nHwdlkJypZI0Wogu-z8XSqGfZvfew_TfnBeLkqs7q6SWE87Yw9u1y0Z1LvKh5LfUEHX17ZBYoNYx2Zlb_-S-YQ2ejMb_gv5VrV3nkMLzBrkddlW_fAV3_tlKg-OZy5IVMu9Y3OBYS1ngG2GpQCJR00dLpKIwOO2T/s320/Schermata%202022-08-03%20alle%2019.44.03.png" width="320" /></a></div>Giacomo Borghi, Michael Herty, Lorenzo Pareschi (preprint <a href="https://arxiv.org/abs/2208.01362" target="_blank">arXiv:2208.01362</a>)<p></p><p style="text-align: justify;">In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space by using energy-based measures to quantify the diversity of the system. The resulting metaheuristic algorithm is mathematically analyzed using a mean-field approximation and convergence guarantees towards optimal points is rigorously proven. <span></span></p><a name='more'></a>In addition, a gradient flow structure in the parameter space for the adaptive method is revealed and analyzed. Several numerical experiments shows the validity of the proposed stochastic particle dynamics and illustrate the theoretical findings.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-60655769120590918002022-08-01T08:04:00.001+02:002022-10-10T16:14:55.175+02:00Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAkotEWktSdxTPNKcSqWVbytP-7sRefAOzMD6JCtyDR7fXcvdqlUPBv-oJU0QYDPtipEo0k25KaNDWperQgyBQN7z4hc71c2fByIp6DdtlQ2J9nRKsZ045-rgvTrS5cF6SvDHczWfs8ZcHwitskenagiRJ4hAWrnVjU5B2STJ47ddR3U_Mn8ihB40a/s1310/Schermata%202022-08-02%20alle%2008.03.08.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1040" data-original-width="1310" height="254" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAkotEWktSdxTPNKcSqWVbytP-7sRefAOzMD6JCtyDR7fXcvdqlUPBv-oJU0QYDPtipEo0k25KaNDWperQgyBQN7z4hc71c2fByIp6DdtlQ2J9nRKsZ045-rgvTrS5cF6SvDHczWfs8ZcHwitskenagiRJ4hAWrnVjU5B2STJ47ddR3U_Mn8ihB40a/s320/Schermata%202022-08-02%20alle%2008.03.08.png" width="320" /></a></div>Andrea Medaglia, Lorenzo Pareschi, Mattia Zanella (preprint <a href="https://arxiv.org/abs/2208.00692" target="_blank">arXiv:2208.00692</a>)<p></p><p style="text-align: justify;">The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle methods for collisional kinetic models of plasmas under the effect of uncertainties. This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles' position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG projection and to obtain the corresponding system for the gPC coefficients. <span></span></p><a name='more'></a>We show that the sG particle method preserves the main physical properties of the problem, such as conservations and positivity of the solution, while achieving spectral accuracy for smooth solutions in the random space. Furthermore, in the fluid limit the sG particle solver is designed to possess the asymptotic-preserving property necessary to obtain a sG particle scheme for the limiting Euler-Poisson system, thus avoiding the loss of hyperbolicity typical of conventional sG methods based on finite differences or finite volumes. We tested the schemes considering the classical Landau damping problem in the presence of both small and large initial uncertain perturbations, the two stream instability and the Sod shock tube problems under uncertainties. The results show that the proposed method is able to capture the correct behavior of the system in all test cases, even when the relaxation time scale is very small.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-37538186300607964072022-07-15T08:13:00.005+02:002022-07-15T08:13:43.357+02:00Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg01gWtEzfPwUSpr1JGoGtL7blaODy0eKwYj9vtfA3dVS6YDsBslHY94dDbqtaF3tHFX8mzJjiSdPdfQUpQpCZfE7yqIkRu83bXzVCogMTYakEEM0u0gtGqgzhj9h1U92IZ9Sob33ZidiQxGbJAOfGjM8o_AUvDVIHpIEtq4IYisejMJxPBunjQMa9d/s1648/Schermata%202022-07-15%20alle%2008.11.53.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1352" data-original-width="1648" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg01gWtEzfPwUSpr1JGoGtL7blaODy0eKwYj9vtfA3dVS6YDsBslHY94dDbqtaF3tHFX8mzJjiSdPdfQUpQpCZfE7yqIkRu83bXzVCogMTYakEEM0u0gtGqgzhj9h1U92IZ9Sob33ZidiQxGbJAOfGjM8o_AUvDVIHpIEtq4IYisejMJxPBunjQMa9d/s320/Schermata%202022-07-15%20alle%2008.11.53.png" width="320" /></a></div>Giacomo Dimarco, Lorenzo Pareschi, Mattia Zanella (preprint <a href="https://arxiv.org/abs/2207.06494" target="_blank">arXiv:2207.06494</a>)<p></p><p style="text-align: justify;">Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation has often to face with physical forces having a significant random component or with particles living in a random environment which characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. <span></span></p><a name='more'></a>In this work, to address the problem of effectively solving stochastic Fokker-Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker-Planck model, leads to highly accurate description of the uncertainty dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-33673987755432413592022-06-28T09:14:00.006+02:002022-07-15T08:23:17.775+02:00Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-9Fc90p5fNPLj57Ti3a13EC5n_5rHLlHoaZ_ffUn4FMBnFxJOD7H8GJlj19NiMgEcYoUVkNIQYKo6haucMEYO6_-pfGjlFWxnrJWB2R9fp-_80AV-QpNRoKPIyZeD72nGrGVockzSeQMaEZLEvfVEal0GE7InoRTWYOadvc8Kqdh78G_CYo-3snlu/s2148/Schermata%202022-06-28%20alle%2009.13.01.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1604" data-original-width="2148" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-9Fc90p5fNPLj57Ti3a13EC5n_5rHLlHoaZ_ffUn4FMBnFxJOD7H8GJlj19NiMgEcYoUVkNIQYKo6haucMEYO6_-pfGjlFWxnrJWB2R9fp-_80AV-QpNRoKPIyZeD72nGrGVockzSeQMaEZLEvfVEal0GE7InoRTWYOadvc8Kqdh78G_CYo-3snlu/s320/Schermata%202022-06-28%20alle%2009.13.01.png" width="320" /></a></div>Giulia Bertaglia, Chuan Lu, Lorenzo Pareschi, Xueyu Zhu (to appear in <i>Mathematical Models and Methods in Applied Sciences</i>, preprint <a href="https://arxiv.org/abs/2206.12625" target="_blank">arXiv:2206.12625</a>)<p></p><p></p><div style="text-align: justify;">When investigating epidemic dynamics through differential models, the parameters needed to understand the phenomenon and to simulate forecast scenarios require a delicate calibration phase, often made even more challenging by the scarcity and uncertainty of the observed data reported by official sources. In this context, Physics-Informed Neural Networks (PINNs), by embedding the knowledge of the differential model that governs the physical phenomenon in the learning process, can effectively address the inverse and forward problem of data-driven learning and solving the corresponding epidemic problem. <span><a name='more'></a></span>In many circumstances, however, the spatial propagation of an infectious disease is characterized by movements of individuals at different scales governed by multiscale PDEs. This reflects the heterogeneity of a region or territory in relation to the dynamics within cities and in neighboring zones. In presence of multiple scales, a direct application of PINNs generally leads to poor results due to the multiscale nature of the differential model in the loss function of the neural network. To allow the neural network to operate uniformly with respect to the small scales, it is desirable that the neural network satisfies an Asymptotic-Preservation (AP) property in the learning process. To this end, we consider a new class of AP Neural Networks (APNNs) for multiscale hyperbolic transport models of epidemic spread that, thanks to an appropriate AP formulation of the loss function, is capable to work uniformly at the different scales of the system. A series of numerical tests for different epidemic scenarios confirms the validity of the proposed approach, highlighting the importance of the AP property in the neural network when dealing with multiscale problems especially in presence of sparse and partially observed systems.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-43927711134115500972022-06-20T09:24:00.006+02:002022-06-20T09:25:44.772+02:00Locally Structure-Preserving div-curl operators for high order Discontinuous Galerkin schemes<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLAL_pfv7S93kuQdPSsG0A5U54sb4UvHdRVPlEMzy8PABROxd3wH5n9QUWawgXFPpnZnJdYdyWgoKkiBq_KPE8q3lYewAl6rOVyIbb7-dBf1Ghk54Ib38T8pr60jMOFolaE_24r_YqdPmkBJUu5JygXIG6ga305OyHLXplw_V0Y-hOdBdzQR96kiWE/s1342/Schermata%202022-06-20%20alle%2009.22.57.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1158" data-original-width="1342" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLAL_pfv7S93kuQdPSsG0A5U54sb4UvHdRVPlEMzy8PABROxd3wH5n9QUWawgXFPpnZnJdYdyWgoKkiBq_KPE8q3lYewAl6rOVyIbb7-dBf1Ghk54Ib38T8pr60jMOFolaE_24r_YqdPmkBJUu5JygXIG6ga305OyHLXplw_V0Y-hOdBdzQR96kiWE/s320/Schermata%202022-06-20%20alle%2009.22.57.png" width="320" /></a></div>Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (preprint <a href="https://arxiv.org/abs/2206.08609" target="_blank">arXiv:2206.08609</a>) <p></p><p style="text-align: justify;">We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. <span></span></p><a name='more'></a>Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-29695294406339701212022-03-30T15:04:00.002+02:002022-07-16T14:55:13.353+02:00A consensus-based algorithm for multi-objective optimization and its mean-field description<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSYzx08Rc7f2P4Pu9NXsNSiUPxyalH0zaL50ZmGojdQEXha9mX7gwHUi7rvyXOI3VaHufL9aWv6kLP7GYj7ApqtnJAlEcAc4qdB6SaVpciXGW-fmNa-jfbT2Gu9KdwarOlz4DyyddoLDYrUeMJj1NWCrYdGybR3zSFYhNSW4AIL0etxibjoWDzAkiE/s1204/Schermata%202022-03-31%20alle%2015.06.08.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="712" data-original-width="1204" height="189" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSYzx08Rc7f2P4Pu9NXsNSiUPxyalH0zaL50ZmGojdQEXha9mX7gwHUi7rvyXOI3VaHufL9aWv6kLP7GYj7ApqtnJAlEcAc4qdB6SaVpciXGW-fmNa-jfbT2Gu9KdwarOlz4DyyddoLDYrUeMJj1NWCrYdGybR3zSFYhNSW4AIL0etxibjoWDzAkiE/s320/Schermata%202022-03-31%20alle%2015.06.08.png" width="320" /></a></div>Giacomo Borghi, Michael Herty, Lorenzo Pareschi (to appear in Proceedings of the 61st IEEE Conference on Decision and Control. Preprint <a href="https://arxiv.org/abs/2203.16384" target="_blank">arXiv:2203.16384</a>)<p></p><p style="text-align: justify;">We present a multi-agent algorithm for multi-objective optimization problems, which extends the class of consensus-based optimization methods and relies on a scalarization strategy. The optimization is achieved by a set of interacting agents exploring the search space and attempting to solve all scalar sub-problems simultaneously. We show that those dynamics are described by a mean-field model, which is suitable for a theoretical analysis of the algorithm convergence. Numerical results show the validity of the proposed method.</p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-63255793671838558862022-02-23T22:07:00.008+01:002022-10-03T15:25:55.236+02:00From agent-based models to the macroscopic description of fake-news spread: the role of competence in data-driven applications<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiISXANFS6loTEcZNASyctOhls_s9eQSK7NSzLGlyIs_tI1KsmhIRKqoGn8L-wNIPictFDPLvm9PuDIIqeg4SxPB5Q8myuxHALlpAxWXHtMQXoaBdNF-qGX_Ho-toN5sHq-Ge_0WWWKrE1885SL87bGr0GwurZyZgyWonxz9oXQIuPmGeMZg92FNIK6=s1294" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1054" data-original-width="1294" height="261" src="https://blogger.googleusercontent.com/img/a/AVvXsEiISXANFS6loTEcZNASyctOhls_s9eQSK7NSzLGlyIs_tI1KsmhIRKqoGn8L-wNIPictFDPLvm9PuDIIqeg4SxPB5Q8myuxHALlpAxWXHtMQXoaBdNF-qGX_Ho-toN5sHq-Ge_0WWWKrE1885SL87bGr0GwurZyZgyWonxz9oXQIuPmGeMZg92FNIK6=s320" width="320" /></a></div>Jonathan Franceschi, Lorenzo Pareschi, Mattia Zanella (<i>Partial Differential Equations and Applications, </i>3(68):1-26, 2022. Preprint <a href="https://arxiv.org/abs/2202.10809" target="_blank">arXiv:2202.10809</a>)<p></p><span style="text-align: justify;">Fake news spreading, with the aim of manipulating individuals' perceptions of facts, is now recognized as a major problem in many democratic societies. Yet, to date, little has been understood about how fake news spreads on social networks, what the influence of the education level of individuals is, when fake news is effective in influencing public opinion, and what interventions might be successful in mitigating their effect. In this paper, starting from the recently introduced kinetic multi-agent model with competence by the first two authors, we propose to derive reduced-order models through the notion of social closure in the mean-field approximation that has its roots in the classical hydrodynamic closure of kinetic theory.</span><div style="text-align: justify;"><span><a name='more'></a></span>This approach allows to obtain simplified models in which the competence and learning of the agents maintain their role in the dynamics and, at the same time, the structure of such models is more suitable to be interfaced with data-driven applications. Examples of different Twitter-based test cases are described and discussed.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-19929718544740890622022-01-10T18:20:00.006+01:002022-01-30T18:39:37.027+01:00Effects of vaccination efficacy on wealth distribution in kinetic epidemic models<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjIDAJBaUYI2fj0eCiTGYGrhC5_3qUXWzYrtQ54j9LeLbgT3J5NFWBC-XNQzIKyZMDn-FeQtLFgZ_Vne5npNFSUAzr_3yYbKm_pVcElxT4kVyUAlVJrhbuuBbQ-f0yISeetGMeHYPjDkQDx6r8sGIEhKhbT-o59g24IkAN2cmmB7roetKOHrBdQhUCP=s2020" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1518" data-original-width="2020" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEjIDAJBaUYI2fj0eCiTGYGrhC5_3qUXWzYrtQ54j9LeLbgT3J5NFWBC-XNQzIKyZMDn-FeQtLFgZ_Vne5npNFSUAzr_3yYbKm_pVcElxT4kVyUAlVJrhbuuBbQ-f0yISeetGMeHYPjDkQDx6r8sGIEhKhbT-o59g24IkAN2cmmB7roetKOHrBdQhUCP=s320" width="320" /></a></div>Emanuele Bernardi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (<a href="https://www.mdpi.com/1099-4300/24/2/216" target="_blank"><i>Entropy </i>24(2):216, 2022</a>)<p></p><p style="text-align: justify;">The spreading of Covid-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modelling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. <span></span></p><a name='more'></a>The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming on the one hand that individuals in different compartments act differently in the economic process and on the other hand that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society.<p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-67074881747200271712022-01-08T20:55:00.006+01:002022-07-15T08:20:34.879+02:00Dinamiche sociali ed equazioni alle derivate parziali in ambito epidemiologico<p><span style="text-align: justify;"></span></p><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHczepWhuA-zGLykrrauF8uoiHOh2mrvPrEizMpsxsSC4ocHqR4Irj8PtyisqnyvUckbtH1-_5umcP0VNozq_VaMMFBj7ck5Kr7vr5ltNJnofjn58laaRHrhF9xKpEAE7HzuG8Jvd6BaxKirDkPIZR7wghu9kXn1JJ5n9WkVl2gUbXZMEFegqbf6lB/s782/Schermata%202022-06-24%20alle%2021.05.04.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="620" data-original-width="782" height="254" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHczepWhuA-zGLykrrauF8uoiHOh2mrvPrEizMpsxsSC4ocHqR4Irj8PtyisqnyvUckbtH1-_5umcP0VNozq_VaMMFBj7ck5Kr7vr5ltNJnofjn58laaRHrhF9xKpEAE7HzuG8Jvd6BaxKirDkPIZR7wghu9kXn1JJ5n9WkVl2gUbXZMEFegqbf6lB/s320/Schermata%202022-06-24%20alle%2021.05.04.png" width="320" /></a></div>Lorenzo Pareschi, Giuseppe Toscani (<a href="https://drive.google.com/file/d/1IG2qtZoMM2TGbuftt5uzfmXb7s79zdZ6/view?usp=sharing" target="_blank">Matematica, Cultura e Società - Rivista dell'Unione Matematica Italiana</a>, Serie I, Volume 6, No.3, 2021)<p></p><p style="text-align: justify;"><span style="text-align: justify;">In questo breve sunto divulgativo discuteremo l'importanza delle dinamiche sociali in ambito epidemico e la loro modellizzazione matematica tramite equazioni alle derivate parziali. Presenteremo inizialmente modelli di interazione tra individui in cui le caratteristiche sociali, come l'età degli individui, il numero di contatti sociali e la loro ricchezza economica, giocano un ruolo chiave nella diffusione di un'epidemia. Successivamente, accenneremo a modelli che tengono conto anche di caratteristiche aggiuntive quali la carica virale e le difese immunitarie dell'individuo.<span></span></span></p><a name='more'></a><span style="text-align: justify;"> Infine, analizzeremo alcuni modelli alle derivate parziali per la descrizione degli spostamenti degli individui, sia su scala urbana che extra urbana, ed evidenzieremo come le dinamiche di movimento giochino un ruolo chiave sull'avanzamento dell'epidemia.</span><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-77786427122516186522021-12-03T09:00:00.005+01:002022-01-26T08:53:36.059+01:00Multi-fidelity methods for uncertainty propagation in kinetic equations<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-XmJ3X1aAUWs/YanOWXfFfrI/AAAAAAAADR4/m3OgiqEQwOsISpOrR64808rMMaSDVQNAwCNcBGAsYHQ/s2048/Schermata%2B2021-12-03%2Balle%2B08.58.56.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1704" data-original-width="2048" height="266" src="https://1.bp.blogspot.com/-XmJ3X1aAUWs/YanOWXfFfrI/AAAAAAAADR4/m3OgiqEQwOsISpOrR64808rMMaSDVQNAwCNcBGAsYHQ/s320/Schermata%2B2021-12-03%2Balle%2B08.58.56.png" width="320" /></a></div>Giacomo Dimarco, Liu Liu, Lorenzo Pareschi, Xueyu Zhu (to appear in <i>Panoramas & Synthèses</i>, Société Mathématique de France, preprint <a href="https://arxiv.org/abs/2112.00932" target="_blank">arXiv:2112.00932</a>)<p></p><p></p><div style="text-align: justify;">The construction of efficient methods for uncertainty quantification in kinetic equations represents a challenge due to the high dimensionality of the models: often the computational costs involved become prohibitive. On the other hand, precisely because of the curse of dimensionality, the construction of simplified models capable of providing approximate solutions at a computationally reduced cost has always represented one of the main research strands in the field of kinetic equations.<span><a name='more'></a></span> Approximations based on suitable closures of the moment equations or on simplified collisional models have been studied by many authors. In the context of uncertainty quantification, it is therefore natural to take advantage of such models in a multi-fidelity setting where the original kinetic equation represents the high-fidelity model, and the simplified models define the low-fidelity surrogate models. The scope of this article is to survey some recent results about multi-fidelity methods for kinetic equations that are able to accelerate the solution of the uncertainty quantification process by combining high-fidelity and low-fidelity model evaluations with particular attention to the case of compressible and incompressible hydrodynamic limits. We will focus essentially on two classes of strategies: multi-fidelity control variates methods and bi-fidelity stochastic collocation methods. The various approaches considered are analyzed in light of the different surrogate models used and the different numerical techniques adopted. Given the relevance of the specific choice of the surrogate model, an application-oriented approach has been chosen in the presentation.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-36414041006896592732021-11-23T09:08:00.012+01:002022-09-23T10:28:08.367+02:00Constrained consensus-based optimization<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vQLTVQIgtRw/YZyhXGY1_wI/AAAAAAAADQE/LLBJ8d3RZ90b07Hn1wTa1C4T9boeXuNjACLcBGAsYHQ/s1686/Schermata%2B2021-11-23%2Balle%2B08.40.54.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1522" data-original-width="1686" height="289" src="https://1.bp.blogspot.com/-vQLTVQIgtRw/YZyhXGY1_wI/AAAAAAAADQE/LLBJ8d3RZ90b07Hn1wTa1C4T9boeXuNjACLcBGAsYHQ/s320/Schermata%2B2021-11-23%2Balle%2B08.40.54.png" width="320" /></a></div>Giacomo Borghi, Michael Herty, Lorenzo Pareschi (<i>SIAM J. Optimization </i>to appear. Preprint <a href="https://arxiv.org/abs/2111.10571" target="_blank">arXiv:2111.10571</a>)<p></p><p></p><div style="text-align: justify;">In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings the class of CBO methods. <span><a name='more'></a></span>Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. <span></span>Using a mean-field description of the the many particle limit of the arising CBO dynamics, we are able to show convergence of the proposed method to the minimum for general nonlinear constrained problems. Properties of the new algorithm are analyzed. Several numerical examples, also in high dimensions, illustrate the theoretical findings and the good performance of the new numerical method.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-76242899012540806952021-10-28T14:15:00.010+02:002022-01-11T18:28:13.662+01:00Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties<div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-1UwxSO6EUwU/YX1iDkJVYOI/AAAAAAAADOQ/O-iJkmOu7tAq1SZszIhC66mtpxAKmtBhwCLcBGAsYHQ/s1616/Schermata%2B2021-10-30%2Balle%2B17.17.09.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1226" data-original-width="1616" height="243" src="https://1.bp.blogspot.com/-1UwxSO6EUwU/YX1iDkJVYOI/AAAAAAAADOQ/O-iJkmOu7tAq1SZszIhC66mtpxAKmtBhwCLcBGAsYHQ/s320/Schermata%2B2021-10-30%2Balle%2B17.17.09.png" width="320" /></a></div>Giulia Bertaglia, Liu Liu, Lorenzo Pareschi, Xueyu Zhu (to appear in <i>Network and Heterogeneous Media</i>, preprint <a href="https://arxiv.org/abs/2110.14579#" target="_blank">arXiv:2110.14579</a>)</div><div><br /></div><div style="text-align: justify;">Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models.<span><a name='more'></a></span> The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model. <span></span>In particular, we will consider the class of multiscale transport models recently introduced in Bertaglia, Boscheri, Dimarco & Pareschi, Math. Biosci. Eng. (2021) and Boscheri, Dimarco & Pareschi, Math. Mod. Meth. App. Scie. (2021) as the high-fidelity reference and use simple two-velocity discrete models for low-fidelity evaluations. Both models share the same diffusive behavior and are solved with ad-hoc asymptotic-preserving numerical discretizations. A series of numerical experiments confirm the validity of the approach.</div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-6809732211838863102021-10-04T18:30:00.013+02:002022-09-26T15:43:08.284+02:00Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-waCwZKBAK-4/YVsr_q0SMwI/AAAAAAAADLA/HppFhREQStMq6xyym7GZ3cRiS9ITOSLaQCLcBGAsYHQ/s1468/Schermata%2B2021-10-04%2Balle%2B18.29.26.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1264" data-original-width="1468" height="276" src="https://1.bp.blogspot.com/-waCwZKBAK-4/YVsr_q0SMwI/AAAAAAAADLA/HppFhREQStMq6xyym7GZ3cRiS9ITOSLaQCLcBGAsYHQ/s320/Schermata%2B2021-10-04%2Balle%2B18.29.26.png" width="320" /></a></div>Giacomo Albi, Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella (<i>Predicting Pandemics in a Globally Connected World,</i> Vol. 1, N. Bellomo and M. Chaplain Editors, Springer-Nature, 43-108, 2022. Preprint <a href="https://arxiv.org/abs/2110.00293" target="_blank">arXiv:2110.00293</a>)<p></p><p></p><div style="text-align: justify;">In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. <span><a name='more'></a></span>Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. <span></span>Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-61854533882101676742021-09-28T08:28:00.006+02:002021-12-08T08:32:27.954+01:00Spreading of fake news, competence, and learning: kinetic modeling and numerical approximation<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-sVzJqlzMS8Q/YVarGAD-D2I/AAAAAAAADKY/z1UNxRt9Py8QLSYyQKk2KbBNAOp_VWlIwCLcBGAsYHQ/s2048/Schermata%2B2021-10-01%2Balle%2B08.30.25.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1587" data-original-width="2048" height="248" src="https://1.bp.blogspot.com/-sVzJqlzMS8Q/YVarGAD-D2I/AAAAAAAADKY/z1UNxRt9Py8QLSYyQKk2KbBNAOp_VWlIwCLcBGAsYHQ/s320/Schermata%2B2021-10-01%2Balle%2B08.30.25.png" width="320" /></a></div> Jonathan Franceschi, Lorenzo Pareschi (to appear in <i>Philosophical Transactions of the Royal Society A</i>. Preprint <a href="https://arxiv.org/abs/2109.14087" target="_blank">arXiv:2109.14087</a>)<p></p><p style="text-align: justify;"></p><div style="text-align: justify;">The rise of social networks as the primary means of communication in almost every country in the world has simultaneously triggered an increase in the amount of fake news circulating online. This fact became particularly evident during the 2016 U.S. political elections and even more so with the advent of the COVID-19 pandemic. Several research studies have shown how the effects of fake news dissemination can be mitigated by promoting greater competence through lifelong learning and discussion communities, and generally rigorous training in the scientific method and broad interdisciplinary education. <span><a name='more'></a></span>The urgent need for models that can describe the growing infodemic of fake news has been highlighted by the current pandemic.</div><span></span><div style="text-align: justify;">The resulting slowdown in vaccination campaigns due to misinformation and generally the inability of individuals to discern the reliability of information is posing enormous risks to the governments of many countries. In this research using the tools of kinetic theory we describe the interaction between fake news spreading and competence of individuals through multi-population models in which fake news spreads analogously to an infectious disease with different impact depending on the level of competence of individuals. The level of competence, in particular, is subject to an evolutionary dynamic due to both social interactions between agents and external learning dynamics. The results show how the model is able to correctly describe the dynamics of diffusion of fake news and the important role of competence in their containment.</div><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-2309808095338073402021-08-04T18:10:00.009+02:002021-12-08T08:36:55.604+01:00Mean-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 (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. <span></span></p><a name='more'></a>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 style="text-align: left;">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.</span><p></p><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-14214444051947655482021-07-20T12:52:00.003+02:002022-04-25T20:22:36.612+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" 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 (to appear in <i>J. Comp. Phys.</i> 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-31905035451934552242021-07-16T08:13:00.012+02:002022-03-18T19:20:37.657+01:00On 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 <i>Journal of Scientific Computing </i>90(3), 97, 2022 (<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. <span></span></p><a name='more'></a>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 style="text-align: left;">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.</span><p></p><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-34531837848806135922021-06-15T11:49:00.015+02:002021-12-08T08:33:14.289+01:00Spatial 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 (<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-43221385687612839032021-06-10T12:45:00.004+02:002021-12-08T08:33:28.578+01:00Modelling 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 (<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-35396917772432470772021-05-31T09:06:00.011+02:002022-01-18T09:53:03.332+01:00Hyperbolic 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 (<i>Math. Mod. Meth. Appl. Scie. </i>31(12): 2495-2531, 2021. 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. <span></span></p><a name='more'></a>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 style="text-align: left;">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.</span><p></p><p></p>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-54726935112076844942021-05-28T12:32:00.002+02:002022-08-01T16:48:14.424+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" 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 (<i>SIAM J. Number. Anal.</i> to appear. Preprint arXiv:2105.13158)<div><br /><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></div>Lorenzo Pareschihttp://www.blogger.com/profile/14028797363227049635noreply@blogger.comtag:blogger.com,1999:blog-1941919530562637402.post-62946147269921172072021-05-07T09:53:00.008+02:002022-08-22T18:39:19.544+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 (<i>Applied Math. and Optim</i>. 86(9):1-41, 2022. 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. <span><a name='more'></a></span>For the latter methods, convergence to the global minimizer can be shown for a large class of functions. 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-67928826772774938602021-04-01T21:21:00.010+02:002022-08-22T18:38:14.260+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 (<i>SIAM J. Optim. </i>32(3):1984-2012, 2022. 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.com