Mean-field particle swarm optimization

 

Sara Grassi, Hui Huang, Lorenzo Pareschi, Jinniao Qiu (3/8/2021 to appear in Modeling and Simulation for Collective Dynamics, IMS Lecture Note Series, World Scientific, preprint arXiv:2108.00393)

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.

A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs

 

Liu Liu, Lorenzo Pareschi, Xueyu Zhu (20/07/2021 preprint arXiv:2107.09250)

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.

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs

Walter Boscheri, Maurizio Tavelli, Lorenzo Pareschi (14/07/2021 preprint arXiv:2107.06956)

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.

Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty

Giulia Bertaglia, Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (15/6/2021, Math. Biosci. Engin. 18(5): 7028-7059, 2021)

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.

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella (10/6/2021, Math. Biosci. Engineer. 18(6):7161-7190, 2021)

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.