An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation

Giacomo Borghi, Michael Herty, Lorenzo Pareschi (preprint arXiv:2208.01362)

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.

Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties

Andrea Medaglia, Lorenzo Pareschi, Mattia Zanella (preprint arXiv::2208.00692)

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.

Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs

Giacomo Dimarco, Lorenzo Pareschi, Mattia Zanella (preprint arXiv:2207.06494)

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.

Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread

Giulia Bertaglia, Chuan Lu, Lorenzo Pareschi, Xueyu Zhu (to appear in Mathematical Models and Methods in Applied Sciences, preprint arXiv:2206.12625)

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.

Locally Structure-Preserving div-curl operators for high order Discontinuous Galerkin schemes

Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2206.08609) 

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.

A consensus-based algorithm for multi-objective optimization and its mean-field description

Giacomo Borghi, Michael Herty, Lorenzo Pareschi (to appear in Proceedings of the 61st IEEE Conference on Decision and Control. Preprint arXiv:2203.16384)

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.