Multi-scale control variate methods for uncertainty quantification in kinetic equations

Giacomo Dimarco, Lorenzo Pareschi 
(25/10/2018. preprint arXiv:1810.10844)

Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge.

Uncertainty Quantification for Hyperbolic and Kinetic Equations

Shi Jin, Lorenzo Pareschi (Eds.)
SEMA SIMAI Springer Series
277 pages, 2018

This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Giacomo Albi, Michael Herty, Lorenzo Pareschi
(24/7/2018, preprint arXiv:1807.08547)

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy.

Structure preserving schemes for the continuum Kuramoto model: phase transitions

José A. Carrillo, Young-Pil Choi, Lorenzo Pareschi  (13/3/2018 J. Comp. Phys. 376, (2019), 365-389. arXiv:1803.03886)

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies.