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Calendario degli incontri a.a. 2023-2024
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Venerdì 26 settembre 2023, ore 15.00, Sala di Consiglio
Stephan Gerster
University of Mainz
Haar-type stochastic Galerkin formulations for random hyperbolic systems
Abstract:
The idea to represent stochastic processes by orthogonal polynomials has been employed in uncertainty quantification and inverse problems. This approach is known as stochastic Galerkin formulation with a generalized polynomial chaos (gPC) expansion. The gPC expansions of the stochastic input are substituted into the governing equations. Then, they are projected by a Galerkin method to obtain deterministic evolution equations for the gPC coefficients. Applications of this procedure have been proven successful for diffusion and kinetic equa- tions. So far, results for general hyperbolic systems are not available. A problem is posed by the fact that the deterministic Jacobian of the projected system differs from the random Jacobian of the original system and hence hyperbolicity is not guaranteed. Applications to hyperbolic conservation laws are in general limited to linear and scalar hyperbolic equations. We analyze the loss of hyperbolicity for isentropic Euler equations. In particular, hy- perbolicity depends on the choice of gPC expansion. In particular, the dependency on a random input is described by Haar-type wavelet systems. Theoretical results are illus- trated numerically by CWENO-type reconstructions combined with a numerical entropy indicator that allow also for higher-order discretizations of balance laws.
Venerdì 22 settembre 2023, ore 11.00, Sala di Consiglio
Diogo Gomes
KAUST
Functional Analytic Insights into Mean Field Game Theory
Abstract:
Monotonicity conditions are crucial in Mean Field Game (MFG) theory, highlighted by the uniqueness results of Larry and Lions. This talk introduces a functional analytic framework to understand MFGs that satisfy monotonicity conditions. By leveraging ideas introduced in Hessian-Riemannian flows from optimization, we propose regularized versions of MFGs and construct contracting flows that can be used for numerical approximation. Our findings present a consolidated view of our prior works and give a different perspective on this class of problems.
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