In this session we concentrate on the latest research insights for uncertainty quantification in transport problems and high-dimensional systems under structural uncertainties, with focus on kinetic and hyperbolic PDEs and multiscale interacting particle systems.
16:30
- NEW - On new developments for Hyperbolic Moment Models of rarefied gases and the connection to Uncertainty Quantification
Julian Köllermeier | KU Leuven | Belgium
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Julian Köllermeier | KU Leuven | Belgium
In this talk, I will present recent developments of hyperbolic moment models for rarefied gases and show possible connections to Uncertainty Quantification. Moment models have been used for kinetic modeling of rarefied gases for decades. However, only recent developments regarding improved hyperbolic models and numerical methods are paving the way for successful applications in science and engineering. In the meantime Uncertainty Quantification is using similar tools, for example the Polynomial Chaos Expansion or filters that allow for an efficient approximation of the uncertainty in the equations. I will present a transformation of the variable space for an efficient, yet accurate discretization of the uncertain parameter, a hyperbolic regularization of the resulting model, and the use of filters for higher order moments to improve the convergence of the method.
17:00
Mean-field feedback stabilization of collective behavior with uncertainty
Giacomo Albi | University of Verona | Italy
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Giacomo Albi | University of Verona | Italy
In this talk we will focus on the control of a large system of N-interacting agent with random inputs. Direct methods for these problems typically fail due to the high-dimensionality and non-linearities of the dynamics. In order to tame the order of complexity, we first compute a class of feedback controls by solving a state-dependent Riccati equation associated to a linearized system. Second we derive the mean-field limit for N to infinity, obtaining a coupled system of a nonlinear Vlasov-type equation, for the agents, and an ODE system for the Riccati equation. These combined approximations reduce considerably the computational cost for the stabilization of this system. In particular, we will show that such approach control the propagation of the uncertainty, and is able to drive the system state towards a desired region. Finally, we will further improve the efficiency of this control strategy introducing predictor-corrector algorithm based on a forward error analysis.
Numerical experiments in the context of mean-field models for swarming and flocking dynamics will be presented showing the validity and efficiency of the approach.
17:30
Micro-macro generalized polynomial chaos techniques for kinetic equations
Giacomo Dimarco | University of Ferrara | Italy
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Giacomo Dimarco | University of Ferrara | Italy
Kinetic equations play a major role in the modelling of large systems of interacting particles/agents with a proved effectiveness in describing real world phenomena ranging from plasma physics to biological dynamics. Their formulation has often to face with physical forces deduced through experimental data and of which we have at most statistical information. Hence, we consider the presence of random inputs in the form of uncertain parameters as a structural feature of the kinetic modelling. In this talk,we will consider uncertainty quantification for Vlasov-Fokker-Planck equations through a micro-macro numerical approach based on stochastic Galerkin methods which preserve the large time distribution of the system.
18:00
- CANCELED - A consensus-based global optimization method for high dimensional machine learning problems
Yuhua Zhu | Stanford University | United States
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Yuhua Zhu | Stanford University | United States
We improve recently introduced consensus-based optimization method, proposed in [R. Pinnau, C. Totzeck, O. Tse and S. Martin, Math. Models Methods Appl. Sci., 27(01):183–204, 2017], which is a gradient-free optimization method for general non- convex functions. We first replace the isotropic geometric Brownian motion by the component-wise one, thus removing the dimensionality dependence of the drift rate, making the method more competitive for high dimensional optimization problems. Secondly, we utilize the random mini-batch ideas to reduce the computational cost of calculating the weighted average which the individual particles tend to relax toward. For its mean-field limit–a nonlinear Fokker-Planck equation–we prove, in both time continuous and semi-discrete settings, that the convergence of the method, which is exponential in time, is guaranteed with parameter constraints independent of the dimensionality. We also conduct numerical tests to high dimensional problems to check the success rate of the method. This is a joint work with Jose A. Carrillo, Shi Jin and Lei Li.