Projet ANR : Stab

Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.

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project_description [2012/12/20 22:11]
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project_description [2012/12/20 22:12] (Version actuelle)
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 In the present proposal we wish to go beyond this academic setting and study the stability and stabilization for both more realistic and more complex evolution equations, and in particular non-reversible or/and degenerate : instances are kinetic Fokker-Planck type hypoelliptic equations, Navier-Stokes type equations, for polymers, mean field equations and systems of reaction-diffusion equations modeling chemical reactions. For such models the techniques used in the reversible setting must be refined and extended, but overall new analytic, probabilistic and numerical techniques must be developed : in particular we intend to stress on the numerical study of such models. The stability of numerical schemes is a fundamental issue whose study is based on discrete functional inequalities and on associated (jump) stochastic processes. Estimating the constants in these inequalities is a main question which we intend to study using tools such as the curvature of discrete spaces, extremal functions, or Liapunov functions as developed by members of the present proposal in other In the present proposal we wish to go beyond this academic setting and study the stability and stabilization for both more realistic and more complex evolution equations, and in particular non-reversible or/and degenerate : instances are kinetic Fokker-Planck type hypoelliptic equations, Navier-Stokes type equations, for polymers, mean field equations and systems of reaction-diffusion equations modeling chemical reactions. For such models the techniques used in the reversible setting must be refined and extended, but overall new analytic, probabilistic and numerical techniques must be developed : in particular we intend to stress on the numerical study of such models. The stability of numerical schemes is a fundamental issue whose study is based on discrete functional inequalities and on associated (jump) stochastic processes. Estimating the constants in these inequalities is a main question which we intend to study using tools such as the curvature of discrete spaces, extremal functions, or Liapunov functions as developed by members of the present proposal in other
 settings. The issues under consideration have analytic, probabilistic and numerical aspects : this is why the present proposal gathers researchers in the fields of PDEs, discrete and continuous Markov processes and numerical analysis. settings. The issues under consideration have analytic, probabilistic and numerical aspects : this is why the present proposal gathers researchers in the fields of PDEs, discrete and continuous Markov processes and numerical analysis.
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 +The project is divided into three interconnected main parts.
 +  -  Nonlinear stability of numerical schemes and particle systems : asymptotic preserving schemes, semi-lagrangian methods, splitting techniques, mean field particle systems.
 +  - Stabilization for non-reversible models including hypocoercivity,​ systems of PDEs, jump processes, discrete time dynamics and acceleration of convergence.
 +  - Perturbation and stability of functional inequalities,​ including the computation of explicit bounds, qualitative properties of optimal functions like symmetry breaking, boundary conditions, non-local operators.
project_description.txt · Dernière modification: 2012/12/20 22:12 par gentil