Wednesdays at 4:15 PM in 380C.
This is a joint work with Anne Nouri. Introducing a Dirac Potential instead of the standard Poisson potential in the Vlasov equation change it into a very singular problem at a cross road of different subjects. In particular Penrose's method differenciates between linearly well posed and ill posed problems and leads to a result of ill posedness for the full non linear problem. On the other hand when restricted to kinetic hydrodynamic it gives rises to a well posed problem which locally in times is related to the semi classical limit of the Non Linear Schrodinger Equation. Hence it can be approached by inverse scattering or by Madelung transform. This point of view is closely related to the properties (integrability and infinite set of conserved quantities) which have been observed for the Benney equation.
We explain the mechanism of the singularity formation for the dynamics of patches. The estimates we obtain are double-exponential for all time and they are sharp. The lower bounds for the true Euler equation will be discussed as well.
Standard Malliavin calculus deals mostly within the paradigm of It^o calculus. An extension of Malliavin calulus beyond this paradigm and a related stochastic "Taylor" expansion will be introduced. Applications of these approximations to solutions of randomly forced Navier-Stokes and Burgers equations will be discussed.
A natural question in the title was (surprisingly) ignored in the theory of open systems. An answer occurred to be a counterintuitive and allowed to:
1.Better understand a role of periodic orbits in chaotic dynamics.
2.Give dynamics (rather than only static used) characterizations of elements and links of dynamical networks.
3.Open up a completely unexploded (and seemingly non-existed) area of finite time QUALITATIVE properties of dynamical and of stochastic systems.
Abstract: We propose a dynamically bi-orthogonal method (DyBO) to study time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen-Loeve expansion minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. In this talk, we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigen-decomposition. We further present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. Several numerical experiments will be provided to demonstrate the effectiveness of the DyBO method.
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