Professor Grégoire Allaire (CMAP, Ecole Polytechnique)
Title
Long time homogenization of the wave equation in periodic media
Abstract
Abstract: We report on a joint work with A. Lamacz-Keymling and J. Rauch. We study the homogenization of the wave equation in a periodic medium for long times of the order of any inverse power of the period. The unknown can be either a scalar or a vector field, while the coefficients can be purely periodic or locallyperiodic tensors. We obtain high order homogenized equations which include dispersive corrections that are crucial for long time accuracy. Our main tools are (i) a so-called "criminal ansatz", which generalizes to the hyperbolic setting an idea of Bakhavalov and Panasenko in the elliptic setting, (ii) an elimination process for the higher order time derivatives in the high order homogenization equation, (iii) a stability estimate for the corresponding homogenized solutions, based on frequency filtering(iv) an error estimate valid for any long times. The importance of considering high order homogenized equations to catch dispersive effects in the context of the wave equation was first recognized by Santosa and Symes and rigorously analyzed by Lamacz. Our work gives a systematic and complete analysis for all time scales and all high order corrective terms.
In this talk we shall present the construction of a stochastic process, which is related to the parabolic p-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) Laplacian.
This is a joint work with Viorel Barbu and Marco Rehmeier.
Nonlocal modeling, analysis and computation: some recent development
Abstrac
Nonlocality has become increasingly prominent in nature, leading to the development of new mathematical theories to model and simulate its impact. In this lecture, we will concentrate on nonlocal models that involve interactions with a finite horizon, examining their significance in understanding phenomena involving potential anomalies, singularities, and other effects that arise from nonlocal interactions. Furthermore, we will present recent analytical studies that explore nonlocal operators and function spaces, discussing how they contribute to the development of robust numerical algorithms.
On three classical beta ensembles on the real line
Abstract
Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles are three beta ensembles on the real line associated with the Gaussian weight, the Laguerre weight and the Jacobi weight, respectively. Here the parameter beta is regarded as the inverse temperature of the system. It is noted that these three beta ensembles are now realized as eigenvalues of certain random tridiagonal matrices.
This talk gives a brief introduction to the study of global spectral properties of these ensembles.
