We study a general viral infection model with spatial diffusion in virus and two types of infection mechanisms: cell-free and cell-to-cell transmissions. The model is a partially degenerate reaction-diffusion system, whose solution map is not compact. We identify the basic reproduction number and explore its properties when the virus diffusion parameter varies from zero to infinity. Moreover, we demonstrate that the basic reproduction number is a threshold parameter for the global dynamics of our model system: the infection and virus will be cleared out if the basic reproduction number is no more than one. On the other hand

We propose a novel approach for variable screening in sparse multivariate additive models with random effects by use of null-beamforming on the data. The new approach includes two stages. In Stage 1, we approximate each nonparametric component by a linear combination of spline basis functions. Consequently, we convert the above problem to that of selecting groups of coefficients in a multivariate regression model with vector-valued covariates. In Stage 2, we conduct a series of filtering operations (called beamforming) by projections of the multiple response observations into each covariate space; each filter is tailored to a particular covariate and resistant to interferences originating from other covariates and from background noises

Graphical models are commonly used in representing conditional independence between random variables, and learning the conditional independence structure from data has attracted much attention in recent years. However, almost all commonly used graph learning methods rely on the assumption that the observations share the same mean vector. In this paper, we extend the Gaussian graphical model to the setting where the observations are connected by a network and propose a model that allows the mean vectors for different observations to be different. We have developed an efficient estimation method for the model and demonstrated the effectiveness of the proposed method using simulation studies. Further, we prove that under the assumption of "network cohesion", the proposed method can estimate both the inverse covariance matrix and the corresponding graph structure accurately.

We develop a modified Poisson-Nernst-Planck model to include Coulomb many-body properties in electrolytes, which also takes the ion-size effect into account and is expected to provide more accurate prediction for ion dynamics with microscopic confinement. Asymptotic expansions are performed to remove the multiscale properties of the equations and also used to understand dielectric properties near interfaces. Furthermore, we discuss numerical strategies to solve the resulted PDEs and show numerical results to demonstrate the performance of our numerical methods.

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e., the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large- scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method

We discuss an arbitrary-order discontinuous Galerkin method for second order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L2 norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems. The application of this method to plate bending problem will also be addressed. This is a joint work with Ruo Li, Ziyuan Sun and Zhijian Yang.

In this talk, we discuss an optimal investment and consumption problem with exponential utility function in a financial market where the asset prices follows a cointegrated model. After applying the dynamic programming method, we derive a Hamilton-Jacobi-Bellman (HJB) equation, then we obtain an optimal investment and consumption strategies and the corresponding value function in a closed form. A verification theorem is proved to demonstrate that under certain growth conditions the solution of the HJB equation is indeed the one of our original problem.

The talk will deal with the following question. Given a homeomorphism f of a surface S and a set X (of S) of fixed points of f, can one find a continuous path f_t (t/in [0,1])of homeomorphisms of S joining the identity to f, so that f_t fixes X pointwise for every t? If such a path (f_t) exist and X is maximal, then a theorem of Le Calvez allows to describe the dynamics of f. This is a joint work with Sylvain Crovisier (Université Paris-Sud) and Frédéric Le Roux (Université Pierre et Marie Curie - Paris)