We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain

In this talk, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. Multiscale models for complex nonlinear systems require nonlocal multicontinua approaches. These rigorous techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters (boundary and source terms) and computing effective properties derived by nonlinear nonlocal multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques.

The semiclassical Schrodinger equation with multiscale and random potentials often appearswhen studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. We propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space.

We provide error analysis of a mixed method with stabilization along mesh interface for the Quad-Curl problem. We present a new discrete embedding inequality. With the help of this embedding inequality, we show the mixed method converges optimally in energy norm even when the solution has only low regularity.

Let H_q be the semisimple Hecke algebra of type D_n. To each simple module V over H_q(D_n), we explicitly construct a quasi-idempotent whose associated idempotent is a primitive idempotent associated to V. A matrix unit basis of H_q(D_n) is also obtained. We generalize some earlier work of Dipper, James, Murphy and Mathas to the type D_n setting.

The theme of this lecture centers around two famous books, one by Hermann Weyl on “The Classical Groups: Their Invariants and Representations” (1939), another the six-volume work by I.M. Gelfand et al. on “Generalized Functions” (1960’s). I will discuss some broad ideas related to the theme of these two books, as well as some recent development in classical groups and their smooth representations. The talk is aimed at a general

基于对丛代数的Poisson结构与其量子化的关系的研究，我们通过对量子丛代数的Poisson结构的刻画，提出了二次量化的概念，并对不含系数的量子丛代数，给出了它的二次量子化的形态，指出了在这个情形下，其二次量子化具有相对的平凡性。

In this talk, we consider the global existence of Yamabe flow and study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. We also use the variational method to study the existence problem of metrics with constant scalar curvature on complete noncompact Riemannian manifolds and we can give a partial affirmative answer to a question posed by J.Kazdan in 1982.