• Speaker: Gerard Awanou, University of Illinois, Chicago
• Title: Discrete Aleksandrov solutions of the Monge-Ampere equation
• Date: 08/31/2018
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

A discrete analogue of the Dirichlet problem of the Aleksandrov theory of the Monge-Amere equation is derived in this paper. The discrete solution is not required to be convex, but only discrete convex in the sense of Oberman. We prove that the uniform limit on compact subsets of discrete convex functions which are uniformly bounded and which interpolate the Dirichlet boundary data is a continuous convex function which satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. We obtain the first proof of convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition,  as proposed by Benamou and Froese, can be used to get a good initial guess for geometric methods solving optimal transport between two measures. 

• Speaker: Dongwook Lee, University of California, Santa Cruz
• Title: New Polynomial-free, Variable High-order Methods using Gaussian Process Modeling for CFD
• Date: 09/14/2018
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

In this talk, an entirely new class of high-order numerical algorithms for computational fluid dynamics is introduced. The new method is based on the Gaussian Processes (GP) modeling that generalizes the Gaussian probability distribution. The new approach is to adopt the idea of the GP prediction technique which utilizes the covariance kernel functions and use it to interpolate and/or reconstruct high-order approximations for computational fluid dynamics simulations. The new GP high-order method is proposed as a new numerical high-order formulation in finite difference and finite volume frameworks, alternative to the conventional polynomial-based approaches.

• Speaker: Akil Narayan, University of Utah
• Title: Sampling techniques for building computational emulators and high-dimensional approximation
• Date: 10/26/2018
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

We present an overview of techniques for building mathematical emulators of parametrized scientific models. We will primarily discuss forward emulation, where one seeks to predict the output of a model given a parametric input. We will emphasize methods that boast stability, accuracy, and computational efficiency. The focus will be on emulators built from non-adapted polynomials, and time permitting we will also explore adapted approximations and reduced order modeling. The talk will highlight some recent notable advances made in the field of building emulators from sample data, and will identify frontiers where mathematical or computational advances are needed.

• Speaker: Wenrui Hao, Pennsylvania State University
• Title: Computational modeling for cardiovascular risk evaluation
• Date: 11/09/2018
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

Atherosclerosis, the leading cause of death in the United State, is a disease in which a plaque builds up inside the arteries. The LDL and HDL concentrations in the blood are commonly used to predict the risk factor for plaque growth. In this talk, I will describe a recent mathematical model that predicts the plaque formation by using the combined levels of (LDL, HDL) in the blood. The model is given by a system of partial differential equations within the plaque with a free boundary. This model is used to explore some drugs of regression of a plaque in mice, and suggest that such drugs as used for mice may also slow plaque growth in humans. Some mathematical questions, inspired by this model, will also be discussed. I will also mention briefly some related projects about abdominal aortic aneurysm (AAA) and red blood cell aggregation, which would have some potential blood biomarkers for diagnosis of AAA.

• Speaker: Rongrong Lin , Sun Yat-sen University
• Title: Machine Learning with Reproducing Kernels
• Date: 11/30/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions on which point evaluation functionals are continuous. Thanks to the existence of an inner product, RKHSs are well-understood in functional analysis. Successful and important machine learning methods based on RKHSs include support vector machines, regularization networks and kernel-based approximation. In the past decade, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBSs) for applied and theoretical purposes for instance sparse approximation. Recently, we propose a generic definition of RKBS and a framework of constructing RKBSs that unifies existing constructions in the literature, and leads further to new RKBSs. As a by-product, the space C([0,1]) of all continuous functions on the interval [0,1] is an RKBS. Motivated by sparse multi-task learning, we constructed a class of vector-valued RKBSs with the l1 norm based on multi-task admissible kernels. The relaxed linear representer theorem holds for regularization networks in the obtained spaces if and only if the Lebesgue constant of kernels is uniformly bounded. A class of translation-invariant kernels of limited smoothness admissible for construction are given. Numerical experiments demonstrate the advantages of the proposed construction and regularization models. This talk is based on two joint papers with Prof. Guohui Song (Clarkson University), Haizhang Zhang (Sun Yat-sen University), and Jun Zhang (University of Michigan).