Spring 2022

Speaker: Dr. Nam Le from Indiana University Bloomington

Date and Time: Wednesday, March 24, 2021, 10am – 11am via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Liouville type theorems for the Monge-Ampere equation

Abstract: Liouville type theorems in Partial Differential Equations are concerned with classification of global solutions. The Monge-Ampere equation consists of prescribing the determinant of the Hessian of a convex function. In this talk, we will review some Liouville type theorems for the Monge-Ampere equation in the whole space, on the half space, and we will discuss in depth the case in the first quadrant in the plane. Application will be given to global second order derivative estimates for the non-degenerate Monge-Ampere equation in convex polygonal domains in the plane. This talk is based on joint work with Ovidiu Savin. The talk will be accessible to general mathematical audience.

Speaker: Dr. Xiaochuan Tian from the University of California, San Diego

Date and Time: Thursday, December 10, 2020, 12pm – 1pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Reproducing kernel collocation methods for nonlocal models: asymptotic compatibility and numerical stability

Abstract: Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. Aiming at both rigorous numerical analysis and computational efficiency, we present the reproducing kernel collocation methods, a class of meshfree methods, for approximating nonlocal models characterized by a length parameter that may change with the models. A central idea is to design asymptotic compatible schemes that are robust under the change of the nonlocal length parameter.

Speaker: Dr. Kin Yau Wong from the Hong Kong Polytechnic University

Date and time: Friday, November 20, 2020, 10am – 11am via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Score Tests with Incomplete Covariates and High-Dimensional Auxiliary Variables

Abstract: Analysis of modern biomedical data is often complicated by the presence of missing values. To improve statistical efficiency, it is desirable to make use of potentially high-dimensional observed variables to impute or predict the missing values. Although many methods have been developed for prediction using high-dimensional variables, it is challenging to perform valid inference based on the predicted values. In this presentation, we develop an association test for an outcome variable and a potentially missing covariate, where the covariate can be predicted using selected variables from a set of high-dimensional auxiliary variables. We establish the validity of the test under general model selection procedures. We demonstrate the validity of the proposed method and its advantages over existing methods through extensive simulation studies and provide an application to a major cancer genomics study.

Speaker: Dr. Jennifer Alonso Garcia from Université Libre de Bruxelles

Date and time: Friday, November 13, 2020, 11am – 12pm via Zoom. Please contact Qingning Zhou to obtain the Zoom link.

Title: Taxation and Policyholder Behavior: The Case of Guaranteed Minimum Accumulation Benefits

Abstract: This paper considers variable annuity contracts embedded with guaranteed minimum accumulation benefit (GMAB) riders when policyholder’s proceeds are taxed. These contracts promise the return of the premium paid by the policyholder, or a higher stepped up value, at the end of the investment period. A partial differential valuation framework, which exploits the numerical method of lines, is used to determine fair fees that render the policyholder and insurer profits neutral. Two taxation regimes are considered; one where capital gains are allowed to offset losses and a second where gains do not offset losses, reflecting stylized institutional arrangements in Australia and the US respectively. Most insurance providers highlight the tax-deferred feature of a variable annuity. We show that the regime under which the insurance provider is taxed significantly impacts supply and demand prices. If losses are allowed to offset gains then this enhances the market, narrowing the gap between fees, and even producing higher demand than supply fees. On the other hand, when losses are not allowed to offset gains, then the demand-supply gap increases. When charging the demand price, we show that insurance companies would be profitable on average. Low (high) Sharpe ratios are not as profitable as policyholders are more likely to stay long (surrender).

Speaker: Dr. Linquan Bai (Department of Systems Engineering & Engineering Management, UNC Charlotte)

Date/Time/Place: 2:00–3:00pm, March 27 (Friday), Fretwell 315.

Title: Transition to Risk Driven Power Grid Operation and Management

Abstract: Affordable and reliable electricity is fundamental to society. Today’s power grid is facing more risks than ever from uncertain renewable power generation, more frequent severe weather events, and cyber-attacks. Integrating emerging technologies such as renewable energy, energy storage, and electric vehicles, the management system of the power grid should evolve to a risk-driven paradigm to effectively manage, hedge and mitigate the system risks. The transition of the power grid management system necessitate advanced risk-theory to analyze probability and uncertainty quantification of renewable generation and vulnerability in power grid topology, optimization approaches for risk-averse optimal power grid scheduling, machine learning for cyberattack detection, etc. I will present the challenges today’s power grid is facing and applications of these new theories and methodologies in addressing the challenges.

Conference room, 11:00 am-12:00 pm

Dr. Tien-Khai Nguyen, NC State University.

Title: Differential Game Models of Optimal Debt Management 
Abstract: In this talk, I will present recent results on game theoretical formulation of optimal debt management problems in infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. Here, the yearly income of the borrower is governed by a stochastic process and bankruptcy instantly occurs when the debt-to-income ratio reaches a threshold. Since the borrower may go bankrupt in finite time, the risk-neutral lenders will charge a higher interest rate in order to compensate for this possible loss of their investment. Thus, a “solution” must be understood as a Nash equilibrium, where the strategy implemented by the borrower represents the best reply to the strategy adopted by the lenders, and conversely. This leads to highly nonstandard optimization processes.

Date: February 21, 2020

Time and location:  11:10am-12:10pm, Fretwell 305

Speaker:  Mingyao Li, Ph.D, Department of Biostatistics, Epidemiology & Informatics, University of Pennsylvania Perelman School of Medicine

Title:   Translation of single-cell genomics into human health: methods and applications

Abstract:   Recent technological breakthroughs have made it possible to measure gene expression at the single-cell level, thus allowing biologists and clinicians to better understand cellular heterogeneity and modify cell behavior through targeted molecular therapies. However, single-cell RNA sequencing protocols are complex. Even with the most sensitive platforms, the data are often noisy owing to a high frequency of dropout events, and the phenomenon of transcriptional bursting in which pulses of transcriptional activity are followed by inactive refractory periods. In this talk, I will present several statistical and machine learning methods that aim to tackle these challenges for a better understanding of cellular heterogeneity. I will illustrate our methods by showing results from ongoing collaborations on age-related macular degeneration and Alzheimer’s disease. With the growing interest in utilizing single-cell RNA sequencing in biomedical research, our methods will aid biomedical researchers to answer medically related questions and make exciting discoveries.

Dr. Thi-Phong Nguyen, University of Minnesota (hosted by Loc Nguyen)

Title: Sampling methods for inverse elastic scattering problems in finite solid bodies. 
Abstract: I will present and discuss in this talk sampling methods to reconstruct partially-closed fractures in rock from surface seismic measurements. For the mathematical viewpoint, the fracture is characterized by a second order tensor synthesizing its special stiffness – relation the jump in seismic field cross the fracture to the contact traction. This method relies on a careful analysis of so-called Generalized Linear Sampling Method to build an indicator function for detecting the fracture. The first challenging of this problem is tensorial quantity intrinsically dependent on the fractures’ aperture and the surface roughness, which makes it partially heterogeneous and difficult to solve. The second one is to deal with the near-filed measurement (which has very little reference) and to numerically approximate the gradient of Green tensor in finite space, which can not be expressed by an analytic formula. Some related numerical results will be presented at the end. 

Speaker: Dr. Lu Lu from Brown University (hosted by Xingjie Li).

Title: Learning dynamical systems and differential equations with deep learning: physics-informed and data-driven
Abstract: Deep learning has achieved remarkable success in diverse applications; however, its use in learning dynamical systems and partial differential equations (PDEs) has emerged only recently. These learning approaches can be either physics-informed or data-driven. In the physics-informed approach, I have improved the physics-informed neural networks (PINNs) and developed the library DeepXDE for solving different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. In the data-driven approach, I have developed the deep operator network (DeepONet) based on the universal approximation theorem of operators to learn dynamical systems accurately and efficiently from a relatively small dataset. In addition, I will present my work on the deep learning theory of optimization and generalization, and the application of applying multi-fidelity neural networks to predict mechanical properties of solid materials.

Dr. Ling Xu from North Carolina Agricultural and Technical State University (hosted by Professor Xingjie Li).

Title: Vortex dynamics computed by Lagrangian methods

Abstract: The work is built upon the research on two-dimensional inviscid fluid dynamics like the merger, filamentation, instability using magnetized electron columns in the laboratory (Durkin & Fajans, PRL, 2000). These fundamental fluid dynamics phenomena find applications in hurricanes, turbulence, and oceanography. In the work, we are using two numerical methods, particle-panel vortex method and contour dynamics, to study the vortex filamentation and mergers. Techniques such as adaptive-point-insertions and tree-code are employed to increase the accuracy and speed up the simulations. In the end, I will talk about an on-going project on studying material transport using Machine Learning.