Title: Theory and Algorithms for Nonlinear Eigenvector Problems with Affine-Linear Structures



Abstract: Eigenvector-dependent Nonlinear Eigenvalue Problems (NEPv) have long played critical roles in computational physics and chemistry and are becoming increasingly important in numerous data science applications. In this talk, we consider a class of NEPv where the coefficient matrices have a special affine-linear structure. One important origin of affine-linear NEPv is the Rayleigh-quotient-related optimization, including the trace-ratio optimization for dimension reduction and robust Rayleigh-quotient optimization for handling data uncertainties. We will establish variational characterizations for particular affine-linear NEPv, and then provide a geometric interpretation of a Self-Consistent Fields (SCF) iteration for solving the NEPv. The geometric interpretation reveals the global convergence of SCF in many cases and explains its potential non-convergence issues in others. New improvements to SCF, including the local acceleration schemes and the global verification techniques, are also discussed. Numerical experiments demonstrate the effectiveness of our approach.

Title: An Adaptive Formation Control Architecture for A Team of Quadrotors with Performance and Safety Constraints

Abstract: We propose a novel adaptive formation control architecture for a group of quadrotor systems, under line-of-sight (LOS) distance and relative distance constraints, where the constraint requirements can be both asymmetric and time-varying in nature. Universal barrier functions are adopted in the controller design and analysis, which is a generic framework that can address system with different types of constraints in a unified controller architecture. Furthermore, each quadrotor’s mass is unknown, and the system dynamics are subjected to time-varying external disturbance. Through rigorous analysis, an exponential convergence rate can be guaranteed on the distance tracking errors, while the constraints are satisfied during the operation. A simulation example further demonstrates the efficacy of the proposed control framework.

Title: Eigenvalue solution via the use of a single random vector

Abtract: In this talk, we show the design of reliable and efficient eigensolvers based on the use of a single random vector in eigenvalue detection strategies. Given a region of interest, some randomized estimators applied to a spectral projector are used to detect the existence of eigenvalues. The reliability of the estimators with a single random vector are studied so as to obtain robust thresholds for eigenvalue detection. This is then combined with repeated domain partitioning to find eigenvalues to a desired accuracy. Preconditioned Krylov subspace methods are used to solve multiple shifted linear systems in the eigenvalue detection scheme and Krylov subspaces are reused for multiple shifts. We also show how another randomized strategy can be used to obtain eigenvectors reliably with little extra costs.

Title: Uncovering potential interventions for pancreatic cancer patients via mathematical modeling

Abstract: While any cancer diagnosis is life-altering, pancreatic cancer is among the most discouraging to receive because of its extreme difficulty to overcome. Recent literature suggests that the surrounding environment of pancreatic cancer cells could play a key role in their therapeutic response. Thus, there is a growing need for the discovery of intervention strategies that can attack these cancer cells and the microenvironment that protects them. To address this problem, we have built a mathematical model to computationally predict patient outcomes and test discovered control targets. Using amenable control approaches, we were able discover novel control targets as well as validate previously known results. Further, we were able to predict a hierarchy of disease aggression based on which mutations were present, in the sense that some combinations may be more difficult to treat or that the patient might see a faster decline. This is a step forward in aiding the development of personalized medicine, as treatment protocols progress in becoming more patient-specific.

Title: Low-rank Structured Data Analysis

Abstract: In modern data analysis, the datasets are often represented by large-scale matrices or tensors (the generalization of matrices to higher dimensions). To have a better understanding of the data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss how CUR decompositions are applied to develop efficient algorithms or models to robust decomposition and completions and show the efficiency of the algorithms on some real and synthetic datasets.