Abstract:
To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. We then consider large networks under the same framework. Using a combination of asymptotic methods and first passage time calculations, we find that when the network is sufficiently large, most agents decide correctly irrespective of whether the first agent’s decision is right or wrong. Interestingly, individuals in networks with both hasty and deliberate agents can make the right choice more quickly and more often than in networks of identical agents: Observing the choices of a small group of hasty agents can allow the more deliberate agents to make accurate decisions. Our model is tractable and readily generalizable, paving the way for the future study of different social network topologies. We conclude that diverse groups make quicker, more accurate decisions than homogenous groups.
Title: Extrapolation for eigenvalue problems: Upcycling data for faster convergence
Abstract: We will discuss accelerating convergence to numerical solutions of eigenvalue problems using a simple post-processing step applied to standard eigensolver techniques. First we will consider accelerating the standard power iteration, one of the most basic and powerful but sometimes very slow iterative methods. We will review some recent results on how we can make the power iteration faster by recombining previous iterates to form our next approximation to a solution; and, we will discuss why this works. We can also apply a similar technique to a restarted Arnoldi method to boost its performance with little additional computational cost. Numerical examples will illustrate the theory.
Title: A C0 finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain
Abstract:
In this talk, we discuss the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.
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.
Abstract: To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. We then consider large networks under the same framework. Using a combination of asymptotic methods and first passage time calculations, we find that when the network is sufficiently large, most agents decide correctly irrespective of whether the first agent’s decision is right or wrong. Interestingly, individuals in networks with both hasty and deliberate agents can make the right choice more quickly and more often than in networks of identical agents: Observing the choices of a small group of hasty agents can allow the more deliberate agents to make accurate decisions. Our model is tractable and readily generalizable, paving the way for the future study of different social network topologies. We conclude that diverse groups make quicker, more accurate decisions than homogenous groups.
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.
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.
