ISU PADS Seminar
Feb 10 (4:10pm) - Ananda Weerasinghe (ISU)
Title: Diffusion approximations and optimal control of processing systems in heavy traffic
Abstract: A running maximum control of a diffusion process is considered. To find an optimal control, we consider the associated two-dimensional degenerate Hamilton-Jacobi-Bellman(HJB) equation and construct a sufficiently smooth solution. This in turn leads to a bounded, feed-back type optimal control.
This diffusion control problem(DCP) is motivated by the service rate control of an infinite-server processing system in heavy traffic where the control mechanism is enforced to limit the storage capacity. The DCP is obtained as the diffusion approximation of the processing system control problem. An asymptotically feed-back type optimal control mechanism for the processing system is obtained using the solution to DCP.
Feb 17 (4:10pm) - Qi Feng (USC)
Title: Accelerating Convergence of Stochastic Gradient MCMC: algorithm and theory
Abstract: Stochastic Gradient Langevin Dynamics (SGLD) shows its advantages in multi-modal sampling and non-convex optimization, which implies broad application in machine learning, e.g. uncertainty quantification for AI safety problems, etc. The core issue in this field is about the acceleration of the SGLD algorithm and convergence rate of the continuous time Langevin diffusion process to its invariant distribution. In this talk, I will present the general idea of entropy dissipation and convergence rate analysis. In the first part, I will show stochastic gradient MCMC algorithms based on replica exchange Langevin dynamics, and empirical studies of our algorithm on optimization and uncertainty estimates for synthetic experiments and image data. In the second part, I will talk about the convergence rate analysis of reversible/non-reversible degenerate Langevin dynamics (i.e. variable coefficient underdamped Langevin dynamics). The talk is based on a series of joint works with W. Deng, L. Gao, G. Karagiannis, W. Li, F. Liang, and G. Lin.
Feb 24 (4:10pm) - Ananda Weerasinghe (ISU)
Title: A diffusion control problem with a path-dependent cost functional
Abstract: We analyze and obtain the solution to a diffusion control problem (DCP) with a cost structure consists of two types of costs: a cost due to increase of the running maximum of the state process and a control cost where the control term effects the linear drift of the state process and it helps to slow down the growth of the running maximum. This DCP is obtained as the diffusion approximation of an infinite-server controlled stochastic processing system. The solution to this DCP enables us to find nearly optimal controls for the control problem associated with the processing system.
Mar 3 (4:10pm) - Nicole Buczkowski (University of Nebraska-Lincoln)
Title: Stability of solutions with respect to changes in data and parameters of nonlocal models
Abstract: Mathematical models that are physically relevant must guarantee existence and uniqueness of solutions, as well as continuity with respect to the data. Thus, small changes in data or parameters will lead to appropriate changes in the solution. These continuity properties are also relevant in numerical analysis, where small variations that are beyond computer precision should not be reflected in the numerical solutions produced. Nonlocal models, with their capability to handle discontinuities, record long range interactions through a kernel which gives additional flexibility. By varying data given by external forces, boundary conditions, the kernel, or the type of nonlinearity used in the model the solution changes accordingly. These changes are quantified through stability or continuity results, which we obtained in linear and nonlinear settings.
Mar 31 (4:10pm) - Hayley Olson (University of Nebraska-Lincoln)
Title: Tempered and Truncated Fractional Operators - Exploring Reduction of Computational Costs
Abstract: Fractional calculus operators have been around nearly as long as the familiar integer-order derivatives. The introduction of an exponential tempering to the fractional operator has recently found applications in geophysics and finance. However, the infinite radius of interaction of the operator makes it computationally hefty. In this talk, we explore the use of a truncated fractional operator -- restricted to a bounded domain of integration -- in order to capture the same action of the tempered operator but with an easier computational load.