Math/Stat Colloquium

September 14, 2023 at 3:30 pm

TLC 029

Dr. David Andrew Smith (Yale-NUS College - The National University of Singapore)

Fokas Diagonalization

Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

September 21, 2023 at 3:30 pm

TLC 029

Tuan Phan (University of Idaho, Institute for Interdisciplinary Data Science)

Mathematical modeling of some biological and medical problems in cancer research and physiology

Abstract: In this presentation, I will discuss two mathematical models utilizing stochastic differential equations (SDEs) and/or ordinary differential equations (ODEs). In the first one, we studied human papillomavirus (HPV) infection by proposing a 5-dimensional Ito's SDE system. The theoretical and numerical analyses of the system were conducted to reveal insights intot he progression from HPV infection to cervical cancer. The second one involves cooperative activation of force in human myocardium in which a 3-dimensional ODE system was proposed to capture the relationship between the rate of force redevelopment and relative force in mechanical data from porcine and murine myocardium. The fitting results of the model led to deep understanding of cooperative mechanisms that underlie differences in myocardial contractile dynamics between large and small mammals.

September 28, 2023

TLC 029

Rodolfo Blanco Rodriguez (University of Idaho)

Unraveling multiscale stochastic disease transmission from within-host dynamics to between-host spread

Abstract: In this research, we employ a multiscale modeling approach to unravel the intricate dynamics of infectious disease transmission. Our study spans various scales, from within-host viral interactions to population-level disease dissemination: By employing mathematical models, we explore the replication of viral particles, their impact on infected cells, and the vital role of immune responses, particularly T cells. Moving up the scale, we investigate the transmission of viruses, such as influenza, through different respiratory organs. Zooming out, we analyze disease transmission between individuals in contact networks. Utilizing a stochastic transmission model, we determine the likelihood of infection based on viral levels and immune responses within the contagious host. The timing of host encounters emerges as a critical factor in disease spread.

October 5, 2023

TLC 029

Chencheng Cai (Washington State University)

KoPA: Automated Kronecker Product Approximation

Abstract: We propose to approximate a given matrix by the sum of a few Kronecker products of matrices, which we refer to as the Kronecker product approximation (KoPA). Comparing with the low-rank matrix approximation, KoPA also offers a greater flexibility, since it allows the user to choose the configuration, which are the dimensions of the two smaller matrices forming the Kronecker product. On the other hand, the configuration to be used is usually unknown, and needs to be determined from the data in order to achieve the optimal balance between accuracy and parsimony. We propose to use extended information criteria to select the configuration. Under the paradigm of high dimensional analysis, we show that the proposed procedure is able to select the true configuration with probability tending to one, under suitable conditions on the signal-to-noise ratio. We demonstrate the superiority of KoPA over the low rank approximations through numerical studies, and several benchmark image examples.

October 26, 2023

Via Zoom Only

February 23, 2023 at 3:30 pm

TLC 029

Dr. Alex Woo (University of Idaho)

The Stanley-Stembridge conjecture and cohomology of Hessenberg varieties

Abstract: The chromatic symmetric function of a graph encodes all of its possible colorings. Richard Stanley and John Stembridge conjectured almost 30 years ago that the chromatic symmetric functions of certain graphs can be written as a positive sum of products of elementary symmetric functions. Each symmetric function can be associated to a formal linear combination of representations of the symmetric group. John Shareshian and Michelle Wachs conjectured that these chromatic symmetric functions are associated to a representation of the symmetric group on the cohomology ring of geometric objects known as Hessenberg varieties; this conjecture was proven by Patrick Brosnan and Timothy Chow and independently by Mathieu Guay-Paquet. By work of Julianna Tymoczko, this cohomology ring can be represented as a vector space on sequences of polynomials satisfying certain relations; the Stanley-Stembridge conjecture, still unsolved, reduces to a statement that this vector space has a basis that is permuted when the sequences of polynomials are themselves permuted in a certain way.

My aim in this talk will be to make the previous paragraph at least somewhat comprehensible. If I talk about any new work, it will be joint work with Erik Insko (Florida Gulf Coast University) and Martha Precup (Washington University in St. Louis).

March 30, 2023 at 3:30 pm

TLC 029

Dr. Xiongzhi Chen (Washington State University)

Geometry, Topology, Statistics and Learning

Abstract: Doubtlessly, we are now in the “Data Era”. However, unconventional data types, such as shapes, social networks, phylogenetic trees and persistence diagrams, have brought new challenges that learning theory rooted in Euclidean spaces is inadequate to address. Further, the recent rise and success of machine learning techniques have pushed learning into the realm of “Big Models”. These models have complexities and deal with sample sizes that are respectively orders more complicated and larger than those of “high-dimensional models”. This prompts us to rethink about the foundations of statistical learning and its implementation in practice. In this talk, I will give an overview on learning, discuss inference and predictive modelling in general, and explain the role of geometry and topology in statistical learning.

 April 27, 2023 at 3:30 pm

TLC 029

 Dr. Abhishek Kaul (Washington State University)

An Efficient Two Step Algorithm for High Dimensional Change Point Regression Models Without Grid Search

Abstract: We propose a two step algorithm based on L1/L0 regularization for the detection and estimation of parameters of a high dimensional change point regression model and provide the corresponding rates of convergence for the change point as well as the regression parameter estimates. Importantly, the computational cost of our estimator is only 2·Lasso(n, p), where Lasso(n, p) represents the computational burden of one Lasso optimization in a model of size (n, p). In comparison, existing grid search based approaches to this problem require a computational cost of at least n · Lasso(n, p) optimizations. Additionally, the proposed method is shown to be able to consistently detect the case of ‘no change’, i.e., where no finite change point exists in the model.  We then characterize the corresponding effects on the rates of convergence of the change point and regression estimates.  Simulations are performed to empirically evaluate performance of the proposed estimators. The methodology is applied to community level socio-economic data of the U.S., collected from the 1990 U.S. census and other sources.

 May 4, 2023 at 3:30 pm

TLC 029

 Dr. Alejandro Anderson (University of Idaho Postdoctoral Fellow)

Set-control theory: model predictive control and controllable state-space topologies

Abstract: Model Predictive Control (MPC) is a form of control that solves on-line, at each sampling instant, a finite horizon optimal control problem, and only the first input of the optimal control sequence is applied to the real system. The MPC technique has the ability to cope with hard constraints on controls and states and it handles large multivariable systems. Lyapunov theory provides a theoretical framework to prove asymptotic stability of a system controlled by an MPC by employing general concepts of permanence regions of dynamical systems. The permanence regions play an important role in the controllability and stability analysis since such regions (i.e. equilibrium sets, invariant sets, limit cycles, etc.) are the only ones that can be formally stabilized by a controller. In this lecture the importance of general permanence regions on the formulation of stable control strategies such as MPC will be discussed and a methodology for computing these type of regions in the state-space will be presented.