Seminar in Applied Mathematics

Department of Mathematics and Statistics

Texas Tech University

Fall 2017 - Wednesday, 4-5 pm - Room MATH 112

image Wednesday, September 6
Is the data worth of learning? Data Analysis by Scale Dependent Anisotropic Random Walks
Dimitri Volchenkov
The most important question of data analysis is: what data is worth of learning? What news are fake? What news are not fake although meaningless? Catching a structural pattern in the data (graphs, texts, musical scores, etc.) involves lifting of an empirical metric defined on elements to a metric defined on their distributions and might be understood as a version of the famous Monge-Kantorovich (MK) transportation problem, searching for minimizing the transportation costs over all available transportation plans. Not surprisingly, many empirical metrics known in measure theory, Markov chains, electrical circuits, music (tonality scale), land use (land prices), etc. are the special cases of the MK-transportation metric. In general, the data allows for an infinite number of transportation metrics at different scales (the data patch sizes). Uncertainty of an element at every data scale (entropy) can be decomposed into a sum of conditional mutual informations fully describing the information flows between the element and the whole. The data is worth of learning if a bit of information learned about a single element might gain more than one bit of information about the whole. In case of a time series or a dynamical system, such a decomposition fully describes information relations between the past, future, and present states, allowing for the robust estimation of our ability to predict the future at every time scale of the system evolution.
image Monday, September 11, 4pm, room Math 112
Noisy rolling ball: non-holonomic constraints perturbed by the noise, conservation of integrals of motion and rolling friction
Vakhtang Putkaradze, University of Alberta
We will discuss some examples of mechanical systems with non-holonomic constraints are modified by the presence of noise. The modification introduces and interesting type of stochasticity in the equations of motion, which will be illustrated on the example of a Routh (Chaplygin) sphere rolling on a flat surface. This is a classical example of a non-holonomic system possessing three integrals of motion, namely the energy, Jellet and Routh. We will show that depending on the type of noise introduced in the rolling constraint, one can either preserve either energy only, both energy and Jellet, or only Jellet integrals. We also discuss contrast these results with the dynamics of non-holonomic systems with the drift noise, derive the general theory of motion of non-holonomic systems of the semidirect product type, and discuss general results on energy preservation. We conclude with a discussion of the relevance of this work for rolling friction in dynamics due to random slipping as originally suggested by Reynolds (1876).
image Tuesday, September 12, 3:30pm, room Chem 107 (Colloquium)
Dynamics and control of flexible solar towers
Vakhtang Putkaradze, University of Alberta
The use of solar chimneys for energy production has been suggested more than 100 years ago. Unfortunately, this technology has not been realized on a commercial scale, in large part due to the high cost of erecting tall towers using traditional methods of construction. Recently, the author and collaborators have suggested a radical decrease in tower cost using an inflatable self-supported tower, built out of stacked toroidal bladders. While the statics deflections of such towers under constant wind have been investigated before, the key for further development of this technology lies in the analysis of dynamics, which is the main point of this talk. Using Lagrangian reduction by symmetry, we develop a fully three dimensional theory of motion for such towers and study the tower's stability and dynamics. Next, we derive a geometric theory of optimal control for the tower dynamics using variable pressure inside the bladders, and perform detailed analytical and numerical studies of the control in two dimensions. Finally, we report on the results of experiments demonstrating the remarkable stability of the tower in real-life conditions, showing good agreement with theoretical results. This work has been supported by NSERC and the University of Alberta.
image Wednesday, September 20
Maximally Informative Next Experiments for nonlinear models
Reginald L. McGee II, Mathematical Biosciences Institute, Ohio State University
Mathematical modeling is a powerful tool in systems biology; we focus here on improving the reliability of model predictions by reducing the uncertainty in model dynamics through experimental design. Model-based experimental design is a process by which experiments can be systematically chosen to reduce dynamic uncertainty in a given model. We discuss the Maximally Informative Next Experiment (MINE) method for experimental design and present a convergence result for MINE with nonlinear models. As an application, we illustrate the method on polynomial regression and an ODE model for immune system dynamics. The MINE criterion sequentially determines experiments that can be conducted to best refine model dynamics.
image Thursday, September 28, 3:30pm, room Chem 107 (SIAM Colloquium)
A coupling strategy for nonlocal and local models with applications to static peridynamics and classical elasticity
Marta D'Elia, Sandia National Laboratories
The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local Partial Differential Equation (PDE) models motivates and drives the interest in this type of simulations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. In particular, a complete nonlocal simulation remains computationally untenable for many science and engineering applications. As a result, it is important to develop local-to-nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model. We develop and analyze an optimization-based method for the coupling of nonlocal and local problems in the context of nonlocal elasticity. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit. Numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method; these numerical tests provide the groundwork for the development of efficient and effective engineering analysis tools. As an application, we present results for the coupling of static peridynamics and classical elasticity.
image Thursday, October 5, 3:30pm, room Chem 107 (Colloquium)
A model of natural selection predicts treatment resistance in prostate cancer
John Nagy, Arizona State University
Standard of care treatment for recurrent and advanced prostate cancer includes chemical castration. Inevitably, however, such treatment results in hormone-refractory tumors with dire prognosis. Clearly, a predictive mathematical model of this process would greatly improve our understanding and ability to mitigate castration resistance in this tumor. Here I develop an adaptive dynamics model of androgen-ablation therapy and show that it predicts progression of treatment resistance in a significant subset of prostate cancer patients. The model assumes that castration resistance evolves due to natural selection on androgen receptor (AR) expression. Formulation and parameterization of the model was completed based on a sample of 25 patients treated with intermittent androgen ablation therapy. The model was then used to predict PSA dynamics in an independent set of 30 patients from the same clinical study. Predictions were reasonably accurate typically for one cycle, and for some patients up to 4 cycles. However, there were significant exceptions, in some cases the model exhibited no predictive power. These observations are consistent with the conclusion that the model accurately reflects castration resistance arising via natural selection acting on AR expression, but fails for cases in which resistance is caused by a different mechanism, like “outlaw" or AR bypass pathways. This modeling approach therefore may provide a noninvasive method to identify emerging resistance mechanisms in nascent hormone-refractory tumors and to plan treatment to delay development of castration resistance.
image Wednesday, October 11, 3:30pm, room Chem 107 (Dayawansa Memorial Lecture Series)
Optimal Co-Design of Scheduling and Control for Networked Systems
Sandra Hirche
Robotic systems in intelligent environments and smart infrastructure systems are examples of networked cyber-physical systems where communication and/or computational resources are constrained or information acquisition/processing is at least costly. The scientific challenge is to design scheduling and control schemes taking into account such resource constraints and to preferably include fair resource sharing mechanisms among different control applications. In this talk we present a framework for the optimal co-design of scheduling and control for networked control systems with resource constraints. We consider multiple control loops, which transmit their measurements over a shared communication channel. Only a limited number of those control loops may close their feedback loop at a time. As a result the dynamics of the individual control loops are coupled through the resource constraint. The scientific question is, when the transmission of a measurement should take place and what the appropriate control law is. We approach the problem from an optimality point of view with the scheduling and control policies being the optimization variables. It turns out that a threshold-based event-triggered control scheme is optimal, i.e. information should be transmitted as soon as a certain threshold is exceeded. Furthermore, we can show that the certainty equivalent controller is optimal. The scheduling decisions among multiple control loops are coordinated by a price-based mechanism. In addition, we provide stability results linking the resource constraints with the system dynamics. Finally, we discuss extensions of the scheme to distributed systems and recent trends towards the characterization of a value of information for control.
image Thursday, October 12, 3:30pm, room Chem 107 (Dayawansa Memorial Lecture Series)
Data-driven Modeling and Control
Sandra Hirche
Most contemporary control methods are model-based, i.e. control design and analysis is performed based on a parametric model of the process, which is often derived from first principles. As the model is only an approximation of the true process dynamics, the performance is often degraded compared to desired specifications. This is particularly true if the control system is complex and operates in an a priori unknown environment such as in autonomous systems and human-machine interaction. In such situations, purely model-based control approaches are likely to fail. Given the technological advances in massive sensing and storage, data-driven techniques to modeling and control appear very promising. However, there are many obstacles to overcome as the theory of data-driven control is still in its infancy. One of the main obstacles is the lack of verifiability of behavior of data-driven control systems: How to provide the required guarantees that the automation system behaves safe and within the desired specifications if its behavior is described by data only? In this talk we discuss approaches towards data-driven modeling and control. As starting point we focus on non-parametric models provided by Gaussian Processes (GPs) as a flexible tool for nonlinear function approximation. One of the main advantages of GPs compared to other non-parametric modeling approaches such as neural networks is their ability to represent model confidence as function of the distance to the training points. We will introduce approaches for feedback linearization based on GPs and analyze their stability properties. Furthermore, we discuss the modeling of dynamic systems based on GPs. Applications in robotics illustrate the results.
image Friday, October 13, 3:30pm, room Chem 107 (Dayawansa Memorial Lecture Series)
Control Challenges in Human-Robot Interaction
Sandra Hirche
Human-robot interaction gains more and more relevance in application domains such as elderly care, collaborative manufacturing, and rehabilitation. The general trend is to increase robot autonomy with the ultimate goal of a human-robot partnership. Desirably, the interaction is intuitive and of course safe. From psychological studies it is well-known, that anticipation and therefore also behavior prediction of the interaction partner is a key aspect of joint action. However, modeling human behavior is challenging as models based on first order principles are not available and furthermore, human exhibit high variability even in repeated tasks. Data-driven probabilistic models have turned out to be promising in the literature. Any prediction based on such human behavior model will be uncertain, due to sparsely available training data and inherent human variability. This uncertainty in prediction is quite crucial to be considered when designing control mechanisms for human-robot interaction. The focus in this talk will be on physical human-robot interaction (pHRI), i.e. interaction involving physical contact between human and robot. During physical contact apart from information also energy is exchanged which poses extra challenges for real-time human-adaptive and safe decision making and control. In this talk we will discuss some of the main control challenges for pHRI, introduce human behavior modeling paradigms and propose a class of control algorithms suitable for assistive control in pHRI.
image Wednesday, October 18
A high order conservative semi-Lagrangian discontinuous Galerkin method for the Vlasov-Poisson simulations
Wei Guo
In this talk, we will introduce a high order conservative semi-Lagrangian (SL) discontinuous Galerkin (DG) method for the Vlasov-Poisson (VP) system. The proposed method relies on a characteristic Galerkin weak formulation and a high order characteristics tracing mechanism. Unlike many existing SL methods, the high order accuracy and mass conservation of the method are realized in a non-splitting manner. Thus, the detrimental splitting error, which could significantly contaminate long term Vlasov simulations, will be not incurred. One key ingredient in the scheme formulation is the use of Green's theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the numerical performance, we benchmark the proposed SLDG schemes for simulating several transport problems and the VP system. The efficiency and efficacy are numerically verified when compared with other prominent Vlasov solvers such as the Eulerian DG methods combined with Runge-Kutta time integrators.
image Thursday, October 26, 3:30pm, room Chem 107 (Colloquium)
Models of Hormone Treatment for Prostate Cancer: Can Mathematical Models Predict the Outcomes?
Yang Kuang, Arizona State University
Prostate cancer is commonly treated by a form of hormone therapy called androgen suppression. This form of treatment, while successful at reducing the cancer cell population, adversely affects quality of life and typically leads to a recurrence of the cancer in an androgen-independent form. Intermittent androgen suppression aims to alleviate some of these adverse affects by cycling the patient on and off treatment. Clinical studies have suggested that intermittent therapy is capable of maintaining androgen dependence over multiple treatment cycles while increasing quality of life during off-treatment periods. We present several mathematical models of prostate cancer growth to study the dynamics of androgen suppression therapy and the production of prostate-specific antigen (PSA), a clinical marker for prostate cancer. Biologically crude preliminary models were based on the assumption of an androgen independent (AI) cell population with constant net growth rate. These models gave poor accuracy when fitting clinical data during simulation. The biologically more refined models presented hypothesizes an AI population with increased sensitivity to low levels of androgen and these models generate high levels of accuracy in fitting clinical data. In general, we found that biologically more plausible models can forecast future PSA levels more accurately. These findings suggest that including more realistic mechanisms of resistance development may help predict the timing of androgen resistance.
image Wednesday, November 1
Operator Splitting and Lie Group Methods for Geometric Integration
Josh Padgett
Geometric integration is the discipline concerned with the discretization of differential equations while conserving exactly their invariants. The motivation for developing structure-preserving algorithms for certain classes of differential equations originates from diverse areas such as astronomy, molecular dynamics, mechanics, control theory, theoretical physics, and numerical analysis. If the equations of interest evolve on Lie groups, then geometric integration methods will guarantee that all approximations remain in the appropriate Lie group. In this talk we will consider Lie group methods for designing structure-preserving schemes with special emphasis on operator splitting methods. Operator splitting in a Lie group setting is primarily concerned with effectively approximating the exponential map and guaranteeing that such approximations map elements from the appropriate Lie algebra to the corresponding Lie group. The talk will consider concrete examples throughout, removing the need for a deep understanding of Lie group theory, and will conclude with several examples of how such methods may be applied to problems of interest.
image Wednesday, November 8
Mixed boundary value problems for degenerate elliptic equations at infinity (Joint Applied Math - Analysis Seminar)
Dat Cao
We study qualitative properties of the solutions of the Zaremba type problem in unbounded domains in non-divergent form. We obtain Phragmen-Lindelof type principle on the growth and decay of the solutions in the domain which is narrowing at infinity with respect to designated direction $x_1$. Equation is considered to be elliptic in the finite part of the domain but may be degenerate at infinity. Main results formulated in terms of the so called $s$-capacity of the Dirichlet portion of the boundary, with Neumann boundary satisfying certain ``admissibility'' condition. This is a joint work with Akif Ibragimov (Texas Tech University) and Alexander I. Nazarov (St. Petersburg Department of Steklov Institute).
image Thursday, November 9, 3:30pm
Distributed control and optimization with application to power systems (Colloquium)
Tao Yang
Network system is a fascinating research field that is evolving rapidly across many domains. The goal in networked control of multi-agent systems is to derive desirable collective behavior through distributed control algorithms based on local interaction with neighboring agents. In this talk, I will share some of my recent work on distributed control and optimization for network systems with applications in power systems. In particular, we consider an optimal coordination problem for distributed energy resources (DERs) including distributed generators and energy storage devices. We first propose an algorithm based on the push-sum and gradient method to optimally coordinate DER in a distributed manner. In the proposed algorithm, each DER only maintains a set of variables and updates them through information exchange with a few neighbors over a time-varying directed communication network. We show that the proposed distributed algorithm solves the optimal DER coordination problem if the time-varying directed communication network is uniformly jointly strongly connected, which is a mild condition on the connectivity of communication topologies. We then consider the case where the communication links suffer from the packet drops. For this case, we first show that the previously proposed distributed algorithm fails. We then equip this algorithm with the running sum method and show that this newly proposed algorithm is robust to packet-dropping communications and solves the optimal DER coordination problem almost surely.
image Thursday, November 16 (Colloquium)
Differential geometry, statistics, and gravitational waves
Ben Owen
Differential geometry is the mathematical foundation of Einstein’s general theory of relativity. Obviously it is crucial to gravitational waves, the hottest prediction of that theory these days, because it affects wave propagation and wave generation from regions of curved spacetime. Not so obviously, it is crucial to statistical methods to search large amounts of data for faint signals. I describe some of these obvious and not so obvious applications of differential geometry to the groundbreaking discovery of gravitational waves. TBD
image Thursday, November 30
Capturing Nonlinear Interaction between Fluids and Fiber-Reinforced Composite Materials: A Unified Mathematical Framework (Colloquium)
Suncica Canic, University of Houston
Mesh-reinforced structures and fiber-reinforced structures arise in many engineering and biological applications. Examples include air domes and space inflatable habitats, vascular stents supporting a compliant vascular wall, and aortic valve leaflets. In all these examples a mesh or a collection of fibers is used to support an elastic structure, and the resulting composite structure has novel mechanical characteristics preferred over the characteristics of each individual component. These structures interact with the surrounding deformable medium, e.g., blood flow or air flow, or another elastic structure. Modeling and computer simulation of this class of problems is important for manufacturing and design of novel materials, for a design of space habitats, and for a design of novel medical constructs. In this talk we present a unified approach to the study of mesh or fiber-reinforced shells and their interaction with incompressible, viscous fluids and 3D elastic structures. The approach is based on dimension reduction and operator splitting strategy. Mesh-like structures are modeled as mathematical 1D hyperbolic nets using dimension reduction. The resulting model is coupled to the elastodynamics of a membrane or a shell, and this mesh-reinforced shell model is then coupled to the flow of a viscous incompressible fluid and/or the elastodynamics of a 3D structure. Two-way coupling is enforced between all the different physical models in this multi-physics problem. An operator-splitting strategy is employed to solve the problems numerically, and in the main steps of the existence proof. This talk will provide an overview of the modeling, numerical method development and analysis of this class of problems. Numerical simulations will be presented to illustrate the results. This is a joint work with J. Tambaca, B. Muha and M. Bukac.