Physical Sciences and Mathematics Commons^™

Geometric Approach To Convex Subdifferential Calculus, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.

Transients Of Platoons With Asymmetric And Different Laplacians, Ivo Herman, Dan Martinec, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover, …

On The Coupling Of Dpg And Bem, Thomas Führer, Norbert Heuer, Michael Karkulik

Mathematics and Statistics Faculty Publications and Presentations

We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. An essential feature of our method is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical experiments to validate our theory.

On A Convex Set With Nondifferentiable Metric Projection, Shyan S. Akmal, Nguyen Mau Nam, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property.

A Tent Pitching Scheme Motivated By Friedrichs Theory, Jay Gopalakrishnan, Peter Monk, Paulina Sepulveda

Mathematics and Statistics Faculty Publications and Presentations

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

Dynamics Of Locally Coupled Oscillators With Next-Nearest-Neighbor Interaction, J. Herbrych, A. G. Chazirakis, N. Christakis, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

A theoretical description of decentralized dynamics within linearly coupled, one-dimensional oscillators (agents) with up to next-nearest-neighbor interaction is given. Conditions for stability of such system are presented. Our results indicate that the stable systems have response that grow at least linearly in the system size. We give criteria when this is the case. The dynamics of these systems can be described with traveling waves with strong damping in the high frequencies. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with …

Monomials And Basin Cylinders For Network Dynamics, Daniel Austin, Ian H. Dinwoodie

Mathematics and Statistics Faculty Publications and Presentations

We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner fan that finds monomials that define cylinders and the other on primary decomposition. Both methods are applied to current examples of gene networks.

Stability Of A Circular System With Multiple Asymmetric Laplacians, Ivo Herman, Dan Martinec, J. J. P. Veerman, Michael Sebek

Mathematics and Statistics Faculty Publications and Presentations

We consider an asymptotic stability of a circular system where the coupling Laplacians are different for each state used for synchronization. It is shown that there must be a symmetric coupling in the output state to guarantee the stability for agents with two integrators in the open loop. Systems with agents having three or more integrators cannot be stabilized by any coupling. In addition, recent works in analysis of a scaling in vehicular platoons relate the asymptotic stability of a circular system to a string stability. Therefore, as confirmed by simulations in the paper, our results have an application also …

Periodic State Revivals In Commensurate Waveguide Arrays, Jovan Petrovic, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Emerging optical and quantum computers require hardware capable of coherent transport of and operations on quantum states. Here, we investigate finite optical waveguide arrays with linear coupling as means of efficient and compact coherent state transfer. Coherent transfer with periodic state revivals is enabled by engineering coupling coefficients between neighbouring waveguides to yield commensurate eigenvalue spectrum. Particular cases of finite arrays have been actively studied to achieve the perfect state transfer by mirroring the input into the output state.

We explore a much wider scope of coherent propagation and revivals of both the state amplitude and phase. We analytically solve …