## Computational and Applied Mathematics Seminar

**Fall 2018**

**Mondays at 4:10 p.m. in 401 Carver
**

The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.

Archive of previous CAM seminars

11/12 (Cancelled, To Be ReScheduled)

**Title: TBD **

**Michael Catanzaro **

Iowa State University

**ABSTRACT:** TBD

11/26

**Title: TBD**

**Maurice S. Fabien **

Rice University

**ABSTRACT:** TBD

12/03

**Title: Stability Properties of Nondissipative Compressible Flow-Structure PDE Models**

**Pelin Guven Geredeli **

University of Nebraska–Lincoln

**ABSTRACT:** In this talk, we present recently derived results of uniform stability for a coupled partial differential equation (PDE) system which models a compressible fluid-structure interaction of current interest within the mathematical literature. The coupled PDE model under discussion will involve a linearized compressible, viscous fluid flow evolving within a 3-D cavity, and a linear elastic plate--in the absence of rotational inertia—which evolves on a portion of the fluid cavity wall. Since the fluid equation component is the result of a careful linearization of the compressible Navier-Stokes equations about an arbitrary state, this interactive PDE component will include a nontrivial ambient flow profile, which tends to complicate the analysis. Moreover, there is an additional coupling PDE which determines the associated pressure variable of the fluid-structure system. Under a suitable assumption on the ambient vector field, and by obtaining an appropriate estimate for the associated fluid-structure generator on the imaginary axis, we provide a result of exponential stability for finite energy solutions of the fluid-structure PDE system.

8/20

**Applied Mathematics Orientaiton **

8/27

**Two-phase Flow in Porous Media with Hysteresis**

**Haitao Fan **

Georgetown University

**ABSTRACT:** Two-phase flow through a porous medium with hysteresis effects is considered. The model consists of a system of two coupled nonlinear equations: a transport equation for the water saturation and an evolution equation for the hysteresis variable. The latter is not in conservation form and contains discontinuous functions of the two unknown variables as coefficients. Some qualitative properties of piecewise smooth solutions of the system are proved. In particular, the hysteresis variable is shown to satisfy a maximum principle, and that its total variation is bounded by the total variation of its initial value. The traveling waves are investigated under the assumption that the convective term is convex. Riemann solvers for the inviscid system are constructed. Non-uniqueness due to hysteresis loops is finally discussed; several solutions are discarded by the maximum principle for the hysteresis variable.

9/03

**Labor day **

9/10

**Canalizing functions and their impact on the
robustness of gene regulatory networks**

**Claus Kadelka **

Iowa State University

**ABSTRACT:** Gene regulatory networks (GRNs) are a collection of genes and other molecules that govern what is happening within a cell. These networks are surprisingly robust to noise and mutations. GRNs are frequently modeled using finite (i.e., time- and state-discrete) dynamical systems, in the simplest case Boolean networks. I present a collection of mathematical and computational tools for the study of robustness of finite dynamical systems. The focus is on networks governed by Boolean k-canalizing functions, a recently introduced class of Boolean functions that is well-suited to model gene regulation. Two measures, the activity and sensitivity
of a function quantify the impact of input changes on the function output. I present a generalization of the latter concept, called c-sensitivity, and provide formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular
measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of generalized Boolean networks, deterministic or stochastic, that does not require simulation.

9/17

**Equilibrium Distributions of Populations of
Biological Species on Biological or Social Networks**

**Zhijun Wu **

Iowa State University

**ABSTRACT:** Under certain principles, populations of biological
species often spread in given biological or social networks in ways so they can
increase (or in some cases, decrease) their biological or social interactions.
We investigate the distributions of such populations on given networks, which are of great research interest in biological and social sciences.
We formulate the problem as an evolutionary game with
the payoff function defined in terms of the network connection matrix multiplied by
possible weights on the nodes or links of the network. We derive network conditions for equilibrium distributions and
their evolutionary stabilities. We show that the
distribution on a subnetwork is equilibrium when any changes
in the distribution would reduce the interactions in the population. Such a distribution may or may not be on a network clique, but cliques support more stable distributions than non-clique subnetworks. These
results provide important theoretical insights into how
populations would distribute over biological or
social networks and whether they are evolutionarily stable.
This is joint work with former students Min Wang and Wen Zhou.

9/24

**Title: Pest Control Via "Additional" Food**

**Rana Parshad **

Iowa State University

**ABSTRACT:** Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the predators efficacy can be boosted by providing it with an additional food source. In the current literature it is proved that if the additional food is of sufficient constant quantity and quality then pest eradication is possible in finite time. We show to the contrary that pest eradication is not possible in finite time, and will occur only in infinite time to which end we derive decay rates. However for a density dependent quantity of additional food, we show pest eradication in finite time is indeed possible. We also consider the case of both predator and prey evolution taking place, and its effect on the dynamics of the system. Our results have many consequences for designing effective biological control strategies.

10/01

10/08

**An embedding theorem: geometric analysis behind data analysis**

**Chen-Yun Lin **

Duke University

**ABSTRACT:** High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis.

In this talk, I will present a theoretical analysis of the VDM for its mathematical foundation. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

10/15

10/22

10/29

**Title: Rotation-invariant features and orbits of the regular representation**

**Joseph Iverson **

Iowa State University

**ABSTRACT:** Data sets of three-dimensional objects often feature measurements taken without a common reference for alignment. This rotational freedom poses a significant barrier to data analysis, especially in the high-noise regime. We connect this issue with a fundamental problem in abstract harmonic analysis, and describe a partial solution.
This is a preliminary report of joint work with Dustin Mixon and Tom Needham.

11/05

**Title: Symmetric direct DG (DDG) method for elliptic interface problems **

**Jue Yan **

Iowa State University

**ABSTRACT:** We first review recent development and advantages of direct discontinuous Galerkin (DDG) method on: (1) 3rd order bound-preserving and (2) super convergence to solution’s gradient approximation. We then discuss recent studies of symmetric DDG method to elliptic interface problems with none zero solution jump and none zero flux jump interface conditions. We focus on the case that mesh is partitioned aligning with the curved interface and on high order approximations. Numerical flux is carefully designed to incorporate the given jump conditions. Optimal order error estimate under energy norm is obtained with $P^k$ polynomial approximations. A sequence of numerical tests are carried out to illustrate the optimal $(k+1)th$ order convergence of symmetric DDG method and its capability to uniformly handle different interface conditions and complicated geometries.