Branwen Purdy prepares hands-on activities for kids at the OMSI Meet-A-Scientist Day in Portland, to share hands-on learning experiences about her research in topological data analysis.

Join us for these events hosted by the Department of Mathematics, including colloquia, seminars, graduate student defenses and outreach, or of interest to Mathematicians hosted by other groups on campus.

We consider a nonlinear coupled system of parabolic variational (in)equalities modeling biofilm– nutrient dynamic in porous media coupled to Brinkman flow model. Microbial growth in porous media is affected by nutrient supply and ambient fluid flow. It is also constrained by available space. We approximate the system using a mixed finite element method of the lowest order of the Raviart-Thomas element due to the low regularity of solutions of parabolic variational inequalities. We show the well-posedness of the discrete model and derive an optimal error estimate of the first order, which is validated experimentally. Moreover, we conduct 2D simulations to investigate the behavior of biofilm growth in simple and complex pore-scale geometries.This is joint work with Prof. Malgorzata Peszynska from Oregon State University and Dr. Choah Shin from Ab Initio Software. Read more.

ABSTRACT: Liquid crystals are important materials that are used in several technological applications. The most common usage is in the omnipresent liquid crystal displays which uses the birefringence property of the material to create images on a screen. However, liquid crystals also respond to other external stimuli, e.g. magnetic, mechanical, chemical, which can be used to induce complex shape changes in the material used for applications in biomedical devices, robotics, optics, textiles, and sensors.Liquid crystals exhibit intermediate phases between solid and liquid. One such phase is the nematic phase which possess the microscopic orientational order of a crystalline solid, however, the molecules have no positional order but flow freely past each other and thus display macroscopic properties of a liquid. Models of liquid crystals usually represent molecules as rods or disks and use some parameter to describe the orientation of the molecules.In this presentation I will discuss the… Read more.

We will talk about some of the issues that arise in solving eigenvalue problems numerically, and introduce some recent ideas on how to address them. First we briefly demonstrate why we want to use methods different from how we first learn to solve eigenvalue problems by hand in linear algebra. Then we will talk about the power method, one of the most basic and powerful but sometimes very slow iterative methods. We will introduce momentum-type extrapolation methods which recycle older information in our approximation sequence to accelerate an iterative method. Finally, we will show some recent results on accelerating the power method using a dynamically chosen momentum term. Read more.

ABSTRACT: The momentum balance equation for glacier flow can be expressed as a convex optimization problem for the ice velocity. The geometry of real glaciers, however, evolves in time as the terminus advances and retreats. The orthodox approaches to solving this free boundary problem are to either (1) set a minimum thickness and accept the resulting mass balance errors, or (2) apply the level-set method or related approaches that "turn off" the physics where the ice thickness is zero. Both of these approaches have their drawbacks. Here, I'll show an alternative formulation of the momentum balance equation that I derived using the principle of convex duality. This dual form adds the stresses as unknowns, but it remains solvable even when the ice thickness is equal to 0. Solvability at zero thickness allows us to simulate terminus advance and retreat -- including iceberg calving -- in a simple way. I'll show some numerical demonstrations and discuss what solution methods were viable. I… Read more.

ABSTRACT: Dense kernel matrices arise in a broad range of disciplines, such as potential theory, molecular biology, statistical machine learning, etc. To reduce the computational cost, low-rank or hierarchical low-rank techniques are often used to construct an economical approximation to the original matrix. In this talk, we consider general kernel matrices associated with possibly high dimensional data. We perform analysis to provide a straightforward geometric interpretation that answers a central question: what kind of subset is preferable for skeleton low-rank approximations. Based on the theoretical findings, we present scalable and robust algorithms for black-box dense kernel matrix computations. The efficiency and robustness will be demonstrated through experiments for various datasets, kernels, and dimensions, including benchmark comparison to the state-of-the-art packages for N-body simulations.BIO: Difeng Cai received his BS in math from University of Science and Technology… Read more.

ABSTRACT: At the terminus of tidewater glaciers an interplay of connected processes result in the melt of ice. From both field and laboratory observations, it has been suggested that both bubbles and sediments could be important yet neglected contributors to ice melt at the submarine tidewater glacier terminus. In the laboratory it has been shown that as glacier ice melts, air trapped in pores inside of the ice is released creating flow transpiration at the boundary and buoyant bubble rise at the ice-ocean interface, leading to increased melt (Wengrove et al.,2023). Additionally, during separate laboratory experiments, sediments entrained in the subglacial discharge plume are shown to increase the entrainment of warm ocean water toward the ice leading to higher melt rates (McConnochie andCenedese, 2023). In July 2024, we made the first ever video observations of both bubbles rising and sediments mixing and falling from a stationary-bolted platform to an Alaskan tidewater glacier… Read more.