Applied Mathematics and Computation Seminars

Coupled Nonlinear Boundary Conditions in PDE-ODE Models discretized with Finite Elements: Analysis and Implementation 

Speaker: Tyler Fara

We study a nonlinear PDE-ODE system arising in bioheat modeling of localized cold exposure in which a parabolic heat equation is coupled through thermoregulatory exchange to an ODE governing core temperature, with nonlinear boundary fluxes modeling radiation, convection, and evaporation. We discuss a fully implicit finite element discretization and develop a nonlinear elliptic projection operator that accommodates the boundary coupling; we prove that this operator has uniform stability and approximation properties. These results lead next to optimal-order a priori error estimates for the backward Euler-Galerkin scheme. In the talk we focus on the construction and analysis of the projection operator, illustrate its behavior numerically, and apply it to the derivation of a priori error estimates. We illustrate with numerical results including those on multiscale modeling of the exchange terms. Read more.

TBA

Speaker: Nick Marshall

Read more.

TBA

Speaker: Kamrul Chowdury

Read more.

Comparison of Parameter Recovery Methods from Stochastic and Continuous Data Assimilation

Speaker: Elizabeth Carlson

ABSTRACT: One of the fundamental challenges of accurate simulation of turbulent flows is that initial data is often incomplete, which for said flows is a strong impediment to accurate modeling due to sensitive dependence on initial conditions. Data assimilation is the study of different methods used to adaptively correct models towards the data. These methods can be modified to solve the inverse problem of finding incorrect parameters. Parameter recovery is a popular research area, the methods for which, to the best of the speaker’s knowledge, almost exclusively use data uncertainty as a driving factor in finding such parameters. A continuous data assimilation method, known as the Azouani-Olson-Titi (AOT) or Continuous Data Assimilation (CDA) algorithm, introduced a linear feedback control term to dissipative systems, giving a simple and rigorous deterministic method by which to understand the underpinnings of more complex data assimilation algorithms used in the geosciences for, e.g… Read more.

No seminar today

Read more.

Structure-Preserving Algorithms for Hyperbolic Balance Laws and Related PDE-Based Models

Speaker: Yekaterina Epshteyn

ABSTRACT: Hyperbolic conservation/balance laws and related PDE-based models are essential mathematical apparatus for modeling a variety of complex physical phenomena, including but not limited to wave propagation, fluid flow, biological and materials science phenomena. Over the past few decades there has been enormous progress in designing stable, robust, structure-preserving numerical schemes (such as positivity-preserving and/or well-balanced schemes) that have enabled high-fidelity simulation of phenomena described by nonlinear hyperbolic PDEs. The main goal of our recent work is to extend these capabilities to systems with a stochastic component, which are relevant models in practical real-world situations since precise knowledge of an environment or operating conditions is frequently absent in such scenarios, and naturally results in models where randomness is used to describe the ignorance.In this talk, we will discuss progress in the design of structure-preserving numerical… Read more.

TBA

Speaker: Kyle Niemeyer

Read more.

TBA

Speaker: Malgorzata Peszynska

Read more.

TBA

Speaker: Praveeni Mathangadeera

Read more.

TBA

Speaker: Stefanie Fazekas and Eddie Ramos-Arteaga

Read more.

TBA

Speaker: Daniel Fust

Read more.