PDE Seminar
University of Houston
For further information or to suggest a speaker for the PDE Seminear contact:
Misha Perepelitsa (maperepelitsa AT uh DOT edu)
Gabriela Jaramillo (gtjaramillo AT uh DOT edu)
William Fitzgibbon (fitz AT uh DOT edu)
Speaker: Jeff Morgan (University of Houston)
Title: Investigations of Reaction-Diffusion Systems with Mixed Local and Nonlocal Diffusion
Speaker: Andrea Welsh (Texas A & M)
Title: Patterns in Physiology- A study in Wave and Wave Break-Up
Abstract:
Modeling in physiology involves describing a wide array of biological processes and determining how each part fits together to reproduce the complex spatiotemporal dynamics that are observed in experiments. In this talk, I will discuss two systems and their interesting dynamical processes: colon motility and cardiac excitation. Colon motility, the spontaneous self-generated movement and motion of the colon muscle and its cells, is produced by activity in different types of cells such as myenteric neurons of the enteric nervous system (ENS), smooth muscle contraction, and interstitial cells of Cajal (ICC). Two colon motor patterns measured experimentally are colonic motor complexes (CMC) often associated with the propulsion of fecal contents, and ripple contractions which are involved in mixing and absorption. We propose a new model that reproduces the global dynamics observed in optogenetic and calcium measurements of mouse colons. This study presents strong evidence from experimental and computational approaches that cyclical ENS-ICC interactions form the intrinsic pattern generator for rhythmic CMC initiation in the proximal colon. Second, cardiac models have been well studied to reproduce not only the coordinated electrical waves that are necessary to contract and pump blood but also understand the breakdown of these waves into chaos which appears as spiral waves of activity causing the heart no longer to be able to pump blood. The wave and spiral wave breakup often occur due to biological properties such as cell refractory periods but also due to scarred tissue. We investigate the FitzHugh Nagumo model, which can reproduce important dynamical features in cardiac dynamics, with an added extension which creates interesting dynamics when coupled in space due to its bi-stability. This model can be useful for further investigation into wave break up due to barriers in tissue.
Speaker: Tracy Stepien (University of Florida)
Title: Analyzing Model Reduction and the Existence of Traveling Waves and Pulses in Astrocyte Migration During Retinal Development
Abstract:
In embryonic development, formation of blood vessels in the retina of the eye is critically dependent on prior establishment of a mesh of astrocytes. Astrocytes emerge from the optic nerve head and then migrate over the retinal surface in a radially symmetric manner, and they mature from astrocyte precursor cells (APCs) to immature perinatal astrocytes (IPAs) via differentiation. We develop a continuum mechanical PDE model describing the migration and differentiation of astrocytes that directly incorporates data from the rat experimental model. We use approximate Bayesian computation to identify model components that can be removed via model reduction. Informed by the experimentally observed and numerically simulated spatial distribution of APCs and IPAs, we explore the existence of traveling waves and pulses.
Speaker: Olena Burkovska (Oak Ridge National Laboratory)
Title: Spatial and temporal nonlocality in multiscale systems: modeling, discretization, and learning
Abstract:
Many multiscale systems exhibit interactions that are nonlocal in both space and time. This talk focuses on two instances of such nonlocality: spatial nonlocality in phase-field modeling and its efficient numerical realization, and temporal nonlocality through memory effects arising from reduction of high-dimensional systems, where the memory kernel must be inferred from data.
In the first part, we focus on nonlocal phase-field models of Allen-Cahn and Cahn-Hilliard type with non-smooth obstacle potentials. Special structure of the model and nonlocality allows solutions to admit perfectly sharp interfaces without requiring fine spatial resolution but introduces additional computational challenges. We develop energy-stable time-stepping schemes combined with Fourier-based spatial discretization, together with proximal-based representations that allow to bypass the solution of nonlinear variational inequalities. We also discuss a neural network discretization that learns the fully discrete evolution operator for such non-smooth nonlocal systems, incorporating nonlocal interactions directly into the architecture.
In the second part, we turn to temporal nonlocality arising in reduced-order modeling of high-dimensional dynamical systems. Eliminating unresolved degrees of freedom leads to non-Markovian dynamics governed by generalized Langevin equations with memory kernels. We propose to recover the kernel from trajectory data by solving a regularized optimization problem in function space, with the kernel parameterized as a Prony series and sparsity-promoting regularization enforcing robustness to noise and parsimony of the representation. The sparsity of the learned kernel naturally defines a low-dimensional Markovian embedding of the dynamics. We demonstrate the approach on molecular dynamics benchmarks.
Speaker: Manuela Girotti (Emory University) Online via Zoom
Title: Random solitons and soliton gases
Abstract: The concept of a soliton gas was originally introduced by Zakharov in the Seventies and it can be loosely described as a class of solutions of nonlinear integrable PDEs that displays an infinite number of -possibly random- solitons. I will present a collection of recent results from a collaborative effort to rigorously understand various aspects of several soliton gas models: long-time asymptotics, random statistics, and interaction dynamics. In particular, we consider a gas of random N-solitons for the KdV equation and we study the limit as N goes to infinity. We derive the limiting solution and we prove that its fluctuations are Gaussian random variables. The starting point is the formulation of a multi-soliton solution and of a regular, dense soliton gas solution in terms of Riemann–Hilbert problems. Tools from asymptotic analysis and probability will be used to provide a detailed description of interesting phenomena.
Speaker: Andrei Tarfulea (Louisiana State University) Online via Zoom
Title: Decay Estimates and Smoothing for the Relativistic Landau Equation
Abstract: We consider the spatially inhomogeneous relativistic Landau equation in the far-from-equilibrium regime; an equation modeling gases and plasmas at very high energies and low density, outside the usual regime of hydrodynamics. The relativistic nature of the problem makes the PDE more difficult to analyze compared to the classical Landau equation, but relatively recent breakthroughs indicate that it should ultimately still behave like a degenerate nonlocal parabolic problem with some hypoellipticity. We aim to build a complete theory of local well-posedness, and as first steps in that direction we prove a priori propagation of momentum-decay and parabolic smoothing for the equation. This talk will focus on a few of the specific challenges posed by the relativistic problem, including Schauder estimates adapted to be Lorentz invariant and how to estimate the non-convolutional singular integrals that create the diffusion operator.
Speaker: Christian Glusa (Sandia National Laboratories)
Title: Scalable methods for nonlocal models
Abstract: The naive discretization of nonlocal operators leads to matrices with significant density, as compared to classical PDEs. This makes the efficient solution of nonlocal models a challenging task. In this presentation, we will discuss ongoing research into efficient hierarchical matrix assembly and geometric and algebraic multigrid preconditioners that are suitable for nonlocal models.
Speaker: Tadele Mengesha (The University of Tennessee, Knoxville)
Title: Variational Analysis of a Parametrized Family of Transmission Problems Coupling Nonlocal and Fractional Models
Abstract: I will present a work that examines the coupling of a model based on the regional fractional Laplacian and a nonlocal model employing a position-dependent interaction kernel. Both operators are inherently nonlocal and act on functions defined within their respective domains. The coupling occurs via a transmission condition across a hypersurface interface. The heterogeneous interaction kernel of the nonlocal operator leads to an energy space endowed with a well-defined trace operator. This, combined with well-established trace results of fractional Sobolev spaces, facilitates the imposition of a transmission condition across an interface. The family of problems will be parametrized by two key parameters that measure nonlocality and differentiability. For each pair of parameters, we demonstrate existence of a solution to the resulting variational problems. Furthermore, we investigate the limiting behavior of these solutions as the parameters approach their extreme.
Speaker: Joshua Siktar (TAMU)
Title: Existence of Solutions for Fractional Optimal Control Problems with Minimax Constraint
Abstract:
In this talk we prove the existence of solutions to an optimal control problem where the constraint is an ill-posed, nonlinear equation containing a Fractional Laplacian. For any fixed control data, the constraint equation is known to have multiple solutions by a previous application of the Mountain Pass Theorem. Due to the pointwise nature of the conditions on controls needed to invoke this theorem, we must make substantial adaptations to the usual direct method of calculus of variations in order to prove our main existence result. The main theoretical tools are a thoughtful construction of an admissible set of controls, and a technical lemma that ensures that a minimizing sequence of pairs of controls and corresponding states exhibits convergence to another control-state pair that satisfies the constraint equation.
Speaker: Jimmie Adriazola (SMU)
Title: Data Driven Integrability and Wave Camouflaging
Abstract: This talk will consist of two separate parts both involving some of my recent numerical optimization projects. The first involves the detection of integrability in Hamiltonian dynamical systems. We formulate the discovery of integrability as a numerical optimization problem. That is, we numerically solve the problem of maximizing the compatibility between a pair of Lax operators with the known Hamiltonian of the dynamical system. Our approach is tested on finite degrees of freedom before moving on to the more challenging PDE setting. We find that our approach is robust against non-integrable perturbations of the Hamiltonian and find that, in all examples, our approach reliably confirms or denies the integrability of the dynamics. This is joint work with Panos Kevrekidis (UMass Amherst), Alejandro Aceves (SMU), and Wei Zhu (GT).
I will discuss the problem of camouflaging electromagnetic waves in the second part of the talk. We formulate the problem in terms of optimal control theory as we seek to find a current density that destructively interferes with an outgoing signal subject to Maxwell's equations. I will discuss some of the numerical and functional analytic challenges that can appear in this problem and show how to resolve them to achieve optimal results. This is joint work with Denis Ridzal (Sandia) and Harbir Antil (GMU).