Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
This talk is concerned with the Uniformization problem: which intrinsic qualities of a metric space allow a good parameterization by a
Euclidean space? In the first part of the talk, we consider geometrically good (conformal, quasiconformal) parameterizations. While
including a wide range of fractal examples, spaces as such enjoy geometric and analytic properties and a great amount of first-order
calculus can be performed on them. In the second part of the talk, we discuss measure-theoretically good (Lipschitz, Holder)
parameterizations. The problem of classifying spaces admitting such parameterizations is is one of the most important problems
in geometric measure theory and it is associated to the famous Traveling Salesman Problem.
Geometric PDEs are concerned with utilizing PDE techniques to study geometric problems. A general theme in my research is the investigation of equilibrium configurations with respect to natural quantities modelling the energy or entropy of geometric objects. Those equilibria enjoy several extremal properties that are usually described by elliptic PDEs. Consequently, understanding these equations would advance our knowledge about the associated geometries. In this talk, I'll describe my contribution to fundamental conjectures in the following concrete directions. First, we'll discuss PDEs on manifolds, exploiting elliptic equations that arise in special manifolds particularly in dimension four. The second direction focuses on geometric eigenvalue problems. Here the geometry of an object is examined through studying appropriate elliptic operators and their eigenvalues. Third, we'll talk about geometric flows and applications in which PDEs arise as a mechanism to change the shape of a manifold.
In this talk, we present our recent development of high order numerical methods on sparse grids. We first review
the sparse grid discontinuous Galerkin (DG) scheme for transport
equations and apply it to kinetic simulations. The method uses the weak formulations
of traditional Runge-Kutta DG schemes for hyperbolic problems and is proven to be $L^2$
stable and convergent. A major advantage of the scheme lies in its low computational
and storage cost due to the employed sparse finite element approximation space. This
attractive feature is explored in simulating linear and nonlinear transport problems including the celebrated Vlasov-Poisson/Maxwell System. We then discuss our very recent development on another novel high order sparse grid method, which is as effective and
efficient as the DG counterpart for box-shaped domains, but more flexible and easier to implement on
unstructured meshes. Some preliminary results are presented to demonstrate the efficiency of our new method.
In this talk I will present some preliminary result obtained from the modified chemotaxis model proposed by Angie Piece. I will present the primitive model of two-species interaction depending on light availability. The Proof of existence and uniqueness for the primitive solution will be presented. From the obtained estimates one can conclude time stability (but not asymptotic) of the constructed solution.
I naively hope that our approach can also highlights aspects of the evolution of the primitive live in reservoir, using PDE techniques. Most likely I'm very wrong! Most of the observation which will be presented are obtained by collaboration and joint discussion with Angie Piece and Eugenio Aulisa.
Multiple recurrent outbreak cycles have been commonly observed in
infectious diseases such as measles and chicken pox. These complex disease outbreak behaviors are rarely captured by deterministic models. In
this talk, we investigate a simple 2-dimensional SI model and propose that
the coexistence of multiple attractors attributes to complex outbreak patterns. We first determine parameter conditions for the existence of an
isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically
prove that the maximum number of the coexisting limit cycles is three,
and provide a corresponding parameter set for the existence of three limit
cycles. Simulation results demonstrate three coexisting attractors, which
contain one unstable endemic equilibrium and two stable limit cycles separated by one unstable limit cycle. Various disease dynamics can be predicted by a single nonlinear deterministic model based on different initial data.We develop and analyze a 2x2 dynamical system describing flow through a single pore to study the dynamics of the appearance and dissolution of gas bubbles during two-component (CO2, H2O), two-phase (gas, liquid) flow. Our analysis indicates that three regimes occur at conditions pertinent to petroleum reservoirs. These regimes correspond to a critical point changing type from an unstable node to an unstable spiral and then to a stable spiral as flow rates increase. Only in the stable spiral case do gas bubbles achieve a steady-state finite size. Otherwise, all gas bubbles that form undergo, possibly oscillatory, growth and then dissolve completely. Under steady flow conditions, this formation and dissolution repeats cyclically. In this talk I will: - present the mathematical model and derive the dynamical system,
- summarize the direction fields, critical points and solution trajectories,
- report the results of numerical solutions to the dynamical system
- provide summary critique of the work.
In recent years, fractional differential equations have become quite prevalent in applied mathematics. When used correctly, these non-local operators can model non-standard transport, such as anomalous diffusion, in many applications of interest (such as porous media). Approximations of fractional operators is still a highly nontrivial process as one must preserve the non-locality of the underlying operators in order of for the method to be valid. In this talk, we will introduce the notion of fractional powers of a class of abstract operators and construct appropriate approximations of these operators via a generalization of a method employed by Caffarelli and Silvestre. These techniques make no assumption of boundedness on the operators, and thus, may be employed in numerous numerical and analytical settings. The stability and convergence of this method can easily be related back to the spectral nature of the operator of interest. Numerical experiments will be presented to further verify the presented results.
The speaker will present a new numerical method which is devised and analyzed for a type of ill-posed elliptic Cauchy problems by using the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is robust and efficient in the sense that the system arising from the scheme is symmetric, well-posed, and is satisfied by the exact solution (if it exists). The speaker will show some numerical results to demonstrate the efficiency of the primal-dual weak Galerkin method as well as the accuracy of the numerical approximations.
In this talk, we will discuss a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions.
The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions.
Structural sensitivity is when the dynamics of a system is strongly dependent on the functional form used in model development. Recently, Cordoleani and colleagues have investigated a chemostat problem in which very similar metabolic response functions, all consistent with experimental data, could produce qualitatively different results (limit cycle or limit point). They introduced a derivative in model space that allowed quantification of local sensitivity to model structure.
We introduce a complementary method for quantification of structural sensitivity in a more global sense. Following Wiener and Feynman, we introduce a measure on the space of possible models. Integration in this measure -- a path integral -- allows assignment of probabilities to the qualitative outcomes. The path integrals can be weighted by a likelihood function based on experimental measurements. We analyze the chemostat problem of Cordoleani et al using this method, and discuss application of the method to model sensitivity problems in galactic dynamics.
This talk will be accessible to graduate and advanced undergraduate students, who will see how topics from analysis and several methods from numerical analysis (function approximation, finite elements for solution of boundary value problems, methods for solution of initial value problems, and numerical integration in high dimensional spaces) can be combined to solve complex problems.
I will discuss our recent work on developing a one-dimensional model for the transient (unsteady) fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it. An Euler--Bernoulli beam equation, which accounts for both transverse bending rigidity and nonlinear axial tension, is coupled to a one-dimensional fluid model obtained from depth-averaging the two-dimensional incompressible Navier--Stokes equations across the channel height. Specifically, the Navier--Stokes equations are scaled in the viscous (lubrication) limit relevant to microfluidics. The resulting set of coupled nonlinear partial differential equations is solved numerically through a segregated approach employing fully-implicit time stepping and second-order finite-difference discretizations. Internal FSI iterations and under-relaxation are employed to handle the stiff nonlinear algebraic problems within each time step. The Reynolds number $Re$ and a dimensionless Young's modulus $\Sigma$ are varied independently to explore the unsteady FSI behaviors in this parameter space. A critical $Re$ is defined by determining when the maximum steady-state deformation of the microchannel's soft wall exceeds a certain a priori threshold. Our numerical results suggest a universal scaling of this critical $Re$ scales as $\Sigma^{3/4}$. Furthermore, the maximum wall displacement and inlet pressures at steady state are shown to correlate with a single dimensionless group, namely $Re/\Sigma^{0.9}$. Finally, the linear stability of the final (inflated) microchannel shape is assessed via a modal eigenvalue analysis, showing the existence of many marginally stable modes, which further highlights the computational challenge of simulating unsteady FSIs.
The Willmore flow of surfaces is a geometric tool known for its aesthetic beauty, and has been studied by researchers including mathematicians, physicists, and computer graphics artists. This talk discusses a finite-element formulation of this flow for closed (possibly self-intersecting) surfaces immersed in $\mathbb{R}^3$, which is amenable to geometric constraints on both surface area and enclosed volume. Inspired by conformal geometry, a post-processing procedure is also presented which ensures that a given surface mesh remains nearly conformal along the Willmore flow despite its initial regularity. This abolishes the mesh degeneration that usually accompanies position-based surface flows, and leads to a robust model that can accommodate variable time steps as well as surface genera.