Applied Mathematics Seminar, Fall 2022

## Mondays 4:00 PM - 5:00 PM MT (unless otherwise noted), Hybrid format: In-Person in LCB 222 (unless otherwise noted), and Online (zoom information will be provided before the seminars)

• Please direct questions or comments about the seminar to Will Feldman (feldman (at) math.utah.edu) or to Akil Narayan (akil (at) sci.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizers for information about how to subscribe to this list.

August 29. In-Person
Speaker: Éduoard Oudet, Laboratoire Jean Kuntzmann, Université Grenoble Alpes
Title: Discrete Geometrical Tools in Shape Optimization
Abstract: The analysis and the fast computation of discrete structures such as Voronoi or Laguerre cells opened new fields in Optimization. More specifically, meshing tools based on centroidal Voronoi tessellation and the approximation of semi-discrete optimal transportation problems focused the attention in the last decade. We illustrate in this talk two new fields of application of the use of these discrete structures : the approximation of Blaschke-Santaló diagrams and Lagrangian shape optimization techniques.

September 12. In-Person
Speaker: Xavier Ros Oton, Departament de Matemàtiques i Informàtica, ICREA
Title: The singular set in the Stefan problem
Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points. The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.

September 19. Online
Speaker: Katy Craig, UCSB
Title: Graph Clustering Dynamics: From Spectral to Mean Shift
Abstract: Clustering algorithms based on mean shift or spectral methods on graphs are ubiquitous in data analysis. However, in practice, these two types of algorithms are treated as conceptually disjoint: mean shift clusters based on the density of a dataset, while spectral methods allow for clustering based on geometry. In joint work with Nicolás García Trillos and Dejan Slepčev, we define a new notion of Fokker-Planck equation on graph and use this to introduce an algorithm that interpolates between mean shift and spectral approaches, enabling it to cluster based on both the density and geometry of a dataset. We illustrate the benefits of this approach in numerical examples and contrast it with Coifman and Lafon’s well-known method of diffusion maps, which can also be thought of as a Fokker-Planck equation on a graph, though one that degenerates in the zero diffusion limit.

September 26. In-person
Speaker: Qi Tang, Applied Math and Plasma Physics Group, Los Alamos National Laboratory
Title: Scalable finite element algorithms and structure-preserving neural networks for fusion modeling
Abstract: This talk will discuss several advanced numerical algorithms related to fusion modeling. In the first part of the talk, we will discuss the development of a high-order stabilized finite-element algorithm for the reduced visco-resistive magnetohydrodynamic equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the Jacobian-free Newton-Krylov method with a physics-based preconditioning strategy. A parallel adaptive mesh refinement scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. In the second part of the talk, we will present several structure-preserving neural networks which can be viewed as machine-learning-based efficient surrogates to assist computations. Two architectures that preserve symplecticity and adiabatic invariants will be discussed. Their improvement over conventional numerical algorithm will be demonstrated through a practical application related to fusion modeling. This research is supported by DOE Advanced Scientific Computing Research (ASCR) and Fusion Energy Sciences (FES) programs.

October 3. In-Person
Speaker: Peter Bates, Michigan State University
Title: TBA
Abstract: TBA

October 17. In-Person
Speaker: Chiu-Yen Kao, Department of Mathematical Sciences, Claremont McKenna College
Title: TBA
Abstract: TBA

October 24. In-Person
Speaker: Zhaoxia Pu, Atmospheric Sciences, University of Utah
Title: TBA
Abstract: TBA

October 31. In-Person
Speaker: Natali Hritonenko, Department of Mathematics, Prairie View A&M University
Title: Optimization Analysis of Dynamic Models in Production and Environmental Economics
Abstract: Mathematical description and analysis of emerging problems related to environmental protection and industrial renovation under technological renovation and scientific innovations require development of new models and advanced methods of their investigation. The talk concentrates on vintage capital models, models with environmental adaptation and pollution mitigation, and models with carbon sequestration and rational forest management under different climate scenarios. The presence of unknowns in the integrand and limits of integration and complex interrelations among model variables significantly complicates investigation of considered problems. A qualitative analysis of models and associated optimal control problems leads to discovering turnpike trajectories, transition and long-term dynamics, replacement and anticipation echoes, zero-investment intervals, bang-bang structure, and other new phenomena and irregularities of solutions. Applied interpretation of all outcomes and their practical implementation are presented. A brief overview of other on-going research directions and future goals is given.

November 7. In-Person
Speaker: Sean Carney, Department of Mathematics, UCLA
Title: Modeling the fabrication of structured microparticles through aqueous two-phase separation
Abstract: Microscale particles--1 to 100 micrometer sized capsules--can enable the cheap and precise analyses of both single cells and individual molecules. Current research uses temperature-induced phase separation of aqueous polymer mixtures to efficiently fabricate microparticles of desired shapes and sizes. To better understand this process, we develop several mathematical models of microscale phase separation of ternary fluid mixtures. While the equilibrium configuration of the microparticles is described by a volume-constrained minimization of a Ginzburg-Landau free energy, a diffuse interface fluid mechanical model reveals a wide landscape of interesting parameter regimes; varying surface tensions, densities, viscosities, and concentrations can all influence the microparticle manufacture. We discuss some key features of a Cahn-Hilliard-Stokes model used to describe these effects, as well as its simulation with a pseudo-spectral method, and we highlight how fluid stresses can influence microparticle evolution.

Seminar organizers: Will Feldman (feldman (at) math.utah.edu), and Akil Narayan (akil (at) sci.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148