Applied Mathematics Seminar, Fall 2022

- 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 (*Paul Fife Memorial Lecture Series*)

**Title: **Reacting systems with nonlocal diffusion: Bifurcation and patterns

**Abstract: ** Many physical and biological processes occur with long-range interaction, giving rise to
equations with nonlocal-in-space operators in place of the usual Laplacian. These
operators are diffusive-like but are bounded rather than unbounded as is the case of
the local diffusion operator. We study systems that include such nonlocal operators and
through a spectral convergence result when a certain scaling parameter becomes small,
show that Turing instabilities also occur, producing patterned stable states. In the scalar
case, bifurcation is shown to occur when this parameter is small but many unanswered
questions persist.

** October 17. In-Person**

Speaker: Chiu-Yen Kao,
Department of Mathematical Sciences, Claremont McKenna College

**Title: **Maximal total population of species in a diffusive logistic model

**Abstract: ** We study the maximization of the total population of a single
species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.
This is a joint work with Seyyed Abbas Mohammadi.

** October 24. In-Person**

Speaker: Zhaoxia Pu,
Atmospheric Sciences, University of Utah

**Title: **Advancing Numerical Weather and Climate Prediction through the Synthesis of
Mathematics and Atmospheric Sciences

**Abstract: ** In the operational and research centers around the world, powerful supercomputers are
used to predict daily weather and project future climate by solving mathematical equations that
model the atmosphere and ocean. Several areas of mathematics play fundamental roles in
weather and climate prediction, including mathematical models and their associated numerical
algorithms, nonlinear computational optimization in very high dimensions, and parallel
computation. After decades of active research with the increasing power of supercomputers, the
forecast skill of numerical models has been improved significantly; yet, improving current
models and developing new models has always been an active area of research.

Operational weather and climate models are based on Navier-Stokes equations coupled
with various interactive earth components such as the ocean, land terrain, and water cycles. The
rapid development in the new technology of massively parallel computation platforms constantly
renews the impetus to investigate better mathematical models using traditional or alternative
numerical grids. However, initial conditions must be generated before one can compute a
solution for numerical prediction. Data assimilation combines observation data and numerical
models to generate initial conditions: its goal is to find the best estimate of the true state of
weather based on observations and prior knowledge (e.g., mathematical models, system
uncertainties, and observational errors); thus, it is treated as an optimization problem. Moreover,
due to inaccurate initial conditions and imperfect numerical models, ensemble forecasting has
been widely used in dealing with uncertainties. Ultimately, developing efficient and accurate
algorithms for numerical integration, data assimilation, and ensemble forecast at high dimensions
of weather and climate models is a long-term challenge for both mathematics and the
atmospheric sciences. The talk will emphasize some critical issues with a broad introduction and
discussion.

** 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.

** November 28. In-Person**

Speaker: Deepanshu Verma,
Department of Mathematics, Emory University

**Title: **Neural Network approached for high-dimensional Optimal Control

**Abstract: ** This talk presents recent advances in neural network approaches for approximating the value function of high-dimensional control problems. A core challenge of the training process is that the value function estimate and the relevant parts of the state space (those likely to be visited by optimal policies) need to be discovered. We show how insights from optimal control theory can be leveraged to achieve these goals. To focus the sampling on relevant states during neural network training, we use the Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. Our approaches can handle both stochastic and deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is self-supervised, in that, it does not require solutions of the control problem.

We will present several numerical experiments for deterministic and stochastic problems with state dimensions of about 100 and compare our methods to existing approaches.

`feldman (at) math.utah.edu`

), and
`akil (at) sci.utah.edu`

).
**Past lectures:**
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Fall 1995.

155 South 1400 East, Room 233, Salt Lake City, UT
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