Applied Mathematics Seminar, Spring 2024

- Please direct
**questions or comments**about the seminar to**Will Feldman**(`feldman (at) math.utah.edu`

), or**Akil Narayan**(`akil (at) sci.utah.edu`

), or to**Kshiteej Deshmukh**(`kjdeshmu (at) math.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.

**Wednesday, January 10 at 4pm. In-person, LCB 222**

Speaker: Justin Baker,
Department of Mathematics, University of Utah

**Title: **Monotone Operator and Representation Theoretic Geometric Deep Learning for Efficient and Reliable Molecular Modeling

**Abstract: **Molecular modeling via machine learning plays an essential role in recent advancements in medical imaging, drug discovery, and composite materials. Neural networks serve as efficient surrogate models, offering alternatives to the traditionally expensive ab-initio simulations. Among these, Message Passing Neural Networks (MPNNs) stand out due to their efficient computational design and expressive power in capturing intricate atomic interactions. In this presentation, we will explore E(3)-equivariant MPNNs which maintain the inherent symmetries of atomic structures, and implicit MPNNs, adept at modeling long-range atomic interactions. We will delve into the advancements of these models, drawing from topological insights, and showcase how these developments enrich our understanding and approach to modeling complex atomic configurations. Through comparative analysis with conventional models, we underscore the enhanced predictive capabilities of these specialized MPNNs.

**Wednesday, January 31 at 4pm. In-person, LCB 222**

Speaker: Raghav Venkatraman,
Courant Institute, New York University

**Title: **The geometry of data through the lens of homogenization

**Abstract: **The ``manifold hypothesis'' posits that many high dimensional data sets that occur in the real world actually are actually scattered on or around a much lower dimensional manifold embedded in the high dimensional space. Estimating attributes of this "ground truth" manifold from finitely many samples (point cloud) is a problem of statistical inference. Given such a point cloud that is modelled as an independent and identically distributed (i.i.d) sample from a (nice) density on a closed manifold, over the past decade there is a body of literature which considers the question: forming a random geometric (weighted) graph on the point cloud (by, for example, joining points that are within a threshold distance by a weighted edge) how well can one estimate the spectrum (eigenvalues, eigenfunctions) of the (weighted) Laplace-Beltrami operator on the ground truth manifold, from that of the graph laplacian associated with the random geometric graph?

After introducing the problem, we will show how this question is one of stochastic homogenization. Warming up with results that are new even for the classical "periodic" homogenization problem, we will describe how one can obtain optimal convergence rates for the spectrum of the graph laplacian using tools from the recent theory of quantitative stochastic homogenization. Briefly: borrowing tools from percolation theory, the argument proceeds by ``coarse-graining'' the random geometry in the problem to large scales, where the environment "appears Euclidean". Then, one adapts arguments from the quantitative theory of homogenization.

This talk represents joint work with Scott Armstrong (Courant).

**Monday, February 5 at 4pm. Virtual over Zoom**

Speaker: Debdeep Bhattacharya,
Department of Mathematics, University of Utah

**Title: **Nonlocal approaches to fracture in continuum and granular media

**Abstract: **Modeling and predicting the evolution of fracture in solids and granular material is of utmost importance in many biological, geophysical, and mechanical applications. Understanding how materials fail under various loading conditions can lead to sustainable design practices by reducing waste. A recent development in modeling deformations in continuum media is the use of a nonlocal approach, specifically peridynamics, which can accurately model the initiation and propagation of fractures as an emergent behavior. In this approach, the displacement field exhibits localized softening zones as jump sets associated with discontinuities in the solution.

We focus on the quasistatic or rate-independent evolution of fracture. Using a fixed-point argument, we prove well-posedness of solutions near the local minima of the energy and show the existence of a stable load path in the suitable Banach space. A numerical scheme involving the tangent stiffness matrix is used to solve the evolution problem and quadratic convergence is established. Next, we introduce the PeriDEM method, a coupling approach between peridynamics and the discrete element method, to model the effect of fracture, grain topology, and deformability in granular media. We describe the massively parallel computational platform perigrain for high-fidelity simulations of large granular aggregates and discuss various applications such as determining bulk strength of brittle granular aggregates, vehicle mobility on dry gravel beds, the hopper flow problem, and the modeling of sea-ice floe dynamics.

**Monday, February 26 at 4pm. In-person, LCB 222**

Speaker: Brendan Keith,
Division of Applied Mathematics, Brown University

**Title: **Proximal Galerkin: A Structure-Preserving Finite Element Method For Pointwise Bound Constraints

**Abstract: **
The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite- dimensional function spaces. In this talk, we will introduce the proximal Galerkin method and apply it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The proximal Galerkin framework is a natural consequence of the latent variable proximal point (LVPP) methodology, which is a stable and robust alternative to the interior point method that will also be introduced in this talk. LVPP can be viewed as a low-iteration complexity, infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout the talk, we will arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and an infinite-dimensional Lie group; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This talk is based on [1].

[1] B. Keith, T.M. Surowiec. Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints arXiv preprint arXiv:2307.12444 2023.

**Monday, March 11 at 4pm. In-person, LCB 222**

Speaker: Yuan Gao,
Department of Mathematics, Purdue University

**Title: **Optimal Control and transition path theory for Markov Jump Processes

**Abstract: **Transition paths connecting metastable states in a stochastic model are rare events which are fundamental but appear with small probability. In this talk, I will present a stochastic optimal control formulation for transition path problems in an infinite time horizon, specifically for Markov jump processes on Polish spaces. An unbounded terminal cost at a stopping time, along with a controlled transition rate, regulates the transitions between metastable states. To maintain the original bridges after control, the running cost adopts an entropic form for the control velocity, contrasting with the quadratic form typically used for diffusion processes. Via the Girsanov transform, this optimal control problem can be framed within a unified approach - converting to an optimal change of measures in càdlàg path space. The unbounded terminal cost however leads to a singular optimal control and brought difficulties in the Girsanov transform. Gamma-convergence techniques and passing limit in the corresponding Martingale problem allow us to obtain a singular optimally controlled transition rate. We demonstrate that the committor function, which solves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. The optimally controlled process realizes the transition paths almost surely but without altering the bridges of the original process.

**Monday, March 18 at 4pm. In-person, LCB 222**

Speaker: Kimberly Ayers,
Department of Mathematics, California State University San Marcos

**Title: **The Stochastic Logistic Map: Theory and Applications

**Abstract: **When dynamicists refer to the logistic map, they are often talking about a one parameter family of maps in one dimension. The logistic map is a famous example of a discrete, chaotic dynamical system in one dimension. The behavior of the logistic map gets qualitatively more complex as its parameter increases, as demonstrated by its well known bifurcation diagram. The logistic map exhibits a period doubling cascade, and then demonstrates a stable period 3 orbit, before entering the "chaotic regime." In addition to its theoretical properties, the logistic map is well known in population dynamics to model the growth of a population with limited resources. In this context, the aforementioned parameter represents the reproduction rate of the population. In this talk, we will consider what happens when this growth rate, rather than remaining fixed, varies stochastically over time. We will explore this stochastic process from a stability perspective as well as a mathematical biological viewpoint.

`feldman (at) math.utah.edu`

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

), and
`kjdeshmu (at) math.utah.edu`

).
**Past lectures:**
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155 South 1400 East, Room 233, Salt Lake City, UT
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