Optimization and Reduced Order Models for Digital Twins

This talk begins by discussing the role of PDE-constrained optimization in the development of digital twins. In particular, applications to identify weaknesses in structures and aneurysms are considered. Next, we analyze a data-driven optimization problem constrained by Darcy’s law to design a permeability that achieves uniform flow properties despite having nonuniform geometries. We establish well-posedness of the problem, as well as differentiability, which enables the use of rapidly converging, derivative-based optimization methods.

Let's count things

Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves.  Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields, which though basic objects of study in number theory, are quite difficult to actually count.  How often does a random polynomial fail to have full Galois group?  How many number fields are there?  We will address both of these questions today.

The Apportionment Problem for the U.S House of Representatives

We focus on the history and the mathematics of the apportionment problem for the US House of Representatives.  An apportionment is a function from {1, 2, ..., s} (where s is the number of states) to the positive integers, A(i), so that the sum of the A(i) is H, the house size.  Today s=51 (including the District of Columbia) and H=435, although s and H have had many values since 1790. Based on census data, one can compute the fair share of representatives for state i, call it f_i, which might turn out to be 4.69435.

Approximating Nonlinear Feedback Controls for Polynomial Systems

The calculation of optimal feedback controllers for nonlinear systems remains elusive since it requires the solution or approximation of the Hamilton-Jacobi-Bellman equations.  By restricting our attention to quadratic regulator problems and polynomial systems, we are able to calculate polynomial feedback laws for systems with hundreds of states.  We describe our approximation algorithm, which relies on introducing a Kronecker structure and provide examples for discretized PDEs such as Burgers, Chafee-Infante, Kuramoto-Sivashinsky, etc.

Scalable High-Order Finite Elements for Compressible Hydrodynamics

The discretization of the Euler equations of gas dynamics (“compressible hydrodynamics”) in a moving material frame is at the heart of many multi-physics simulation codes. The Arbitrary Lagrangian-Eulerian (ALE) framework is frequently applied in these settings in the form of a Lagrange phase, where the hydrodynamics equations are solved on a moving mesh, followed by a three-part “advection phase” involving mesh optimization, field remap and multi-material zone treatment.

Limit shapes in the Abelian Sandpile Model

 In a series of 2-3 talks we will discuss the abelian sandpile
model (ASM for short). The ASM is a growth model in which grains of sand
are placed at the vertices of a graph, and they spread according to a
toppling rule. A site topples if its amount of sand is at least as large
as the degree of the underlying vertex, in which case it sends one grain
of sand to each neighbor.

The first talks will consist of an informal introduction to the model,

Graphs and Networks Seminar

We focus this term on graph expanders, Ramanujan graphs, optimal connected graphs and Schur-convex functions, maximal and minimal complexity, networks that quickly induce brain synchrony, and number theoretic conjectures associated to Hadamard codes. We may or may not cover all this but we are in no rush. The spirit of the Seminar is to give several talks introducing the research topic, present some fundamental results, then lead to new research and open questions.