Spring 2024

4/17/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Michael Kuian
Title: Solution of Linear Ill-Posed Problems by Modified Truncated Singular Value Expansion
Abstract: The numerical solution of linear ill-posed problems generally requires incorporation of regularization to yield a meaningful approximate solution. A common approach to compute a regularized approximate solution is to apply the truncated singular value expansion of the operator. A modification of the truncated singular value expansion for linear discrete ill-posed problems in finite dimensions was shown to furnish approximate solutions of higher quality than the standard truncated singular value expansion. This work extends the modified singular value expansion to ill-posed problems in a Hilbert space setting.

4/10/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. David Gurarie
Title: Modeling atmospheric chemistry-mixing via moment closure scheme
Abstract: Multiple chemical species are released into atmosphere from the earth surface and aloft, where they undergo turbulent mixing and a chain of reactions. Some of them (ozone, nitric oxides) are major air pollutants, subject to air quality control. To understand and predict mixing-chemistry dynamics different modeling approaches are utilized, ranging from simple ‘box chemistry’ (well-mixed concentrations) to large turbulent mixing schemes based on large-eddy simulations (LES). The former allow detailed chemical makeup and processes, but grossly undercount mixing contribution. The latter resolve fine detail of turbulent mixing but have limited scope due to high power computing resources and expertise required to run them.

We developed an intermediate level model that employs moment closure scheme for turbulent mixing coupled to chemistry. It requires minimal resources (desktop), and its predictions are comparable to LES.

The talk will give a general introduction to atmospheric chemistry-mixing. Then I will outline a moment closure scheme (called SOMCRUS), developed with collaborators at NCAR (Boulder, CO). I will elaborate its mathematical setup and computer implementation. Model simulations will be compared to LES case studies, and further applications and development discussed.

4/3/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Abdul-Nasah Soale
Title: Regression graphics for regression with metric-valued response objects
Abstract: As novel data collection becomes increasingly common, traditional dimension reduction and data visualization techniques are becoming inadequate to handle these complex data. A surrogate-assisted sufficient dimension reduction (SDR) method for regression with a general metric-valued response on Euclidean predictors is proposed. The response objects are mapped to a real-valued distance matrix using an appropriate metric and then projected onto a large sample of random unit vectors to obtain scalar-valued surrogate responses. An ensemble estimate of the subspaces for the regression of the surrogate responses versus the predictor is used to estimate the original central space. Under this framework, classical SDR methods such as ordinary least squares and sliced inverse regression are extended. The surrogate-assisted method applies to responses on compact metric spaces such as Euclidean, distributional, functional, and other response types. An extensive simulation experiment demonstrates the superior performance of the proposed surrogate-assisted method on synthetic data compared to existing competing methods where applicable. The analysis of the distributions of county level COVID-19 transmission rates in the United States as a function of demographic characteristics is also provided. The theoretical justifications are included as well.

3/6/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Weihong Guo
Title: Nonnegative and Nonlocal Sparse Tensor Factorization Based Hyperspectral Image Super-Resolution
Abstract: Hyperspectral image (HSI) super-resolution refers to enhancing the spatial resolution of a 3-D image with many spectral bands (slices). It is a seriously ill-posed problem when the low-resolution (LR) HSI is the only input. It is better solved by fusing the LR HSI with a high-resolution (HR)  multispectral image (MSI) for a 3-D image with both high spectral and spatial resolution. In this talk, we propose a novel nonnegative and nonlocal 4-D tensor dictionary learning-based HSI super-resolution model using group-block sparsity. By grouping similar 3-D image cubes into clusters and then conduct super-resolution cluster by cluster using 4-D tensor structure, we not only preserve the structure but also achieve sparsity within the cluster due to the collection of similar cubes. We use 4-D tensor Tucker decomposition and impose nonnegative constraints on the dictionaries and group-block sparsity. Numerous experiments demonstrate that the proposed model outperforms many state-of-the-art HSI super-resolution methods.

1/31/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Anuj Abhishek
Title: An operator learning framework for an inverse problem in Electrical Impedance Tomography
Abstract: Neural network architectures such as Fourier Neural Operators (FNO) and Deep Operator Networks (Deep-O-Net) have been shown to be fairly useful in approximating an operator between two function spaces. In this talk, we will briefly review an inverse problem that arises in Electrical Impedance tomography as well as review such operator learning network architectures. We will then see how we might use similar network architectures to learn (or, approximate) a map that takes in as its input the Dirichlet to Neumann operator and outputs the corresponding conductivity function. This is based on an unfinished ongoing work with my collaborator, Thilo Strauss (Xi’an Jiaotong-Liverpool University).

Spring 2024

4/3/2024, Wed. 2:15-3:15 pm in Olin 314
Speaker: Stephen Andryc
Title: Geometrically Interpreting Edwards’ Addition Formula
Abstract: Elliptic curves are often introduced in the context of real cubic curves, so that the point addition formula can be explained geometrically using intersections with lines. In 2007, Harold Edwards published an incredibly compact and powerful formula for addition on special quartic elliptic curves. In this talk, we will extract a geometric construction underlying Edwards’ formula directly from the familiar process on cubics, in a manner that is made as accessible as possible.

3/6/2024, Wed. 2:15-3:15 pm in Olin 314
Speaker: Dr. David Singer
Title: Confocal Families of Conics in the Hyperbolic Plane
Abstract: Conics in the hyperbolic plane exhibit much more variety than those in the Euclidean plane. Depending on the author, they have been classified as occurring in nine, eleven, or twelve forms. I will describe a classification of confocal families of conics, where a focus can be understood as a source of rays from a point in the plane, an ideal point, or an ultra-ideal point. The presentation will be on an expository level; previous experience with hyperbolic geometry will not be vital for understanding the talk (I hope!)

3/20/2024, Wed. 2:15-3:15 pm in Olin 314
Speaker: Andrew Edwards
Title: Weierstrass-Enneper Representations for Minimal Surfaces
Abstract: A surface in R^3 is called minimal if its mean curvature is zero at all points. The classical examples of minimal surfaces are the catenoid and the helicoid. Constructing more complicated minimal surfaces can be difficult, but a useful tool for doing so is the Weierstrass-Enneper representation, which uses principles of complex analysis to generate minimal surfaces.

3/27/2024, Wed. 2:15-3:15 pm in Olin 314
Speaker: Reeve Johnson
Title: The Problem(s) with Malfatti’s Marble Problem (but also, the Solution(s))
Abstract: In 1803 the Italian mathematician Gian Francesco Malfatti posed a problem: maximize the volume of three circular columns cut out of a triangular slab of marble with fixed height. He also proposed a solution: three circles within the triangle, each tangent to one another, should do the trick. His solution remained unquestioned for many years, simply because it *looked* correct. But, in 1967, somebody clocked his tea- Malfatti’s solution was not only incorrect in some cases, it was incorrect in EVERY case! In 1994, the correct solutions were found and, in 2022, those solutions were fully proved. Ok great! But there’s more… Why do we call his (incorrect) solution the “Malfatti circles” when, 30 years earlier in Japan, Naonobu Ajima examined the geometry of this exact same arrangement of three circles within a triangle? What exactly did Ajima determine about these circles? Did Malfatti know about Ajima? This talk will address these questions along with questions about proofs, sacred Japanese temple geometry, and so much more.