Seminars archive

The scientific computing group at Emory will kick off the new academic year by short overviews of the faculty's ongoing research. This week, Dr. Nagy and Dr. Ruthotto will be in the spotlight. Dr. Nagy will give his overview of his group's efforts aiming at developing more efficient numerical linear algebra techniques for large-scale image processing. Dr. Ruthotto will present recent advances and open problems at the interface between PDEs, optimization, and machine learning. These high-level talks will not be too technical and faculty and students working in other but related fields are encouraged to attend.

Nonlinear inverse problems arise in numerous imaging applications, and solving them is often difficult due to ill-posedness and high computational cost. In this work, we introduce tailored solvers for several nonlinear inverse problems in imaging within a Gauss-Newton optimization framework.\\ \\ We develop a linearize and project (LAP) method for a class of nonlinear problems with two (or more) sets of coupled variables. At each iteration of the Gauss-Newton optimization, LAP linearizes the residual around the current iterate, eliminates one block of variables via a projection, and solves the resulting reduced dimensional problem for the Gauss-Newton step. The method is best suited for problems where the subproblem associated with one set of variables is comparatively well-posed or easy to solve. LAP supports iterative, direct, and hybrid regularization and supports element-wise bound constraints on all the blocks of variables. This offers various options for incorporating prior knowledge of a desired solution. We demonstrate the advantages of these characteristics with several numerical experiments. We test LAP for two and three dimensional problems in super resolution and MRI motion correction, two separable nonlinear least squares problems that are linear in one block of variables and nonlinear in the other. We also use LAP for image registration subject to local rigidity constraints, a problem that is nonlinear in all sets of variables. These two classes of problems demonstrate the utility and flexibility LAP method.\\ \\ We also implement an efficient Gauss-Newton optimization scheme for the problem of phase recovery in bispectral imaging, a univariate nonlinear inverse problem. Using a fixed approximate Hessian, matrix-reordering, and stored matrix factors, we accelerate the Gauss-Newton step solve, resulting in a second-order optimization method which outperforms first-order methods in terms of cost per iteration and solution quality.

Uncovering anatomical changes over time is important in understanding brain development, aging, and disease progression. Data for these studies, image and shape time series, have complex structures and are best treated as elements of non-Euclidean spaces. In this talk, I present our non-Euclidean models for image and shape regression to estimate the time-varying trend of a population by generalizing Euclidean regression and to predict a subject-specific trend by integrating image geometry with deep neural networks. I also introduce a complementary segmentation network that preprocesses image scans and accurately extracts the brain from both normal and pathological images. Our experimental results demonstrated the promise of our models in the study of normal brain aging and Alzheimer’s disease.

Most high-dimensional nonconvex optimization problems cannot be solved to optimality. It has been observed, however, that deep neural networks have a benign geometry that permits standard optimization methods to find acceptable solutions. However, solution times can be exorbitant. In addition, not all minimizers of the neural network loss functions are equally desirable, as some lead to prediction systems with better generalization properties than others. In this talk we discuss classical and new optimization methods in the light of these observations, and conclude with some open questions. BIO: Jorge Nocedal is the Walter P. Murphy Professor in the Department of Industrial Engineering and Management Sciences at Northwestern University. His research is in optimization, both deterministic and stochastic, and with emphasis on very large-scale problems. His current work is driven by applications in machine learning. He is a SIAM Fellow, was awarded the 2012 George B. Dantzig Prize, and the 2017 Von Neumann Theory Prize for contributions to theory and algorithms of optimization.

The Gauss Circle Problem asks how many lattice points are contained in a circle centered at the origin or radius R. A simple geometric argument establishes that this count is approximated by the area $\pi R^2$, with an error bounded by the perimeter $O(R)$. \\ ``Arithmetic statistics" is about arithmetic objects -- number fields, ideal class groups, and so on. Bhargava and many others have recently proved spectacular theorems by parametrizing such objects in terms of lattice points, and then using geometry to counting the lattice points. \\ Meanwhile, harmonic analysts have long known that you can do better than an error of $O(R)$ in Gauss's circle problem. I will describe a program to import such improvements into arithmetic statistics, and give an overview of the number theoretic results we hope to obtain. \\ This is ongoing joint work with Theresa Anderson and Takashi Taniguchi.

Erd\H{o}s, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k \geq 2$: Every graph on $n \geq k-1$ vertices with at least $(k-1)(n-k+2)+(k-2)(k-3)/2$ edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdos et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\epsilon_k)n$ vertices with minimum degree at least $k$, where $\epsilon_k>0$ depends only on $k$. In this talk, we will sketch a proof of this conjecture. The proof relies on ideas from a paper of Mousset, Noever and $\check{S}kori\acute{c}$. We will discuss these ideas and how they can be extended to give a proof of the full conjecture.

The problem of bounding the total number of cycles in a graph is more than a century old. In 1897, Ahrens proved bounds on the number of cycles using the cyclomatic number of the graph and since then many results have appeared on the maximum number of cycles in graphs with different restrictions. <br> <br> In this talk I will consider a problem of maximizing the number of cycles for three classes of graphs: graphs with given number of edges (and unrestricted number of vertices), graphs with a given average degree, and graphs without a clique of a specific size. For the first two classes I will show that the maximum number of cycles in a graph has bounds exponential in the number of edges of the graph. I will also present exponentially tight bounds for the maximum number of cycles in a multigraph with a fixed number of vertices and edges.

For any fundamental discriminant $D > 0$, Hirzebruch and Zagier constructed a modular form of weight two whose Fourier coefficients are corrections of the Hurwitz class number sums $\sum_{r^2 \equiv 4n \, (D)} H((4n - r^2) / D)$. In this talk, we will discuss how one can reinterpret their result and remove the condition that $D$ is fundamental by working instead with vector-valued modular forms for Weil representations.

The gambler's fallacy is the erroneous belief that if (for example) a coin comes up heads often, then in the next toss it is more likely to be tails. In recent work with Robert Lemke Oliver, we found that funnily the primes exhibit a kind of gambler's fallacy: for example, consecutive primes do not like to have the same last digit. I'll show some of the data on this, and explain what we think is going on.