This colloquium is co-coordinated by Daniel Glasscock (email: Daniel_Glasscock@uml.edu), Jong Soo Lee (email: JongSoo_Lee@uml.edu), Hung Phan (email: Hung_Phan@uml.edu), and Joris Roos (email: Joris_Roos@uml.edu). Contact any of them if you would like to speak as part of this seminar.
Abstract: We give an overview of some problems in the area of multilinear singular integrals and discuss their connection with questions on point configurations in large subsets of Euclidean space.
Abstract: In this talk I will give an overview of some interrelated problems and results in Euclidean harmonic analysis that I have been interested in. We will begin with Carleson's theorem on pointwise convergence of Fourier series and end with certain multilinear singular integrals involving curvature.
Abstract: For wave-based inverse problems the resolution is usually limited by the so-called diffraction limit, i.e., the smallest features to be reconstructed cannot be smaller than the smallest wavelength of available data. If one properly restricts the class of features to, for example, point-scatterers, the seminal work of Donoho in the early 90’s demonstrates that the recovery of these sub-wavelength features is tractable. However, algorithms to recover a more general class of structured scatterers containing features below the diffraction limit remains an open question. In this talk we aim to surpass the diffraction limit using deep learning coupled with computational harmonic analysis tools. In particular, I will introduce a new a neural network architecture for inverting wide-band data to recover acoustic scatterers at resolutions finer than the classical limit. The architecture incorporates insights from the butterfly factorization and the Cooley-Tukey algorithm to explicitly account for the physics of wave propagation. The dimensions of the network seamlessly adapt to the desired image resolution, resulting in a number of trainable weights that scale quasi linearly with the image resolution and the data bandwidth. In addition, the data is optimally assimilated across frequencies thus enhancing the stability of the training stage. I will provide the rationale for such construction and showcase its properties for several classes of scatterers with sub-Nyquist features embedded in a known background media. (Joint work with Matthew Li and Laurent Demanet).
Abstract: Projection algorithms are powerful tools to solve feasibility problems consisting in finding a point in the intersection of a family of sets. One of the most widely used among them is the so-called Douglas-Rachford (DR) algorithm, due to its successful behavior both in convex and non-convex scenarios. In this talk, we shall introduce the family of projection methods with special attention to DR and some of its applications such as Sudoku puzzles. Then, we shall show how a slight modification of this method yields to a new algorithm with a completely different dynamics: it permits to solve best approximation problems rather than feasibility ones; that is, the algorithm finds, not just a point in the intersection, but the closest one to any given point in the space. This paradigm will be illustrated as an application on the problem of finding probability density functions with prescribed moments. Some other promising numerical results will be also reported.
Abstract: We explain several fundamental properties of the Douglas-Rachford algorithm based on monotone operator and fixed point theory. An application in civil engineering design will be briefly presented. A part of the discussed material is suitable for both undergraduate and graduate students.
Abstract: Establishing inequalities among graph densities is a central pursuit in extremal graph theory. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares or as a rational sum of squares. We will be giving a pair of consecutive talks about our ongoing research in this area. In this first part, we will explain how one does so and we will then identify simple conditions under which a graph density expression cannot be a sum of squares or a rational sum of squares.
Abstract: In this second part, we will talk about an application to Sidorenko's conjecture on the density of bipartite subgraphs.
Abstract: We consider a subclass of maximization geometric programs (GPs) with known coefficient parameters that in each constraint, sum up to less than unity. For such programs, the stochasticity of the real-world may impose practical limitations on the degree to which one can exactly resolve the remaining (i.e., exponent) parameters that help define each constraint's feasible space. Confronted with this challenge, we employ a chance-constrained approach which entails maximizing the GP's objective function subject to a specified minimum probability of being feasible. Then, we derive some theoretical properties of the GP, and propose a decomposition algorithm for finding good feasible solutions