Friday, 2:30-3:30pm, 458 Campbell Hall, tea to follow.

(unless otherwise noted)

contact: cnavasca@uab.edu

]]>(unless otherwise noted)

contact: cnavasca@uab.edu

Title. Designing Optimal Spectral Filters and Low-Rank Matrices for Inverse Problems

Abstract. Computing reliable solutions to inverse problems is important in many applications such as biomedical imaging, computer graphics, and security. Regularization by incorporating prior knowledge is needed to stabilize the inversion process. In this talk, we develop a new framework for solving inverse problems that incorporates probabilistic information in the form of training data. We provide theoretical results for the underlying Bayes risk minimization problem and discuss efficient approaches for solving the associated empirical Bayes risk minimization problem. Various constraints can be imposed to deal with large-scale problems. Here we describe methods for computing optimal spectral filters, for cases where the SVD is available, and methods for computing an optimal low-rank regularized inverse matrix, for cases where the forward model is not known.

This is joint work with Matthias Chung (Virginia Tech) and Dianne O'Leary (University of Maryland, College Park).

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Title. Maxwell operator. Inception.

Abstract. We'll cover some basics about Maxwell operator in bounded regions and introduce its rigorous mathematical definition. Most of the talk I'll be integrating by parts, so everybody is welcome to participate.

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Abstract. Analyzing data collected from many areas is a challenge facing scientists and engineers. The property of being high-dimensional makes these data sets hard to tackle. Fortunately, one can work with some low-dimensional structures, because in many cases, data concentrates around a low-dimensional subspace or does so in a local neighborhood. After an introduction of the area, I will present in detail a regression problem and use that as an example to show how to capture the low-dimensional structure of data and make decisions based on the learned structure. In the regression problem, more specifically, a set of data points x and the corresponding responses y is given, one wants to find a mapping f such that f(x) approximates y well and this mapping f can be applied to unobserved instances. An algorithm with piecewise linear mappings built on a tree structure is proposed. The proposed method can be applied when both x and y are high-dimensional and can handle it well in particular when the closeness in x is not consistent with that in y. By comparing the proposed method to its competitors in experiments, it is shown to be advantageous.]]>

Title. From Oil Wells to Wall Street: Stories from the land of inverse problems

Abstract. The generic inverse problem arises, as a fundamental part of the modeling process, when one makes external measurements on a physical system with the intention of determining unknown internal properties of the system. One could, for example, send sound waves into a body, measure the output waves, and try to infer the internal density function for non-invasive medical imaging purposes. The list of potential applications is vast, from enhanced oil recovery techniques to land mine detection to probing the earth's interior via natural earthquake waves to estimating future stock market volatility from current option prices. We will look at the background and the mathematics behind some of these inverse problems and note that they may all be handled by adaptions of a common approach.]]>

Title. 3D Mixed Element Discontinuous Galerkin with Shock Capturing and RANS

Abstract. A parallel high-order Discontinuous Galerkin method is developed for mixed elements to solve the Navier-Stokes equations. A PDE-based artificial viscosity equation is implemented to smooth and stabilize shocks. To solve this system of non-linear equations a Newton solver is implemented and preconditioned flexible-GMRES is used to solve the linear system arising from the Jacobian matrix. The preconditioners that are implemented include Jacobi relaxation, Gauss-Seidel relaxation, line implicit Jacobi, and ILU(0). A wide variety of simulations are performed to demonstrate the capabilities of the DG solver. The inviscid simulations include a p-adapted subsonic flow over a cylinder, a p=0 h-adapted hypersonic flow over a sphere, and a large scale p=2 simulation of an aircraft with artificial viscosity to stabilize the shock formed on the wing. Two hypersonic viscous flows of a cylinder and sphere are simulated and compared to the NASA code LAURA. The solution matches closely to LAURA and the shock becomes more resolved as the polynomial degree is increased. The heating rate on the surface matches closely to LAURA at p=3. In the case of turbulent flows the Reynolds Averaged Navier-Stokes (RANS) equations are solved. The new negative-Spalart-Almaras model is implemented and used to solve turbulent flow over a NACA 0012 wing, RAE2822 wing, and a multi-element 30P30N wing. Finally, the parallel scalability is tested and good speed up is obtained using up to 2048 processor cores. As the polynomial degree increases the scalability improves. Although, an ideal speedup was not shown this was contributed to load balancing. These simulations demonstrate the capability of the DG solver to handle strong shocks, RANS, complex geometry, hp-adaption, and parallel scalability.

This is joint work with Dimitri J. Mavriplis.

Figure. M=17.605, Re=376,930 flow over a cylinder, contours of artificial viscosity (left) and contours of Mach number (right)]]>

Title. Universal computation by multi-particle quantum walk in 2D

Abstract. In this talk we discuss a model consisting of

Title. Additivity (or not) of the Fixed Point Property

Abstract. Let each of X, Y, and X intersect Y be a continuum with the fixed point property (fpp).We say that "the fpp is additive for X and Y" if X union Y has the fpp. If G is some class of continua with the fpp, we say that "the fpp is additive for G" provided that whenever X, Y, and X intersect Y are in G, the fpp is additive for X and Y.

Question. For what classes G of continua is the fpp additive?

We discuss the history of this question, reviewing both positive and negative results. We end with recent examples of Hagopian and Marsh that show the fpp is not additive for the class of tree-like continua.]]>