Seminars and Colloquia by Series

Series

Control and Inverse Problems for Differential Equations on Graphs

Series: Applied and Computational Mathematics Seminar
Time: Monday, September 10, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Sergei Avdonin – University of Alaska Fairbanks – s.avdonin@alaska.edu

Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs
is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.

In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more
regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems
for such systems more diffcult.

We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful
method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.

Application of stochastic maximum principle. Risk-sensitive regime switching in asset management.

Series: Applied and Computational Mathematics Seminar
Time: Monday, July 2, 2018 - 01:55 for 1.5 hours (actually 80 minutes)
Location: Skiles 005
Speaker: Isabelle Kemajou-Brown – Morgan State University – elisabeth.brown@morgan.edu

We assume the stock is modeled by a Markov regime-switching diffusion process and that, the benchmark depends on the economic factor. Then, we solve a risk-sensitive benchmarked asset management problem of a firm. Our method consists of finding the portfolio strategy that minimizes the risk sensitivity of an investor in such environment, using the general maximum principle.After the above presentation, the speaker will discuss some of her ongoing research.

Convolutional Neural Network with Structured Filters

Series: Applied and Computational Mathematics Seminar
Time: Monday, April 16, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Xiuyuan Cheng – Duke University – xiuyuan.cheng@duke.edu

Filters in a Convolutional Neural Network (CNN) contain model parameters learned from enormous amounts of data. The properties of convolutional filters in a trained network directly affect the quality of the data representation being produced. In this talk, we introduce a framework for decomposing convolutional filters over a truncated expansion under pre-fixed bases, where the expansion coefficients are learned from data. Such a structure not only reduces the number of trainable parameters and computation load but also explicitly imposes filter regularity by bases truncation. Apart from maintaining prediction accuracy across image classification datasets, the decomposed-filter CNN also produces a stable representation with respect to input variations, which is proved under generic assumptions on the basis expansion. Joint work with Qiang Qiu, Robert Calderbank, and Guillermo Sapiro.

Simulating large-scale geophysical flows on unstructured meshes

Series: Applied and Computational Mathematics Seminar
Time: Monday, April 9, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Prof. Qingshan Chen – Department of Mathematical Sciences, Clemson University – qsc@clemson.edu

Large-scale geophysical flows, i.e. the ocean and atmosphere, evolve on spatial scales ranging from meters to thousands of kilometers, and on temporal scales ranging from seconds to decades. These scales interact in a highly nonlinear fashion, making it extremely challenging to reliably and accurately capture the long-term dynamics of these flows on numerical models. In fact, this problem is closely associated with the grand challenges of long-term weather and climate predictions. Unstructured meshes have been gaining popularity in recent years on geophysical models, thanks to its being almost free of polar singularities, and remaining highly scalable even at eddy resolving resolutions. However, to unleash the full potential of these meshes, new schemes are needed. This talk starts with a brief introduction to large-scale geophysical flows. Then it goes over the main considerations, i.e. various numerical and algorithmic choices, that one needs to make in deisgning numerical schemes for these flows. Finally, a new vorticity-divergence based finite volume scheme will be introduced. Its strength and challenges, together with some numerical results, will be presented and discussed.

Compute Faster and Learn Better: Machine Learning via Nonconvex Optimization

Series: Applied and Computational Mathematics Seminar
Time: Monday, April 2, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Tuo Zhao – Georgia Institute of Technology

Nonconvex optimization naturally arises in many machine learning problems. Machine learning researchers exploit various nonconvex formulations to gain modeling flexibility, estimation robustness, adaptivity, and computational scalability. Although classical computational complexity theory has shown that solving nonconvex optimization is generally NP-hard in the worst case, practitioners have proposed numerous heuristic optimization algorithms, which achieve outstanding empirical performance in real-world applications.To bridge this gap between practice and theory, we propose a new generation of model-based optimization algorithms and theory, which incorporate the statistical thinking into modern optimization. Specifically, when designing practical computational algorithms, we take the underlying statistical models into consideration. Our novel algorithms exploit hidden geometric structures behind many nonconvex optimization problems, and can obtain global optima with the desired statistics properties in polynomial time with high probability.

Fast Phase Retrieval from Localized Time-Frequency Measurements

Series: Applied and Computational Mathematics Seminar
Time: Monday, March 26, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Mark Iwen – Michigan State University – iwenmark@msu.edu

We propose a general phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks. The algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees are presented. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

Joint-sparse recovery for high-dimensional parametric PDEs

Series: Applied and Computational Mathematics Seminar
Time: Monday, March 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Nick Dexter – University of Tennessee – ndexter@utk.edu

We present and analyze a novel sparse polynomial approximation method for the solution of PDEs with stochastic and parametric inputs. Our approach treats the parameterized problem as a problem of joint-sparse signal reconstruction, i.e., the simultaneous reconstruction of a set of signals sharing a common sparsity pattern from a countable, possibly infinite, set of measurements. Combined with the standard measurement scheme developed for compressed sensing-based polynomial approximation, this approach allows for global approximations of the solution over both physical and parametric domains. In addition, we are able to show that, with minimal sample complexity, error estimates comparable to the best s-term approximation, in energy norms, are achievable, while requiring only a priori bounds on polynomial truncation error. We perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.

A characterization of domain of beta-divergence and its connection to Bregman-divergence

Series: Applied and Computational Mathematics Seminar
Time: Monday, February 26, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Prof. Hyenkyun Woo – Korea University of Technology and Education

Please Note: Bio: Hyenkyun Woo is an assistant professor at KOREATECH (Korea University of Technology and Education). He got a Ph.D at Yonsei university. and was a post-doc at Georgia Tech and Korea Institute of Advanced Study and others.

In machine learning and signal processing, the beta-divergence is well known as a similarity measure between two positive objects. However, it is unclear whether or not the distance-like structure of beta-divergence is preserved, if we extend the domain of the beta-divergence to the negative region. In this article, we study the domain of the beta-divergence and its connection to the Bregman-divergence associated with the convex function of Legendre type. In fact, we show that the domain of beta-divergence (and the corresponding Bregman-divergence) include negative region under the mild condition on the beta value. Additionally, through the relation between the beta-divergence and the Bregman-divergence, we can reformulate various variational models appearing in image processing problems into a unified framework, namely the Bregman variational model. This model has a strong advantage compared to the beta-divergence-based model due to the dual structure of the Bregman-divergence. As an example, we demonstrate how we can build up a convex reformulated variational model with a negative domain for the classic nonconvex problem, which usually appears in synthetic aperture radar image processing problems.

Georgia Scientific Computing Symposium

Series: Applied and Computational Mathematics Seminar
Time: Saturday, February 24, 2018 - 09:30 for 8 hours (full day)
Location: Helen M. Aderhold Learning Center (ALC), Room 24 (60 Luckie St NW, Atlanta, GA 30303)
Speaker: Wenjing Liao and others – GSU, Clemson,UGA, GT, Emory

The Georgia Scientific Computing Symposium is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community. This year, the symposium will be held on Saturday, February 24, 2018, at Georgia State University. More information can be found at: https://math.gsu.edu/xye/public/gscs/gscs2018.html

[unusual date and room] Temporal Resolution of Uncertainty and Exhaustible Resource Pricing: A Dynamic Programming Approach

Series: Applied and Computational Mathematics Seminar
Time: Friday, February 23, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location: Skiles 269
Speaker: Prof. Justin Kakeu – Morehouse University

We use a stochastic dynamic programming approach to address the following question: Can a homogenous resource extraction model (one without extraction costs, without new discoveries, and without technical progress) generate non-increasing resource prices? The traditional answer to that question contends that prices should exhibit an increasing trend as the exhaustible resource is being depleted over time (The Hotelling rule). In contrast, we will show that injecting concerns for temporal resolution of uncertainty in a resource extraction problem can generate a non-increasing trend in the resource price. Indeed, the expected rate of change of the price can become negative if the premium for temporal resolution of uncertainty is negative and outweighs both the positive discount rate and the short-run risk premium. Numerical examples are provided for illustration.

Georgia Institute of Technology College of Sciences

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