### Accelerated Optimization in the PDE Framework

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Anthony Yezzi – Georgia Tech, ECE

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- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Anthony Yezzi – Georgia Tech, ECE

Following the seminal work of Nesterov, accelerated optimization methods (sometimes referred to as momentum methods) have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios were second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with an attraction basin large enough to accommodate the initial overshoot. This behavior has made accelerated search methods particularly popular within the machine learning community where stochastic variants have been proposed as well. So far, however, accelerated optimization methods have been applied to searches over finite parameter spaces. We show how a variational setting for these finite dimensional methods (recently formulated by Wibisono, Wilson, and Jordan) can be extended to the infinite dimensional setting, both in linear functional spaces as well as to the more complicated manifold of 2D curves and 3D surfaces.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 17, 2018 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Professor Lourenco Beirao da Veiga – Università di Milano-Bicocca

**Please Note:** This is a joint seminar by College of Engineering and School of Math.

The Virtual
Element Method (VEM), is a very recent technology introduced in [Beirao da
Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the
discretization of partial differential equations, that has shared a good
success in recent years. The VEM can be interpreted as a generalization of the
Finite Element Method that allows to use general polygonal and polyhedral
meshes, still keeping the same coding complexity and allowing for arbitrary
degree of accuracy. The Virtual Element Method makes use of local functions
that are not necessarily polynomials and are defined in an implicit way.
Nevertheless, by a wise choice of the degrees of freedom and introducing a novel
construction of the associated stiffness matrixes, the VEM avoids the explicit
integration of such shape functions.
In addition
to the possibility to handle general polytopal meshes, the flexibility of the
above construction yields other interesting properties with respect to more
standard Galerkin methods. For instance, the VEM easily allows to build discrete
spaces of arbitrary C^k regularity, or to satisfy exactly the divergence-free
constraint for incompressible fluids.
The present
talk is an introduction to the VEM, aiming at showing the main ideas of the
method. We consider for simplicity a simple elliptic model problem (that is
pure diffusion with variable coefficients) but set ourselves in the more
involved 3D setting. In the first part we introduce the adopted Virtual Element
space and the associated degrees of freedom, first by addressing the faces of
the polyhedrons (i.e. polygons) and then building the space in the full
volumes. We then describe the construction of the discrete bilinear form and
the ensuing discretization of the problem. Furthermore, we show a set of
theoretical and numerical results. In the very final part, we will give a
glance at more involved problems, such as magnetostatics (mixed problem with more
complex spaces interacting) and large deformation elasticity (nonlinear
problem).

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

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