Seminars and Colloquia by Series

Controlled SPDEs: Peng’s Maximum Principle and Numerical Methods

Series
SIAM Student Seminar
Time
Friday, November 17, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lukas WesselsGeorgia Tech

In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First, we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs.

In the second part, we present a numerical algorithm that allows the efficient approximation of optimal controls in the case of stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm.

This talk is based on joint work with Wilhelm Stannat and Alexander Vogler. Talk will also be streamed: https://gatech.zoom.us/j/93808617657?pwd=ME44NWUxbk1NRkhUMzRsK3c0ZGtvQT09

Neural-ODE for PDE Solution Operators

Series
SIAM Student Seminar
Time
Friday, September 29, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan GabyGeorgia State University

We consider a numerical method to approximate the solution operator for evolutional partial differential equations (PDEs). By employing a general reduced-order model, such as a deep neural network, we connect the evolution of a model's parameters with trajectories in a corresponding function space. Using the Neural Ordinary Differential Equations (NODE) technique we learn a vector field over the parameter space such that from any initial starting point, the resulting trajectory solves the evolutional PDE. Numerical results are presented for a number of high-dimensional problems where traditional methods fail due to the curse of dimensionality.

Geometric Equations for Matroid Varieties

Series
SIAM Student Seminar
Time
Tuesday, November 15, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashley K. WheelerSchool of Mathematics

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

Sparse Quadratic Optimization via Polynomial Roots

Series
SIAM Student Seminar
Time
Tuesday, October 25, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin ShuSchool of Mathematics

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.

Ergodic theory: a statistical description of chaotic dynamical systems

Series
SIAM Student Seminar
Time
Friday, December 3, 2021 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Alex BlumenthalGeorgia Tech

Dynamical systems model the way that real-world systems evolve in time. While the time-asymptotic behavior of many systems can be characterized by “simple” dynamical features such as equilibria and periodic orbits, some systems evolve in a chaotic, seemingly random way. For such systems it is no longer meaningful to track one trajectory at a time individually- instead, a natural approach is to treat the initial condition as random and to observe how its probabilistic law evolves in time. This is the core idea of ergodic theory, the topic of this talk. I will not assume much beyond some basics of probability theory, e.g., random variables. 

About Coalescence of Eigenvalues for Matrices Depending on Several Parameters

Series
SIAM Student Seminar
Time
Friday, November 12, 2021 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Luca DieciGeorgia Institute of Technology

We review some theoretical and computational results on locating eigenvalues coalescence for matrices smoothly depending on parameters. Focus is on the symmetric 2 parameter case, and Hermitian 3 parameter case. Full and banded matrices are of interest.

Mathematical approaches to Imaging and data

Series
SIAM Student Seminar
Time
Friday, September 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Sung-Ha KangSchool of Math, Georgia Tech,

 

I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding.  This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization.  If time permits, I will introduce identification of differential equation from given noisy data.   

A Self-Limiting Hawkes Process

Series
SIAM Student Seminar
Time
Monday, November 16, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
ONLINE at https://bluejeans.com/703668715
Speaker
John OlindeGT Math

Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time.  Examples of processes that have this property are earthquakes, crime in a neighborhood, or emails within a company.  In 1971, Alan Hawkes first used what is now known as the Hawkes process to model such processes.  Since then much work has been done on estimating the parameters of a Hawkes process given a data set and creating variants of the process for different applications.

 

In this talk, I will be proposing a new variant of a Hawkes process that takes into account the effect of police activity on the underlying crime rate and an algorithm for estimating its parameters given a crime data set.

Post-grazing dynamics of a vibro-impacting energy generator

Series
SIAM Student Seminar
Time
Tuesday, November 3, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Online at https://bluejeans.com/893955256
Speaker
Larissa SerdukovaMathematics & Statistics Department, University of Reading

 

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric. 

Bluejeans link: https://bluejeans.com/893955256

Post-grazing dynamics of a vibro-impacting energy generator--Postponed

Series
SIAM Student Seminar
Time
Friday, March 27, 2020 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Larissa SerdukovaGT Math

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

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