Seminars and Colloquia by Series

Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, February 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna EstrachNew York University
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

An Adaptive Sampling Approach for Surrogate Modeling of Expensive Computer Experiments

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashwin RenganathanGT AE

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary.

GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

An Adaptive Sampling Approach for Surrogate Modeling of Expensive Computer Experiments

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashwin RenganathanGT AE

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary

GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

Synchronization of pendulum clocks and metronomes

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 14, 2019 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Guillermo GoldszteinGT School of Math
In 1665, Huygens discovered that, when two pendulum clocks hanged from a same wooden beam supported by two chairs, they synchronize in anti-phase mode. On the other hand, metronomes synchronize in-phase when oscillating on top of the same movable surface. In this talk, I will describe and analyze a model to help understand the conditions that lead to anti-phase synchronization vs. the conditions that lead to in-phase synchronization.

Finite Dimensional Balian-Low Theorems

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 7, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Dr. Michael NorthingtonGT Math
Gabor systems, or collections of translations and modulations of a window function, are often used for time-frequency analysis of signals. The Balian-Low Theorem and its generalizations say that if a Gabor system obeys certain spanning and independence properties in L^2(R), then the window function of such a system cannot be well localized in both time and frequency. Recently, Shahaf Nitzan and Jan—Fredrik Olsen show that similar behavior extends to Gabor systems of finite length signals in l^2(Z_d). In this talk, I will discuss these finite dimensional results as well as recent extensions proven in collaboration with Josiah Park.

Nonparametric inference of interaction laws in particles/agent systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 3, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fei LuJohns Hopkins University
Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. An interesting question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistic learning theory addressing learnability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved.

Granular sessile drops

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 26, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
005 Skiles
Speaker
Ray TreinenTexas State University

We consider one or more volumes of a liquid or semi-molten material sitting on a substrate, while the vapor above is assumed to have the same medium in suspension. There may be both evaporation and condensation to move mass from one cell to another. We explore possible equilibrium states of such configurations. Our examples include a single sessile drop (or cell) on the plate, connected clusters of cells of the material on the plate, as well as a periodic configuration of connected cells on the plate. The shape of the configurations will depend on the type of energy that we take into consideration, and in settings with a vertical gravitational potential energy the clusters are shown to exhibit a preferred granular scale. The majority of our results are in a lower dimensional setting, however, some results will be presented in 3-D.

Granular sessile drops

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 26, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ray TreinenTexas State University

Please Note: This should be unpublished. Againx3

We consider one or more volumes of a liquid or semi-molten material sitting on a substrate, while the vapor above is assumed to have the same medium in suspension. There may be both evaporation and condensation to move mass from one cell to another. We explore possible equilibrium states of such configurations. Our examples include a single sessile drop (or cell) on the plate, connected clusters of cells of the material on the plate, as well as a periodic configuration of connected cells on the plate. The shape of the configurations will depend on the type of energy that we take into consideration, and in settings with a vertical gravitational potential energy the clusters are shown to exhibit a preferred granular scale. The majority of our results are in a lower dimensional setting, however, some results will be presented in 3-D.

Capture small-noise-induced rare events in differential equations: from variation to sampling

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 12, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Xiaoliang WanLouisiana State University
In this talk, we will discuss some computational issues when applying the large deviation theory to study small-noise-induced rare events in differential equations. We focus on two specific problems: the most probable transition path for an ordinary differential equation and the asymptotically efficient simulation of rare events for an elliptic problem. Both problems are related to the large deviation theory. From a computational point of view, the former one is a variational problem while the latter one is a sampling problem. For the first problem, we have developed an hp adaptive minimum action method, and for the second problem, we will present an importance sampling estimator subject to a sufficient and necessary condition for its asymptotic efficiency.

High-dimensional Covariance Structure Testing using Maximum Pairwise Bayes Factors

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lizhen LinUniversity of Notre Dame
Hypothesis testing of structure in covariance matrices is of significant importance, but faces great challenges in high-dimensional settings. Although consistent frequentist one-sample covariance tests have been proposed, there is a lack of simple, computationally scalable, and theoretically sound Bayesian testing methods for large covariance matrices. Motivated by this gap and by the need for tests that are powerful against sparse alternatives, we propose a novel testing framework based on the maximum pairwise Bayes factor. Our initial focus is on one-sample covariance testing; the proposed test can optimally distinguish null and alternative hypotheses in a frequentist asymptotic sense. We then propose diagonal tests and a scalable covariance graph selection procedure that are shown to be consistent. Further, our procedure can effectively control false positives. A simulation study evaluates the proposed approach relative to competitors. The performance of our graph selection method is demonstrated through applications to a sonar data set.

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