Applied Math Seminar

The Fast Radial Basis Functions Orthogonal Gradients (RBF-OGr) method for solving PDEs on arbitrary surfaces

11/13/2017 - 4:15pm
11/13/2017 - 5:15pm
Speaker: 
Cecile Piret (Michigan Tech University)
Abstract: 

The RBF-OGr method was introduced in [Piret, 2012] to discretize differential operators defined on arbitrary manifolds defined exclusively by a point cloud. The method was designed to take advantage of the meshfree character of RBFs, which offers the flexibility to represent complex geometries in any spatial dimension while providing a high order of accuracy. A large limitation of the original RBF-OGr method was its large computational complexity, which greatly restricted the size of the point cloud. In this talk, a fast version of the RBF-OGr method will be introduced. This latest algorithm makes use of the RBF-Finite Difference (RBF-FD) technique for building sparse differentiation matrices discretizing continuous differential operators such as the Laplace-Beltrami or the surface biharmonic operators.

Where: 
Emmy Noether Rm, Millikan 1021, Pomona College

RNA Interactions - a PDE Story

09/18/2017 - 4:15pm
09/18/2017 - 5:15pm
Speaker: 
Maryann Hohn (UCSD)
Abstract: 

Small non-coding RNAs regulate developmental events through certain interactions with messenger RNA (mRNA). By binding to specific sites on a strand of mRNA, small RNAs may cause a gene to be activated or suppressed, turning a gene "on" or "off". To better understand these interactions, we developed a mathematical model that consists of a system of coupled partial differential equations describing mRNA and small RNA interactions across cells and tissue. In this talk, we will discuss the mathematical models created and numerical simulations using these equations.

Where: 
Emmy Noether Rm, Millikan 1021, Pomona College

Reduced Order Models of Fractured Systems using Graph Theory and Machine Learning

05/09/2017 - 11:00am
05/09/2017 - 12:00pm
Speaker: 
Gowri Srinivasan (LANL)
Abstract: 

Microstructural information (fracture size, orientation, etc.) plays a key role in governing the dominant physics for two timely applications of interest to LANL: dynamic fracture processes like spall and fragmentation in metals (weapons performance) and detection of gas flow in static fractures in rock due to underground explosions (nuclear nonproliferation). Micro-fracture information is only known in a statistical sense, so representing millions of micro-fractures in 1000s of model runs to bound the uncertainty requires petabytes of information. Our critical advance is to integrate computational physics, machine learning and graph theory to make a paradigm shift from computationally intensive grid-based models to efficient graphs with at least 3 orders of magnitude speedup for Discrete Fracture Networks (DFNs).

Where: 
CGU Math South

Stochastic differential equations representing anomalous diffusions

05/05/2017 - 1:00pm
05/05/2017 - 2:00pm
Speaker: 
Kei Kobayashi (Fordham University)
Abstract: 

Brownian motion has been employed to model a number of random time-dependent quantities observed in many different research areas. However, this classical model has several drawbacks; one notable shortcoming is that it does not allow the quantities to be constant over any time interval of positive length. One way to describe such constant periods is to introduce a random time change given by the so-called inverse stable subordinator. The Brownian motion composed with this specific time change is significantly different from the classical Brownian motion; for example, it is non-Markovian and has transition probability densities satisfying a time-fractional order heat equation.

Where: 
CGU Math South

What spatial statistical model is best for predicting fisheries by catch risk?

09/11/2017 - 4:15pm
09/11/2017 - 5:15pm
Speaker: 
Brian Stock (HMC Math Bio '09; UC San Diego)
Abstract: 

Bycatch (i.e. catch of at least some non-targeted species) is an omnipresent problem in commercial and recreational fisheries. High bycatch rates can reduce the efficiency and sustainability of fisheries, but even extremely low bycatch rates can be a problem for protected or rebuilding species. Given these economic and environmental concerns, the fishing community would be well served by tools that predict, and ultimately help avoid, bycatch. I will demonstrate the ability of a new, computationally efficient spatial statistics method, Gaussian Markov Random Fields (GMRFs) implemented in R-INLA software, to produce bycatch risk maps using two large U.S. fisheries observer datasets. I compare the GMRF approach with two other species distribution model frameworks, generalized additive models (GAMs) and random forests (RF), and show how the models' performance differs for species with a broad range of bycatch rates, from leatherback sea turtles (0.7%) to blue sharks (96%) in the Hawaii longline fishery, and yelloweye rockfish (0.3%) to Pacific halibut (29%) in the West Coast groundfish trawl fishery.

I will conclude by highlighting other research opportunities at the intersection of applied math/statistics and fisheries science.

Where: 
Emmy Noether Rm, Millikan 1021, Pomona College

Dynamic Topic Models for the Classification of Music Files

05/01/2017 - 4:15pm
05/01/2017 - 5:15pm
Speaker: 
Rebecca Garnett (NAWCWD China Lake)
Abstract: 

With the advent of large-scale digital music repositories and personalized streaming radio software, there is a growing need for effective, autonomous methods of music categorization. The majority of published research in this area employ the physics of sound propagation and attempt to draw algorithmic parallels to the human auditory system for classification of music into different genres. However, deep neural network architectures are currently the state of the art for many classification problems. These deep networks typically require large amounts of data, long time scales, and extensive computational resources for training, putting constraints on their ability to be effectively implemented. Motivated by Mallat’s Invariant Scattering Convolution Networks (Bruna, Joan, and Stéphane Mallat. "Invariant scattering convolution networks." IEEE transactions on pattern analysis and machine intelligence 35.8 (2013): 1872-1886.), this work presents some preliminary studies to overcome these limitations. Mallat’s work demonstrated that respecting natural symmetries and adding robustness to deformations using non-linear functions can substantially improve classification. This study exploits these ideas to classify musical audio signals based on learned representations of their spectrograms’ dynamics. First, Nonnegative Matrix Factorization (NMF) was used to obtain a representation of spectrograms. Then the nonlinear max-pooling operator was used to add stability and reduce computational complexity. Finally, Hidden Markov Models (HMMs) were built to characterize the signal dynamics for each genre of music, and samples were classified according to how well they fit each HMM. Employing these HMMs induced a time-independent model, while the non-linear pooling step added robustness to deformations. Testing was executed against the well-studied GTZAN genre dataset and classification was performed using a multi-class Support Vector Machine (SVM). An 86% correct classification rate was achieved.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Introduction to Deep Learning and its Applications

03/06/2017 - 4:15pm
03/06/2017 - 5:10pm
Speaker: 
Huiyi Hu (Google)
Abstract: 

Deep learning uses artificial neural networks to uncover intricate pattern and structure in large data set. This type of method has achieved significant improvement in many fields such as visual object detection, object recognition, speech recognition and natural language processing (NLP) problems. As a result, it has been receiving rapidly increasing amount of attention from both academia and industry.

 

In this talk, I will first give an overview of the recent progress in deep learning along with a few examples. Then I will go into more technical details on what a basic deep learning model is made of and how it works (multi-layer neural networks and back propagation). At last I will present 1-2 examples in finer details, to show how deep learning technique is used in various applications.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

A Skew-Product Flow Model for Hybrid Dynamical Systems

04/03/2017 - 4:15pm
04/03/2017 - 5:15pm
Speaker: 
Kimberly Ayers (Pomona)
Abstract: 

In this talk, we consider a finite family of dynamical systems all on the same compact metric space, M, and study what happens when switch between these systems at regular time intervals. We begin by isolating and examining the “switching” dynamics by constructing a space made up of piecewise constant functions, and then study the dynamics of this space under the left shift map. We demonstrate that this function space, when paired with the behavior on M, gives a skew product flow. We then define and generalize various recurrence and limit concepts for this new skew product flow that demonstrates hybrid continuous and discrete behavior.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

organizational discussion

01/23/2017 - 4:15pm
01/23/2017 - 5:15pm
Speaker: 
TBA
Abstract: 

TBA

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Matrix Properties of a Class of Birth-Death Chains and Processes

04/24/2017 - 4:15pm
04/24/2017 - 5:15pm
Speaker: 
Alan Krinik (Cal Poly Pomona)
Abstract: 

We consider various recurrent birth-death chains on state space S1 = {0,1,2,...,H} and its associated dual birth-death chain on state space S2 = {−1,0,1,2,...,H} having absorbing states −1 and H. Assume P and P∗ are the one-step transition probability matrices of each birth-death chain respectively.

Conclusions:

1. P and P∗ have the same set of eigenvalues.

2. An explicit, simple formula for the eigenvalues of P (and P∗) is described as a function of H.

3. Pn and (P∗)n can be expressed exactly for n ∈ N.

Conclusion 2 follows from some nice linear algebra results on certain types of tridiagonal matrices found in Kouachi (2006, 2008). Our conclusion 3 has implications for finite-time gambler’s ruin problems. Some of the preceding results extend beyond birth-death chains to certain Markov chains. Our results for stochastic matrices suggest further ways to generalize Kouachi’s work. Explicit formula for finding transition probability functions of certain birth-death processes is also described.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College
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