Joy Mueller Dissertation Defense

Joy Mueller, Department of Applied Mathematics, University of Colorado Boulder

Dissertation Defense

Chemical reaction networks are fundamental computational models that are used to study the behavior of chemical reactions in well-mixed solutions. They have been used extensively to model a broad range of biological systems, and are primarily used when the more traditional model of deterministic continuous mass action kinetics is invalid due to small molecular counts.

We present a perfect sampling algorithm that can produce error-free samples from the stationary distributions of stochastic models for coupled, linear chemical reaction networks. The state spaces of such networks are given by all permissible combinations of molecular counts for each chemical species, and thereby grow exponentially with the numbers of species in the network. To avoid simulations involving large numbers of states, we propose subsets of chemical species such that coupling of paths started from these states guarantee coupling of paths started from all states in the state space and we show for the well-known Reversible Michaelis-Menten model that the subset does in fact guarantee perfect draws from the stationary distribution of interest. We compare solutions computed in two ways with this algorithm to those found analytically using the chemical master equation and we compare the distribution of coupling times for the two simulation approaches.

We then describe a Jackson queuing network representation for chemical reactions. This approach does not require the state space of the chemical reaction network to be found explicitly, but instead relies on conservation laws intrinsic to the network to describe the set of states in the space. Using properties of Jackson networks, we demonstrate how exact product-form stationary solutions can be computed for the class of unbranched monomolecular reaction networks, and we provide the exact solutions for both the $n$-species linear reactions and $n$-species rings which belong to this class. Finally, we show how the solutions for certain bimolecular reactions can be found by relating the space of chemical species in the network to the corresponding space of chemical complexes.

Image segmentation is an essential component in many image processing and computer vision tasks. The primary goal of image segmentation is to simplify an image for easier analysis, and there are two broad approaches for achieving this: edge based methods, which extract the boundaries of specific known objects, and region based methods, which partition the image into regions that are statistically homogeneous. One of the more prominent edge finding methods is known as the Level Set Method. This algorithm evolves a zero-level contour in the image plane with gradient descent until the contour has converged to the object boundaries. While the classical level set method and its variants have proved successful in segmenting real images, they are susceptible to becoming stuck in noisy regions of the image plane. We propose a level set inspired image segmentation algorithm that can detect object boundaries by making use of random point initialization. We demonstrate the efficacy of our approach on a set of synthetic images and we compare the performance of our method on real image to that of the well-known Canny Method.

Dial-In Information

https://cuboulder.zoom.us/j/93381209313

Monday, June 28, 2021 at 12:00pm to 2:00pm

Virtual Event
Event Type

Colloquium/Seminar

Interests

Science & Technology, Research & Innovation

Audience

Faculty, Students, Graduate Students, Postdoc

College, School & Unit

Engineering & Applied Science

Group
Applied Mathematics
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