Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item DVM: a deep learning algorithm for minimizing functionals(Montana State University - Bozeman, College of Letters & Science, 2022) Bair, Dominic Robert; Chairperson, Graduate Committee: Dominique ZossoThe use of data-driven techniques to solve PDEs is a rapidly developing field. Current deep learning methods can find solutions to high-dimensional PDEs with great accuracy and efficiency. However, for certain classes of problems these techniques may be inefficient. We focus on PDEs with a so-called 'variational formulation'. Here the solution to the PDE is represented as a minimizer or maximizer to a functional. We propose a family of novel deep learning algorithms to find these minimizers with similar accuracy and greater efficiency than techniques using the PDE formulation. These algorithms can be also be used to minimize functionals which do not have an equivalent PDE formulation. We call these algorithms 'Deep Variational Methods' (DVM).Item Adapting archetypal analysis to scientific imaging applications(Montana State University - Bozeman, College of Letters & Science, 2022) Potts, Catherine Gabriel; Chairperson, Graduate Committee: Dominique ZossoScientific imaging applications create large sets of high-dimensional data, which may be difficult to process using traditional supervised machine learning representative models. First, many representative models generate computational elements that are difficult to interpret in terms of the scientific application and second, the high embedding dimension of the images often makes generating the models computationally inefficient. We propose using archetypal analysis (AA) as the representative model for these scientific imaging problems, since the computational elements, so called archetypes, resemble members of the original dataset. Specifically, the archetypes are generated as extreme points to an approximation of the convex hull of the data cloud, which means they maintain the structure of individual data points. To improve the computational task of generating the AA model, we propose a sketch-based AA method which projects the data to a lower embedding dimension before calculating the computational elements, lowering computation time for these high-dimensional problems, while at the same time retaining the geometric structure enough so that the computational elements closely match the results of AA. We also applied a primal-dual hybrid gradient (PDHG) solver to the AA algorithm structure attempting to speed up computation. To verify the significance of the interpretation of AA, we applied AA to transient fluorescent calcium images, recorded in the Kunze Neuroengineering lab as videos, in order to determine whether or not adding different nanoparticles changed the way the neurons in culture communicate. We also applied our sketch-based AA method to other sorts of imaging data sets, exploring the differences between our method and the standard AA method. Our experimentation shows the different ways that AA can be adapted to scientific imaging applications, providing a machine learning representation model that is interpretable in the context of the imaging problem and verifies the benefits of the sketch-based method in terms of computation time.Item Development and applications of particle swarm optimization for constructing optimal experimental designs(Montana State University - Bozeman, College of Letters & Science, 2021) Walsh, Stephen Joseph; Chairperson, Graduate Committee: John J. BorkowskiThe primary objective motivating this dissertation was to illustrate the efficacy of particle swarm optimization (PSO) as the engine of an algorithm to generate optimal design of experiments (DoE). PSO is a wildly popular and successful metaheuristic and machine learning algorithm which makes no assumptions regarding the behavior of the function being optimized. Optimal DoE, in part thanks to modern computing, has become the current dominant paradigm for practitioners to generate a DoE with some desirable property. We bring together these concepts first by extending the PSO to optimizing functions that take matrix inputs. Julia software was developed for this purpose and validated against published results. A detailed benchmarking study of three PSO variants was conducted and a preferred version of the algorithm was identified for further research and application. Next, we implemented the approach to generating G-optimal designs--a difficult mini-max optimization problem. New heretofore unknown G-optimal designs have been produced and the efficacy of PSO in generating efficiently (w.r.t. computing time) highly G-optimal designs is compared to current published results. Next, a new algorithm for generating optimal designs with specified replication structure, and thereby guaranteed a degrees-of-freedom for estimating the pure error variance, is proposed, illustrated, benchmarked and validated. Last, we propose a new algorithm for generating optimal mixture experiment designs which implements a PSO type search using a non-Euclidean geometry (specifically the Aitchison geometry). In this setting the space of candidate matrices is the Cartesian product of standard (K - 1)-simplices. The algorithm is extended to mixture experiments with lower and upper constraints on the mixture proportions; in this setting, the space of candidate matrices is the Cartesian product of high-dimensional irregular convex polytopes. The algorithm is validated against very recent published results. In total, the work presented in this dissertation speaks very favorably to PSO as a tool for generating optimal DoEs. We believe this approach should become part of the standard machine learning and statistical tool box for generating optimal experimental designs.