Theses and Dissertations at Montana State University (MSU)
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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 Identification of distributed parameter systems using finite differences(Montana State University - Bozeman, College of Engineering, 1968) Collins, Paul LeeItem An adaptive stencil finite difference method for first order linear hyperbolic systems(Montana State University - Bozeman, College of Letters & Science, 1995) Hoar, Robert HenryItem On the solution of the transmission line equations(Montana State University - Bozeman, College of Letters & Science, 1951) McMurdo, Robert B.Item Discontinuous Galerkin finite element method for simulation of a transcription process model(Montana State University - Bozeman, College of Letters & Science, 2013) Thorenson, Jennifer Rae; Chairperson, Graduate Committee: Lisa DavisThe classical traffic flow PDE from the 1950s is used to model the biological process of transcription; the process of transferring genetic information from DNA to mRNA, in an E. coli gene. Polymerase elongating along the DNA strand encounter frequent but short pauses which are incorporated into the transcription model as several traffic lights. These pauses result in a delay in the transcription time and a delay function is defined to quantify this effect. Numerical simulations of the PDE model are conducted using a discontinuous Galerkin finite element method (DG) formulation. The entropy satisfying weak solution of the PDE model with a single pause is derived using the method of characteristics. This weak solution is used to show convergence of the DG formulation even though the flux function is not smooth. Once convergence of the DG solution is established for one pause, the numerical simulation for multiple pauses is used to calculate the delay due to the pauses and determine their effect on the overall transcription time. Preliminary parameter studies show a complex relationship between pause location and delay values. To determine the effect of pause clustering on protein production, an ongoing research goal is optimization of the delay function with respect to pause location. For preliminary work on this optimization problem, a DG formulation used to solve a sensitivity equation for a linear hyperbolic PDE with a spatial interface parameter is derived to gain insight for the more complicated nonlinear traffic flow PDE.