{"id":{"repo_id":"colo-mines","oai_identifier":"oai:repository.mines.edu:11124/179142"},"canonical_url":"https://search.dev.ndltd.org/etd/colo-mines/oai:repository.mines.edu:11124/179142","repository":{"repo_id":"colo-mines","name":"Colorado School of Mines","base_url":"https://repository.mines.edu/server/oai/request"},"display":{"title":"Applications of smoothness-increasing accuracy-conserving filtering to particle-in-cell data denoising, enhanced multiresolution analysis, and entropy-correction schemes","abstract":"The numerical solution of partial differential equations can be very challenging, and while different numerical methods have their own advantages, there is no single numerical method without drawbacks. For example, data arising from Particle-In-Cell (PIC) methods have noise in their numerical solutions owing to the use of finitely many particles. The underlying mesh and corresponding approximation in Discontinuous Galerkin (DG) methods may not be of sufficient spatial resolution to capture localized solution features. Lastly, additional conservation equations such as entropy conditions satisfied by analytic solutions to PDEs are often not satisfied by their discrete counterparts. These deficiencies can be ameliorated by auxiliary techniques and method augmentations. It is the purpose of this thesis to detail three such improvements making use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters, proceeding in order of increasing embedment of SIAC methodologies within the underlying numerical process. First, we detail the application of SIAC filters as denoisers of data arising from PIC simulations and demonstrate how careful tuning of these filters enables noise-reduction in the presence of non-periodic boundaries. Furthermore, in the computation of Bohm speed estimates from PIC data, we show that SIAC filters enable a dramatic reduction in the quantity of PIC data needed. Having applied SIAC filters to the equivalent of finite volume data, we next consider higher-order piecewise polynomial data. Here we show that SIAC filtering can be used to increase the resolution of coarse mesh polynomial data in multiple dimensions and over nonuniform meshes. The resulting enhancement procedure can be viewed within the framework of multiresolution analysis (MRA), making it a natural candidate for application in mesh adaptive DG schemes, where it provides a means of providing more accurate approximations with reduced degrees of freedom as compared to uniformly refined data. To conclude, we break the requirement of applying SIAC methodologies to effectively stationary data by fully integrating SIAC filters with DG numerical methods in the context of entropy-correction schemes. In this application, SIAC filters are applied to ensure that approximations produced by the baseline DG method satisfy additional physically motivated conservation conditions, namely energy conservation, for the duration of a simulation.","abstract_html":"The numerical solution of partial differential equations can be very challenging, and while different numerical methods have their own advantages, there is no single numerical method without drawbacks. For example, data arising from Particle-In-Cell (PIC) methods have noise in their numerical solutions owing to the use of finitely many particles. The underlying mesh and corresponding approximation in Discontinuous Galerkin (DG) methods may not be of sufficient spatial resolution to capture localized solution features. Lastly, additional conservation equations such as entropy conditions satisfied by analytic solutions to PDEs are often not satisfied by their discrete counterparts. These deficiencies can be ameliorated by auxiliary techniques and method augmentations. It is the purpose of this thesis to detail three such improvements making use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters, proceeding in order of increasing embedment of SIAC methodologies within the underlying numerical process. First, we detail the application of SIAC filters as denoisers of data arising from PIC simulations and demonstrate how careful tuning of these filters enables noise-reduction in the presence of non-periodic boundaries. Furthermore, in the computation of Bohm speed estimates from PIC data, we show that SIAC filters enable a dramatic reduction in the quantity of PIC data needed. Having applied SIAC filters to the equivalent of finite volume data, we next consider higher-order piecewise polynomial data. Here we show that SIAC filtering can be used to increase the resolution of coarse mesh polynomial data in multiple dimensions and over nonuniform meshes. The resulting enhancement procedure can be viewed within the framework of multiresolution analysis (MRA), making it a natural candidate for application in mesh adaptive DG schemes, where it provides a means of providing more accurate approximations with reduced degrees of freedom as compared to uniformly refined data. To conclude, we break the requirement of applying SIAC methodologies to effectively stationary data by fully integrating SIAC filters with DG numerical methods in the context of entropy-correction schemes. In this application, SIAC filters are applied to ensure that approximations produced by the baseline DG method satisfy additional physically motivated conservation conditions, namely energy conservation, for the duration of a simulation.","abstract_has_math":false,"creators":["Picklo, Matthew John, Jr."],"institution":"Colorado School of Mines. Arthur Lakes Library","degree_name":"Doctor of Philosophy (Ph.D.)","degree_level":"Doctoral","degree_discipline":"Applied Mathematics and Statistics","degree_department":null,"school":null,"contributors":[],"advisors":["Ryan, Jennifer K.","Edoh, Ayaboe"],"committee_chairs":[],"committee_members":["Fasshauer, Gregory","Wakin, Michael B.","Wu Fung, Samy"],"year":2023,"date_issued":"2023","date_published":"2023","updated_at":"2026-07-24T01:42:22Z","subjects":["discontinuous Galerkin","entropy-correction","multiresolution analysis","SIAC filtering"],"languages":["eng","English"],"rights":["Copyright of the original work is retained by the author."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["T 9677"],"render_values":[{"text":"T 9677","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/11124/179142","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ryan, Jennifer K.","Edoh, Ayaboe"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Fasshauer, Gregory","Wakin, Michael B.","Wu Fung, Samy"]},{"key":"dc:creator","label":"Author","values":["Picklo, Matthew John, Jr."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2024-07-12T17:28:35Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-07-12T17:28:35Z"]},{"key":"dc:date.issued","label":"Date","values":["2023"]},{"key":"dc:publisher","label":"Institution","values":["Colorado School of Mines. 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For example, data arising from Particle-In-Cell (PIC) methods have noise in their numerical solutions owing to the use of finitely many particles. The underlying mesh and corresponding approximation in Discontinuous Galerkin (DG) methods may not be of sufficient spatial resolution to capture localized solution features. Lastly, additional conservation equations such as entropy conditions satisfied by analytic solutions to PDEs are often not satisfied by their discrete counterparts. These deficiencies can be ameliorated by auxiliary techniques and method augmentations. It is the purpose of this thesis to detail three such improvements making use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters, proceeding in order of increasing embedment of SIAC methodologies within the underlying numerical process. First, we detail the application of SIAC filters as denoisers of data arising from PIC simulations and demonstrate how careful tuning of these filters enables noise-reduction in the presence of non-periodic boundaries. Furthermore, in the computation of Bohm speed estimates from PIC data, we show that SIAC filters enable a dramatic reduction in the quantity of PIC data needed. Having applied SIAC filters to the equivalent of finite volume data, we next consider higher-order piecewise polynomial data. Here we show that SIAC filtering can be used to increase the resolution of coarse mesh polynomial data in multiple dimensions and over nonuniform meshes. The resulting enhancement procedure can be viewed within the framework of multiresolution analysis (MRA), making it a natural candidate for application in mesh adaptive DG schemes, where it provides a means of providing more accurate approximations with reduced degrees of freedom as compared to uniformly refined data. To conclude, we break the requirement of applying SIAC methodologies to effectively stationary data by fully integrating SIAC filters with DG numerical methods in the context of entropy-correction schemes. 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For example, data arising from Particle-In-Cell (PIC) methods have noise in their numerical solutions owing to the use of finitely many particles. The underlying mesh and corresponding approximation in Discontinuous Galerkin (DG) methods may not be of sufficient spatial resolution to capture localized solution features. Lastly, additional conservation equations such as entropy conditions satisfied by analytic solutions to PDEs are often not satisfied by their discrete counterparts. These deficiencies can be ameliorated by auxiliary techniques and method augmentations. It is the purpose of this thesis to detail three such improvements making use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters, proceeding in order of increasing embedment of SIAC methodologies within the underlying numerical process. First, we detail the application of SIAC filters as denoisers of data arising from PIC simulations and demonstrate how careful tuning of these filters enables noise-reduction in the presence of non-periodic boundaries. Furthermore, in the computation of Bohm speed estimates from PIC data, we show that SIAC filters enable a dramatic reduction in the quantity of PIC data needed. Having applied SIAC filters to the equivalent of finite volume data, we next consider higher-order piecewise polynomial data. Here we show that SIAC filtering can be used to increase the resolution of coarse mesh polynomial data in multiple dimensions and over nonuniform meshes. The resulting enhancement procedure can be viewed within the framework of multiresolution analysis (MRA), making it a natural candidate for application in mesh adaptive DG schemes, where it provides a means of providing more accurate approximations with reduced degrees of freedom as compared to uniformly refined data. To conclude, we break the requirement of applying SIAC methodologies to effectively stationary data by fully integrating SIAC filters with DG numerical methods in the context of entropy-correction schemes. In this application, SIAC filters are applied to ensure that approximations produced by the baseline DG method satisfy additional physically motivated conservation conditions, namely energy conservation, for the duration of a simulation."],"dc:format.medium":["born digital","doctoral dissertations"],"dc:identifier":["Picklo_mines_0052E_12759.pdf","T 9677"],"dc:identifier.uri":["https://hdl.handle.net/11124/179142"],"dc:language":["English"],"dc:language.iso":["eng"],"dc:publisher":["Colorado School of Mines. 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