{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/106319"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/106319","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Efficient solution of the Fokker-Planck Equation via smooth particle hydrodynamics for nonlinear estimation","abstract":"Modeling and predicting the transient behavior of higher dimensional nonlinear dynamical systems subject to Gaussian white noise excitation remains an open problem with broad application to nonlinear estimation, uncertainty quantification, and reliability to name but a few. These problems remain open in large part due to the Curse of Dimensionality, where computational costs tend to increase exponentially, in time and/or memory, with the number of states of the system. This is especially true for sampling based methods that, while robust and often very simple to implement, quickly become computationally cost ineffective in many applications. Parametric models are usually able to scale far more efficiently as state dimension increases, but they are often limited to specific classes of systems due to not being as robust as sampling based methods. Another method to analyze the uncertainty in those dynamical systems is to solve the system's corresponding Fokker-Planck equation. The Fokker-Planck Equation (FPE) is the degenerate parabolic partial differential continuity equation that fully defines the evolution of the states' transition probability density function (PDF) over time. This also makes the transient solution to the FPE an option for estimation purposes. Since any dynamical system that is subject to Gaussian white noise excitation, either additive and/or multiplicative, and satisfies the Markov assumption has a corresponding FPE; this represents an extremely large class of systems of engineering interest. However, finding solutions to the transient Fokker-Planck equation remains difficult for the same reasons mentioned previously; solving partial differential equations in high dimensional state spaces also runs up against the Curse of Dimensionality. Analytical solutions only exist for linear systems and a handful of low dimension nonlinear systems to make evaluating simulation efficacy even more difficult. This work extensively modifies the Smooth Particle Hydrodynamics (SPH) meshfree method in order to rigorously assess its accuracy in simulating up to four state Fokker-Planck equations and explore its suitability for scaling further into higher dimensions. Due to the different potential applications of transient FPE solutions, SPH algorithm performance is analyzed from both an accuracy focused, runtime agnostic standpoint (relevant to reliability analysis) and a runtime focused, accuracy sacrificing standpoint more suited to nonlinear estimation. Rigorous analysis of the convergence characteristics of the algorithm as particle spatial resolution changes is given, as well as the performance impacts of the various accuracy and runtime modifications. The SPH algorithm turns out to be capable of simulating the transient FPE even at extremely low spatial resolution in a stable and robust manner across a variety of systems. However, the mathematical properties of the SPH algorithm make it unable to achieve an arbitrary accuracy concomitant with increasing particle spatial density, especially once one gets to four dimensions. Steep diminishing returns on increased accuracy for computational effort expended are also observed as particle count and state dimension increase. While not very suitable for applications where accuracy is essential far out into the tails of the time-dependent PDF, the SPH algorithm's ability to reliably approximate the probability evolution over time even at very low spatial resolution makes it a good candidate for use in nonlinear estimation. A novel Bayesian Filter based on the SPH simulated FPE is presented in detail, along with a resampling methodology developed to efficiently perform measurement likelihood updates without degeneracy of the SPH particle field occurring. This new FPE-SPH Filter is compared directly to the Extended Kalman Filter and Particle Filter with importance resampling for each system. The FPE-SPH Filter is able to accurately and robustly estimate a variety of two and four state dynamical systems with long times between state measurements, including the bi-modal Duffing oscillator that renders approaches such as the Extended Kalman Filter inconsistent. Lastly, a new method of visualizing the error of high dimensional PDFs is presented, referred to as Sub-Field Signatures (SFSs). These signatures make it easier to analyze at a glance both the distribution and balance of the field error to assess simulation performance at a given time regardless of state dimension. This provides a more information dense way to analyze error than traditional RMS error values, but one that is not nearly as involved as the collections of marginal distributions typically used to examine higher dimensional PDFs.","abstract_html":"Modeling and predicting the transient behavior of higher dimensional nonlinear dynamical systems subject to Gaussian white noise excitation remains an open problem with broad application to nonlinear estimation, uncertainty quantification, and reliability to name but a few. These problems remain open in large part due to the Curse of Dimensionality, where computational costs tend to increase exponentially, in time and/or memory, with the number of states of the system. This is especially true for sampling based methods that, while robust and often very simple to implement, quickly become computationally cost ineffective in many applications. Parametric models are usually able to scale far more efficiently as state dimension increases, but they are often limited to specific classes of systems due to not being as robust as sampling based methods. Another method to analyze the uncertainty in those dynamical systems is to solve the system's corresponding Fokker-Planck equation. The Fokker-Planck Equation (FPE) is the degenerate parabolic partial differential continuity equation that fully defines the evolution of the states' transition probability density function (PDF) over time. This also makes the transient solution to the FPE an option for estimation purposes. Since any dynamical system that is subject to Gaussian white noise excitation, either additive and/or multiplicative, and satisfies the Markov assumption has a corresponding FPE; this represents an extremely large class of systems of engineering interest. However, finding solutions to the transient Fokker-Planck equation remains difficult for the same reasons mentioned previously; solving partial differential equations in high dimensional state spaces also runs up against the Curse of Dimensionality. Analytical solutions only exist for linear systems and a handful of low dimension nonlinear systems to make evaluating simulation efficacy even more difficult. This work extensively modifies the Smooth Particle Hydrodynamics (SPH) meshfree method in order to rigorously assess its accuracy in simulating up to four state Fokker-Planck equations and explore its suitability for scaling further into higher dimensions. Due to the different potential applications of transient FPE solutions, SPH algorithm performance is analyzed from both an accuracy focused, runtime agnostic standpoint (relevant to reliability analysis) and a runtime focused, accuracy sacrificing standpoint more suited to nonlinear estimation. Rigorous analysis of the convergence characteristics of the algorithm as particle spatial resolution changes is given, as well as the performance impacts of the various accuracy and runtime modifications. The SPH algorithm turns out to be capable of simulating the transient FPE even at extremely low spatial resolution in a stable and robust manner across a variety of systems. However, the mathematical properties of the SPH algorithm make it unable to achieve an arbitrary accuracy concomitant with increasing particle spatial density, especially once one gets to four dimensions. Steep diminishing returns on increased accuracy for computational effort expended are also observed as particle count and state dimension increase. While not very suitable for applications where accuracy is essential far out into the tails of the time-dependent PDF, the SPH algorithm's ability to reliably approximate the probability evolution over time even at very low spatial resolution makes it a good candidate for use in nonlinear estimation. A novel Bayesian Filter based on the SPH simulated FPE is presented in detail, along with a resampling methodology developed to efficiently perform measurement likelihood updates without degeneracy of the SPH particle field occurring. This new FPE-SPH Filter is compared directly to the Extended Kalman Filter and Particle Filter with importance resampling for each system. The FPE-SPH Filter is able to accurately and robustly estimate a variety of two and four state dynamical systems with long times between state measurements, including the bi-modal Duffing oscillator that renders approaches such as the Extended Kalman Filter inconsistent. Lastly, a new method of visualizing the error of high dimensional PDFs is presented, referred to as Sub-Field Signatures (SFSs). These signatures make it easier to analyze at a glance both the distribution and balance of the field error to assess simulation performance at a given time regardless of state dimension. This provides a more information dense way to analyze error than traditional RMS error values, but one that is not nearly as involved as the collections of marginal distributions typically used to examine higher dimensional PDFs.","abstract_has_math":false,"creators":["Duffy, Michael"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Aerospace Engineering","degree_department":null,"school":null,"contributors":["Bergman, Lawrence","Chung, Soon-Jo","Masud, Arif","Panesi, Marco"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-03-02T22:12:09Z","date_published":"2020-03-02T22:12:09Z","updated_at":"2026-07-22T22:24:45Z","subjects":["Fokker-Planck Equation","smooth particle hydrodynamics (SPH)","meshfree methods","nonlinear filtering","state estimation","Bayesian filter","high-dimensional analysis","stochastic processes"],"languages":["en"],"rights":["Copyright 2019 - Michael Duffy"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/106319","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bergman, Lawrence","Chung, Soon-Jo","Masud, Arif","Panesi, Marco"]},{"key":"dc:creator","label":"Author","values":["Duffy, Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-03-02T22:12:09Z","2022-03-03T10:15:19Z","2019-09-30","2019-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Aerospace Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fokker-Planck Equation","smooth particle hydrodynamics (SPH)","meshfree methods","nonlinear filtering","state estimation","Bayesian filter","high-dimensional analysis","stochastic processes"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 - Michael Duffy"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/106319"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Modeling and predicting the transient behavior of higher dimensional nonlinear dynamical systems subject to Gaussian white noise excitation remains an open problem with broad application to nonlinear estimation, uncertainty quantification, and reliability to name but a few. These problems remain open in large part due to the Curse of Dimensionality, where computational costs tend to increase exponentially, in time and/or memory, with the number of states of the system. This is especially true for sampling based methods that, while robust and often very simple to implement, quickly become computationally cost ineffective in many applications. Parametric models are usually able to scale far more efficiently as state dimension increases, but they are often limited to specific classes of systems due to not being as robust as sampling based methods. Another method to analyze the uncertainty in those dynamical systems is to solve the system's corresponding Fokker-Planck equation. The Fokker-Planck Equation (FPE) is the degenerate parabolic partial differential continuity equation that fully defines the evolution of the states' transition probability density function (PDF) over time. This also makes the transient solution to the FPE an option for estimation purposes. Since any dynamical system that is subject to Gaussian white noise excitation, either additive and/or multiplicative, and satisfies the Markov assumption has a corresponding FPE; this represents an extremely large class of systems of engineering interest. However, finding solutions to the transient Fokker-Planck equation remains difficult for the same reasons mentioned previously; solving partial differential equations in high dimensional state spaces also runs up against the Curse of Dimensionality. Analytical solutions only exist for linear systems and a handful of low dimension nonlinear systems to make evaluating simulation efficacy even more difficult. This work extensively modifies the Smooth Particle Hydrodynamics (SPH) meshfree method in order to rigorously assess its accuracy in simulating up to four state Fokker-Planck equations and explore its suitability for scaling further into higher dimensions. Due to the different potential applications of transient FPE solutions, SPH algorithm performance is analyzed from both an accuracy focused, runtime agnostic standpoint (relevant to reliability analysis) and a runtime focused, accuracy sacrificing standpoint more suited to nonlinear estimation. Rigorous analysis of the convergence characteristics of the algorithm as particle spatial resolution changes is given, as well as the performance impacts of the various accuracy and runtime modifications. The SPH algorithm turns out to be capable of simulating the transient FPE even at extremely low spatial resolution in a stable and robust manner across a variety of systems. However, the mathematical properties of the SPH algorithm make it unable to achieve an arbitrary accuracy concomitant with increasing particle spatial density, especially once one gets to four dimensions. Steep diminishing returns on increased accuracy for computational effort expended are also observed as particle count and state dimension increase. While not very suitable for applications where accuracy is essential far out into the tails of the time-dependent PDF, the SPH algorithm's ability to reliably approximate the probability evolution over time even at very low spatial resolution makes it a good candidate for use in nonlinear estimation. A novel Bayesian Filter based on the SPH simulated FPE is presented in detail, along with a resampling methodology developed to efficiently perform measurement likelihood updates without degeneracy of the SPH particle field occurring. This new FPE-SPH Filter is compared directly to the Extended Kalman Filter and Particle Filter with importance resampling for each system. The FPE-SPH Filter is able to accurately and robustly estimate a variety of two and four state dynamical systems with long times between state measurements, including the bi-modal Duffing oscillator that renders approaches such as the Extended Kalman Filter inconsistent. Lastly, a new method of visualizing the error of high dimensional PDFs is presented, referred to as Sub-Field Signatures (SFSs). These signatures make it easier to analyze at a glance both the distribution and balance of the field error to assess simulation performance at a given time regardless of state dimension. This provides a more information dense way to analyze error than traditional RMS error values, but one that is not nearly as involved as the collections of marginal distributions typically used to examine higher dimensional PDFs.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2021-12-01","The student, Michael Duffy, accepted the attached license on 2019-09-28 at 01:51.","The student, Michael Duffy, submitted this Dissertation for approval on 2019-09-28 at 02:05.","This Dissertation was approved for publication on 2019-09-30 at 16:30.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14476 on 2020-02-28 at 17:20:27","Made available in DSpace on 2020-03-02T22:12:09Z (GMT). No. of bitstreams: 3 DUFFY-DISSERTATION-2019.pdf: 164402496 bytes, checksum: a0e6a918f5824098d65b1cc1d08738ef (MD5) LICENSE.txt: 4210 bytes, checksum: 49de7330087fcd3a8a7b56c6dd091b00 (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: 185543731f8d3f8f50eaca17af1eb7de (MD5) Previous issue date: 2019-09-30","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:12:26Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:15:21Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:18:25Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 113860 on 2022-03-03T10:15:19Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Efficient solution of the Fokker-Planck Equation via smooth particle hydrodynamics for nonlinear estimation"]}]}],"canonical_facts":{"dc:contributor":["Bergman, Lawrence","Chung, Soon-Jo","Masud, Arif","Panesi, Marco"],"dc:creator":["Duffy, Michael"],"dc:date":["2020-03-02T22:12:09Z","2022-03-03T10:15:19Z","2019-09-30","2019-12"],"dc:description":["Modeling and predicting the transient behavior of higher dimensional nonlinear dynamical systems subject to Gaussian white noise excitation remains an open problem with broad application to nonlinear estimation, uncertainty quantification, and reliability to name but a few. These problems remain open in large part due to the Curse of Dimensionality, where computational costs tend to increase exponentially, in time and/or memory, with the number of states of the system. This is especially true for sampling based methods that, while robust and often very simple to implement, quickly become computationally cost ineffective in many applications. Parametric models are usually able to scale far more efficiently as state dimension increases, but they are often limited to specific classes of systems due to not being as robust as sampling based methods. Another method to analyze the uncertainty in those dynamical systems is to solve the system's corresponding Fokker-Planck equation. The Fokker-Planck Equation (FPE) is the degenerate parabolic partial differential continuity equation that fully defines the evolution of the states' transition probability density function (PDF) over time. This also makes the transient solution to the FPE an option for estimation purposes. Since any dynamical system that is subject to Gaussian white noise excitation, either additive and/or multiplicative, and satisfies the Markov assumption has a corresponding FPE; this represents an extremely large class of systems of engineering interest. However, finding solutions to the transient Fokker-Planck equation remains difficult for the same reasons mentioned previously; solving partial differential equations in high dimensional state spaces also runs up against the Curse of Dimensionality. Analytical solutions only exist for linear systems and a handful of low dimension nonlinear systems to make evaluating simulation efficacy even more difficult. This work extensively modifies the Smooth Particle Hydrodynamics (SPH) meshfree method in order to rigorously assess its accuracy in simulating up to four state Fokker-Planck equations and explore its suitability for scaling further into higher dimensions. Due to the different potential applications of transient FPE solutions, SPH algorithm performance is analyzed from both an accuracy focused, runtime agnostic standpoint (relevant to reliability analysis) and a runtime focused, accuracy sacrificing standpoint more suited to nonlinear estimation. Rigorous analysis of the convergence characteristics of the algorithm as particle spatial resolution changes is given, as well as the performance impacts of the various accuracy and runtime modifications. The SPH algorithm turns out to be capable of simulating the transient FPE even at extremely low spatial resolution in a stable and robust manner across a variety of systems. However, the mathematical properties of the SPH algorithm make it unable to achieve an arbitrary accuracy concomitant with increasing particle spatial density, especially once one gets to four dimensions. Steep diminishing returns on increased accuracy for computational effort expended are also observed as particle count and state dimension increase. While not very suitable for applications where accuracy is essential far out into the tails of the time-dependent PDF, the SPH algorithm's ability to reliably approximate the probability evolution over time even at very low spatial resolution makes it a good candidate for use in nonlinear estimation. A novel Bayesian Filter based on the SPH simulated FPE is presented in detail, along with a resampling methodology developed to efficiently perform measurement likelihood updates without degeneracy of the SPH particle field occurring. This new FPE-SPH Filter is compared directly to the Extended Kalman Filter and Particle Filter with importance resampling for each system. The FPE-SPH Filter is able to accurately and robustly estimate a variety of two and four state dynamical systems with long times between state measurements, including the bi-modal Duffing oscillator that renders approaches such as the Extended Kalman Filter inconsistent. Lastly, a new method of visualizing the error of high dimensional PDFs is presented, referred to as Sub-Field Signatures (SFSs). These signatures make it easier to analyze at a glance both the distribution and balance of the field error to assess simulation performance at a given time regardless of state dimension. This provides a more information dense way to analyze error than traditional RMS error values, but one that is not nearly as involved as the collections of marginal distributions typically used to examine higher dimensional PDFs.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2021-12-01","The student, Michael Duffy, accepted the attached license on 2019-09-28 at 01:51.","The student, Michael Duffy, submitted this Dissertation for approval on 2019-09-28 at 02:05.","This Dissertation was approved for publication on 2019-09-30 at 16:30.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14476 on 2020-02-28 at 17:20:27","Made available in DSpace on 2020-03-02T22:12:09Z (GMT). No. of bitstreams: 3 DUFFY-DISSERTATION-2019.pdf: 164402496 bytes, checksum: a0e6a918f5824098d65b1cc1d08738ef (MD5) LICENSE.txt: 4210 bytes, checksum: 49de7330087fcd3a8a7b56c6dd091b00 (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: 185543731f8d3f8f50eaca17af1eb7de (MD5) Previous issue date: 2019-09-30","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:12:26Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:15:21Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 113860 Lift date: 2022-03-02T22:18:25Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 113860 on 2022-03-03T10:15:19Z."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/106319"],"dc:language":["en"],"dc:rights":["Copyright 2019 - Michael Duffy"],"dc:subject":["Fokker-Planck Equation","smooth particle hydrodynamics (SPH)","meshfree methods","nonlinear filtering","state estimation","Bayesian filter","high-dimensional analysis","stochastic processes"],"dc:title":["Efficient solution of the Fokker-Planck Equation via smooth particle hydrodynamics for nonlinear estimation"],"dc:type":["text"],"thesis:degree_discipline":["Aerospace Engineering"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:45Z"}