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Stochastic Methods for Nonlinear Parabolic-Elliptic Systems
Stochastic Methods for Nonlinear Parabolic-Elliptic Systems
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202105621
- ISBN
- 9798265428493
- DDC
- 000
- 서명/저자
- Stochastic Methods for Nonlinear Parabolic-Elliptic Systems
- 발행사항
- [Sl] : Stanford University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 145 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 87-05, Section: B.
- 주기사항
- Advisor: Tartakovsky, Daniel.
- 학위논문주기
- Thesis (Ph.D.)--Stanford University, 2025.
- 초록/해제
- 요약Nonlinear parabolic-elliptic systems model several physical phenomena, such as electrodiffusion, chemotaxis, and self-gravitating particles. These systems typically arise in transport problems in which particles - ions, biological cells, cosmic dust - generate a potential with field forces that affect their own motion. This potential is described by an elliptic equation that depends on the particle distribution, which, in turn, is governed by a transport parabolic PDE influenced by the potential. This coupling often leads to strong nonlinearity and poses significant challenges to efficient and accurate numerical solutions.In this thesis, we develop three scalable and easily parallelizable classes of stochastic methods for these problems. The first, the particle methods, explicitly simulate an Ito process with space- and time-dependent drift whose evolving density solves the coupled system. The second, the integral kernel methods, introduce new integral kernels to directly time step the density of the stochastic process that solves the PDEs. The third, Feynman-Kac Walk-on-Spheres (FK-WoS) for PDE systems, extends the classical Walk-on-Spheres (WoS) method to solve the full system using coarsely sampled Brownian paths.In addition to these developments, we also present two key innovations that play a crucial role in FK-WoS but have wider applications. These include i) a fast algorithm based on confined stochastic bridges for computation of expectations of path integrals of killed processes and ii) the first WoS method for parabolic PDEs with varying coefficients.Prior to this work, the application of stochastic methods to nonlinear parabolic-elliptic systems has been very limited. We hope not only to demonstrate its potential but also to encourage further research on the topic. With the persistent rise of parallelism, we expect the performance of stochastic methods to continue to improve, challenging the dominance of the conventional, deterministic, mesh-based methods for PDEs.
- 일반주제명
- Spheres
- 일반주제명
- Boundary conditions
- 일반주제명
- Bandwidths
- 일반주제명
- Integrals
- 일반주제명
- Mathematics
- 기타저자
- Stanford University.
- 기본자료저록
- Dissertations Abstracts International. 87-05B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798265428493
■035 ▼a(MiAaPQ)AAI32316502
■035 ▼a(MiAaPQ)Stanfordkt552tx7075
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a000
■1001 ▼aMonteiro, Henrique Bittencourt Netto.
■24510▼aStochastic Methods for Nonlinear Parabolic-Elliptic Systems
■260 ▼a[Sl]▼bStanford University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a145 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 87-05, Section: B.
■500 ▼aAdvisor: Tartakovsky, Daniel.
■5021 ▼aThesis (Ph.D.)--Stanford University, 2025.
■520 ▼aNonlinear parabolic-elliptic systems model several physical phenomena, such as electrodiffusion, chemotaxis, and self-gravitating particles. These systems typically arise in transport problems in which particles - ions, biological cells, cosmic dust - generate a potential with field forces that affect their own motion. This potential is described by an elliptic equation that depends on the particle distribution, which, in turn, is governed by a transport parabolic PDE influenced by the potential. This coupling often leads to strong nonlinearity and poses significant challenges to efficient and accurate numerical solutions.In this thesis, we develop three scalable and easily parallelizable classes of stochastic methods for these problems. The first, the particle methods, explicitly simulate an Ito process with space- and time-dependent drift whose evolving density solves the coupled system. The second, the integral kernel methods, introduce new integral kernels to directly time step the density of the stochastic process that solves the PDEs. The third, Feynman-Kac Walk-on-Spheres (FK-WoS) for PDE systems, extends the classical Walk-on-Spheres (WoS) method to solve the full system using coarsely sampled Brownian paths.In addition to these developments, we also present two key innovations that play a crucial role in FK-WoS but have wider applications. These include i) a fast algorithm based on confined stochastic bridges for computation of expectations of path integrals of killed processes and ii) the first WoS method for parabolic PDEs with varying coefficients.Prior to this work, the application of stochastic methods to nonlinear parabolic-elliptic systems has been very limited. We hope not only to demonstrate its potential but also to encourage further research on the topic. With the persistent rise of parallelism, we expect the performance of stochastic methods to continue to improve, challenging the dominance of the conventional, deterministic, mesh-based methods for PDEs.
■590 ▼aSchool code: 0212.
■650 4▼aSpheres
■650 4▼aBoundary conditions
■650 4▼aBandwidths
■650 4▼aIntegrals
■650 4▼aMathematics
■690 ▼a0405
■71020▼aStanford University.
■7730 ▼tDissertations Abstracts International▼g87-05B.
■790 ▼a0212
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17360796▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
