語系:
繁體中文
English
說明(常見問題)
回圖書館首頁
手機版館藏查詢
登入
回首頁
切換:
標籤
|
MARC模式
|
ISBD
Uncertainty Quantification for Fokke...
~
Zhu, Yuhua.
FindBook
Google Book
Amazon
博客來
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning.
紀錄類型:
書目-電子資源 : Monograph/item
正題名/作者:
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning./
作者:
Zhu, Yuhua.
出版者:
Ann Arbor : ProQuest Dissertations & Theses, : 2019,
面頁冊數:
179 p.
附註:
Source: Dissertations Abstracts International, Volume: 80-12, Section: B.
Contained By:
Dissertations Abstracts International80-12B.
標題:
Computational physics. -
電子資源:
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13882983
ISBN:
9781392162477
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning.
Zhu, Yuhua.
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning.
- Ann Arbor : ProQuest Dissertations & Theses, 2019 - 179 p.
Source: Dissertations Abstracts International, Volume: 80-12, Section: B.
Thesis (Ph.D.)--The University of Wisconsin - Madison, 2019.
This item must not be sold to any third party vendors.
This thesis focuses on studying Fokker Planck type equations in two aspects Mathematical and numerical analysis on uncertain problems and related problems in machine learning.In Chapters 2, 3 and 4, we study the Vlasov-Poisson-Fokker-Planck (VPFP) system with uncertainty and multiple scales. Here the uncertainty, modeled by multi-dimensional random variables, enters the system through the initial data, while the multiple scales lead the system to its high-field or parabolic regimes. We obtain a sharp decay rate of the solution to the global Maxwellian, which reveals that the VPFP system is decreasingly sensitive to the initial perturbation as the Knudsen number goes to zero. We develop a stochastic asymptotic preserving (s-AP) scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime with uncertainty based on the generalized polynomial chaos stochastic Galerkin framework (gPC-SG). The sharp regularity estimates in terms of the Knudsen number lead to the stability of the gPC-SG method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy uniform in the Knudsen number. Numerical examples are given to validate the accuracy and s-AP properties of the proposed method.In Chapters 5, we consider the Vlasov-Fokker-Planck equation with a random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with a suitable assumption on the anisotropy of the randomness, we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in Chkifa, et al., (2014), we develop a residual based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multiscale linear kinetic equation, when using numerical schemes that are time-dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.In Chapters 6, we study the generalizations of mini-batch stochastic gradient descent (SGD) method in deep neural networks. We theoretically justify a hypothesis that large-batch SGD tends to converge to sharp minimizers by providing new properties of SGD in both finite-time and asymptotic regimes. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime and prove that SGD tends to converge to flatter minima in the asymptotic regime (although may take exponential time to converge) regardless of the batch size.
ISBN: 9781392162477Subjects--Topical Terms:
3343998
Computational physics.
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning.
LDR
:03918nmm a2200349 4500
001
2210823
005
20191121124317.5
008
201008s2019 ||||||||||||||||| ||eng d
020
$a
9781392162477
035
$a
(MiAaPQ)AAI13882983
035
$a
(MiAaPQ)wisc:16152
035
$a
AAI13882983
040
$a
MiAaPQ
$c
MiAaPQ
100
1
$a
Zhu, Yuhua.
$3
3437965
245
1 0
$a
Uncertainty Quantification for Fokker Planck Type Equations and Related Problems in Machine Learning.
260
1
$a
Ann Arbor :
$b
ProQuest Dissertations & Theses,
$c
2019
300
$a
179 p.
500
$a
Source: Dissertations Abstracts International, Volume: 80-12, Section: B.
500
$a
Publisher info.: Dissertation/Thesis.
500
$a
Advisor: Li, Qin.
502
$a
Thesis (Ph.D.)--The University of Wisconsin - Madison, 2019.
506
$a
This item must not be sold to any third party vendors.
520
$a
This thesis focuses on studying Fokker Planck type equations in two aspects Mathematical and numerical analysis on uncertain problems and related problems in machine learning.In Chapters 2, 3 and 4, we study the Vlasov-Poisson-Fokker-Planck (VPFP) system with uncertainty and multiple scales. Here the uncertainty, modeled by multi-dimensional random variables, enters the system through the initial data, while the multiple scales lead the system to its high-field or parabolic regimes. We obtain a sharp decay rate of the solution to the global Maxwellian, which reveals that the VPFP system is decreasingly sensitive to the initial perturbation as the Knudsen number goes to zero. We develop a stochastic asymptotic preserving (s-AP) scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime with uncertainty based on the generalized polynomial chaos stochastic Galerkin framework (gPC-SG). The sharp regularity estimates in terms of the Knudsen number lead to the stability of the gPC-SG method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy uniform in the Knudsen number. Numerical examples are given to validate the accuracy and s-AP properties of the proposed method.In Chapters 5, we consider the Vlasov-Fokker-Planck equation with a random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with a suitable assumption on the anisotropy of the randomness, we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in Chkifa, et al., (2014), we develop a residual based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multiscale linear kinetic equation, when using numerical schemes that are time-dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.In Chapters 6, we study the generalizations of mini-batch stochastic gradient descent (SGD) method in deep neural networks. We theoretically justify a hypothesis that large-batch SGD tends to converge to sharp minimizers by providing new properties of SGD in both finite-time and asymptotic regimes. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime and prove that SGD tends to converge to flatter minima in the asymptotic regime (although may take exponential time to converge) regardless of the batch size.
590
$a
School code: 0262.
650
4
$a
Computational physics.
$3
3343998
650
4
$a
Applied Mathematics.
$3
1669109
650
4
$a
Artificial intelligence.
$3
516317
650
4
$a
Computer science.
$3
523869
690
$a
0216
690
$a
0364
690
$a
0800
690
$a
0984
710
2
$a
The University of Wisconsin - Madison.
$b
Mathematics.
$3
2101076
773
0
$t
Dissertations Abstracts International
$g
80-12B.
790
$a
0262
791
$a
Ph.D.
792
$a
2019
793
$a
English
856
4 0
$u
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13882983
筆 0 讀者評論
館藏地:
全部
電子資源
出版年:
卷號:
館藏
1 筆 • 頁數 1 •
1
條碼號
典藏地名稱
館藏流通類別
資料類型
索書號
使用類型
借閱狀態
預約狀態
備註欄
附件
W9387372
電子資源
11.線上閱覽_V
電子書
EB
一般使用(Normal)
在架
0
1 筆 • 頁數 1 •
1
多媒體
評論
新增評論
分享你的心得
Export
取書館
處理中
...
變更密碼
登入