一类随机微分方程的随机源反演方法和性质

陈琛; 冯晓莉; 陈汉章

doi:10.21656/1000-0887.430170

一类随机微分方程的随机源反演方法和性质

doi: 10.21656/1000-0887.430170

西安电子科技大学数学与统计学院，西安 710126

基金项目:

陕西省自然科学基础研究计划项目 2023-JC-YB-054

中央高校基本科研业务费 XJS220702

详细信息

作者简介:
陈琛(1998—)，男，硕士(E-mail: c.chen@stu.xidian.edu.cn)

陈汉章(2000—)，男(E-mail: hzchen2000@foxmail.com)

通讯作者:
冯晓莉(1981—)，女，博士(通讯作者. E-mail: xiaolifeng@xidian.edu.cn)

中图分类号: O175.26
计量
- 文章访问数: 477
- HTML全文浏览量: 158
- PDF下载量: 87
- 被引次数: 0
出版历程
- 收稿日期: 2022-05-19
- 修回日期: 2022-06-28
- 刊出日期: 2023-07-01

The Random Source Inverse Method and Properties for a Class of Stochastic Differential Equations

School of Mathematics and Statistics, Xidian University, Xi'an 710126, P.R.China

摘要

摘要: 该文考虑了一类由分式Brown运动驱动的随机微分方程的随机源反演方法及其性质，其中分式Brown运动对应的Hurst参数H∈(0, 1).该问题可由很多随机模型转化而得，是一种比较广泛的随机问题.对于正问题，通过常数变易法得到方程的温和解，根据温和解的统计性质讨论其适定性.对于反问题，根据终止时刻的随机数据的统计量反演随机源项的部分统计量，证明了反演的唯一性，并讨论了当a(x)在不同范围时反问题的稳定性情况.
- 随机微分方程 /
- 分数阶Brown运动 /
- 反源问题 /
- 唯一性 /
- 不适定性
Abstract: The random source inverse method and properties for a class of stochastic differential equations driven by the fractional Brownian motion with Hurst index H∈(0, 1). This problem can be obtained from the transform of many stochastic models and is a widely followed problem. For the direct problem, the mild solution to the equation was obtained by means of constant variation, and according to the statistical properties of the mild solution, the well-posedness of the direct problem was discussed. For the inverse problem, some statistics of the random source term were determined from the random data at the final moment, to prove the uniqueness of the inverse problem, and the stability of the inverse problem with a(x) in different ranges was discussed.
- stochastic differential equation /
- fractional Brownian motion /
- inverse source problem /
- uniqueness /
- ill-posedness

HTML全文

参考文献(17)

[1]	GROETSH C W. 反问题: 大学生的科技活动[M]. 程晋, 谭永基, 刘继军, 译. 北京: 清华大学出版社, 2006. GROETSH C W. Inverse Problem: Activities for Undergraduates[M]. CHENG Jin, TAN Yongji, LIU Jijun, transl. Beijing: Tsinghua University Press, 2006. (in Chinese)
[2]	刘继军. 不适定问题的正则化方法及应用[M]. 北京: 科学出版社, 2005. LIU Jijun. Regularization Methods and Applications for Ill-Posed Problems[M]. Beijing: Science Press, 2005. (in Chinese)
[3]	王彦飞. 反演问题的计算方法及其应用[M]. 北京: 高等教育出版社, 2007. WANG Yanfei. Computational Methods for Inversion Problems and Their Applications[M]. Beijing: Higher Education Press, 2007. (in Chinese)
[4]	耿肖肖, 程浩. 一类球型区域上变系数反向热传导问题[J]. 应用数学和力学, 2021, 42(7): 723-734. doi: 10.21656/1000-0887.410297 GENG Xiaoxiao, CHENG Hao. The backward heat conduction problem with variable coefficients in a spherical domain[J]. Applied Mathematics and Mechanics, 2021, 42(7): 723-734. (in Chinese) doi: 10.21656/1000-0887.410297
[5]	柳冕, 程浩, 石成鑫. 一类非线性时间分数阶扩散方程反问题的变分型正则化[J]. 应用数学和力学, 2022, 43(3): 341-352. doi: 10.21656/1000-0887.420168 LIU Mian, CHENG Hao, SHI Chengxin. Variational regularization of the inverse problem of a class of nonlinear time-fractional diffusion equations[J]. Applied Mathematics and Mechanics, 2022, 43(3): 341-352. (in Chinese) doi: 10.21656/1000-0887.420168
[6]	RASCANU A, NUALART D. Differential equations driven by fractional Brownian motion[J]. Collectanea Mathematica , 2002, 53(1): 55-81.
[7]	BAI L H, MA J. Stochastic differential equations driven by fractional Brownian motion and Poisson point process[J]. Bernoulli, 2015, 21(1): 303-334.
[8]	MIJENA J B, NANE E. Space-time fractional stochastic partial differential equations[J]. Stochastic Processes and Their Applications, 2015, 125(9): 3301-3326. doi: 10.1016/j.spa.2015.04.008
[9]	TINDEL S, TUDOR C A, VIENS F. Stochastic evolution equations with fractional Brownian motion[J]. Probability Theory and Related Fields, 2003, 127(2): 186-204. doi: 10.1007/s00440-003-0282-2
[10]	LI P J, WANG X. Inverse random source scattering for the Helmholtz equation with attenuation[J]. SIAM Journal of Applied Mathematics, 2021, 81: 485-506. doi: 10.1137/19M1309456
[11]	LI P J, WANG X. An inverse random source problem for Maxwell's equations[J]. Multiscale Modeling & Simulation, 2021, 19(1): 25-45.
[12]	LI P J, WANG X. An inverse random source problem for the one-dimensional Helmholtz equation with attenuation[J]. Inverse Problems, 2021, 37: 015009. doi: 10.1088/1361-6420/abcd43
[13]	NIE Daxin, DENG Weihua. An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion[J/OL]. Journal of Inverse and Ill-Posed Problems, 2023(2023-02-28)[2023-06-28]. https://doi.org/10.1515/jiip-2021-0061.
[14]	FENG X L, ZHAO M X, LI P J, et al. An inverse source problem for the stochastic wave equation[J]. Inverse Problems and Imaging, 2022, 16: 397-415. doi: 10.3934/ipi.2021055
[15]	FENG X L, LI P J, WANG X. An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion[J]. Inverse Problems, 2020, 36(4): 045008.
[16]	赵丽志, 冯晓莉. 一类随机对流扩散方程的反源问题[J]. 应用数学和力学, 2022, 43(12): 1392-1401. doi: 10.21656/1000-0887.420399 ZHAO Lizhi, FENG Xiaoli. An inverse source problem for the stochastic convection-diffusion equation[J]. Applied Mathematics and Mechanics, 2022, 43(12): 1392-1401. (in Chinese) doi: 10.21656/1000-0887.420399
[17]	NUALART D. The Malliavin Calculus and Related Topics[M]//Probability and Its Applications. Springer, 1996.

施引文献

资源附件(0)

访问统计

计量

文章访问数: 477
HTML全文浏览量: 158
PDF下载量: 87
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

留言板

一类随机微分方程的随机源反演方法和性质

doi: 10.21656/1000-0887.430170

作者简介:
陈琛(1998—)，男，硕士(E-mail: c.chen@stu.xidian.edu.cn)

陈汉章(2000—)，男(E-mail: hzchen2000@foxmail.com)

通讯作者:
冯晓莉(1981—)，女，博士(通讯作者. E-mail: xiaolifeng@xidian.edu.cn)

计量

The Random Source Inverse Method and Properties for a Class of Stochastic Differential Equations

计量

目录

留言板

一类随机微分方程的随机源反演方法和性质

doi: 10.21656/1000-0887.430170

作者简介: 陈琛(1998—)，男，硕士(E-mail: c.chen@stu.xidian.edu.cn) 陈汉章(2000—)，男(E-mail: hzchen2000@foxmail.com)

通讯作者: 冯晓莉(1981—)，女，博士(通讯作者. E-mail: xiaolifeng@xidian.edu.cn)

计量

出版历程

The Random Source Inverse Method and Properties for a Class of Stochastic Differential Equations

计量

出版历程

目录

作者简介:
陈琛(1998—)，男，硕士(E-mail: c.chen@stu.xidian.edu.cn)

陈汉章(2000—)，男(E-mail: hzchen2000@foxmail.com)

通讯作者:
冯晓莉(1981—)，女，博士(通讯作者. E-mail: xiaolifeng@xidian.edu.cn)