Abstract
This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11401404
Funding statement: This work is partially supported by the NSF of China under Grant 11401404.
References
[1] G. Bal, Inverse problems for homogeneous transport equations. I. The one-dimensional case, Inverse Problems 16 (2000), no. 4, 997–1011. 10.1088/0266-5611/16/4/308Suche in Google Scholar
[2] G. Bal, Inverse problems for homogeneous transport equations. II. The multidimensional case, Inverse Problems 16 (2000), no. 4, 1013–1028. 10.1088/0266-5611/16/4/309Suche in Google Scholar
[3] G. Bal and A. Tamasan, Inverse source problems in transport equations, SIAM J. Math. Anal. 39 (2007), no. 1, 57–76. 10.1137/050647177Suche in Google Scholar
[4] G. Bao, S.-N. Chow, P. Li and H. Zhou, Numerical solution of an inverse medium scattering problem with a stochastic source, Inverse Problems 26 (2010), no. 7, Article ID 074014. 10.1088/0266-5611/26/7/074014Suche in Google Scholar
[5] L. Cavalier and A. Tsybakov, Sharp adaptation for inverse problems with random noise, Probab. Theory Related Fields 123 (2002), no. 3, 323–354. 10.1007/s004400100169Suche in Google Scholar
[6] P. L. Chow and L. Maestrello, Stochastic inverse problem in the radiation of noise, SIAM J. Appl. Math. 35 (1978), no. 4, 665–677. 10.1137/0135055Suche in Google Scholar
[7] D. Crisan, Y. Otobe and S. Peszat, Inverse problems for stochastic transport equations, Inverse Problems 31 (2015), no. 1, Article ID 015005. 10.1088/0266-5611/31/1/015005Suche in Google Scholar
[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University, Cambridge, 1992. 10.1017/CBO9780511666223Suche in Google Scholar
[9] T. Deck and J. Potthoff, On a class of stochastic partial differential equations related to turbulent transport, Probab. Theory Related Fields 111 (1998), no. 1, 101–122. 10.1007/s004400050163Suche in Google Scholar
[10] H. Gjessing, Wick calculus with applications to anticipating stochastic differential equations, Manuscript, University of Bergen, 1994. Suche in Google Scholar
[11] M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl. 343 (2008), no. 1, 352–365. 10.21236/ADA448486Suche in Google Scholar
[12] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 10: Physical Kinetics, Pergamon Press, New York, 1981. Suche in Google Scholar
[13] X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application, ESAIM Control Optim. Calc. Var. 20 (2014), no. 3, 823–839. 10.1051/cocv/2013085Suche in Google Scholar
[14] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems 28 (2012), no. 4, Article ID 045008. 10.1088/0266-5611/28/4/045008Suche in Google Scholar
[15] Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems 29 (2013), no. 9, Article ID 095011. 10.1088/0266-5611/29/9/095011Suche in Google Scholar
[16] Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim. 51 (2013), no. 1, 121–144. 10.1137/110830964Suche in Google Scholar
[17] Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim. 52 (2014), no. 1, 397–419. 10.1007/978-3-030-82331-3_8Suche in Google Scholar
[18] Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control Optim. 53 (2015), no. 6, 3457–3482. 10.1137/151002605Suche in Google Scholar
[19] Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math. 68 (2015), no. 6, 948–963. 10.1002/cpa.21503Suche in Google Scholar
[20] G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Problems 31 (2015), no. 8, Article ID 085003. 10.1088/0266-5611/31/8/085003Suche in Google Scholar
[21] X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal. 40 (2008), no. 2, 851–868. 10.1137/070685786Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
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- Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn
- Lipschitz stability for a semi-linear inverse stochastic transport problem
- Inverse problem for elastic body with thin elastic inclusion
- Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source
- Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions
- A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification
- A linear regularization method for a parameter identification problem in heat equation
- A study of frozen iteratively regularized Gauss–Newton algorithm for nonlinear ill-posed problems under generalized normal solvability condition
- Numerics of acoustical 2D tomography based on the conservation laws
- Identification of the diffusion coefficient in a time fractional diffusion equation
- Linear least squares method in nonlinear parametric inverse problems
Artikel in diesem Heft
- Frontmatter
- Image reconstruction method for exterior circular cone-beam CT based on weighted directional total variation in cylindrical coordinates
- Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn
- Lipschitz stability for a semi-linear inverse stochastic transport problem
- Inverse problem for elastic body with thin elastic inclusion
- Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source
- Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions
- A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification
- A linear regularization method for a parameter identification problem in heat equation
- A study of frozen iteratively regularized Gauss–Newton algorithm for nonlinear ill-posed problems under generalized normal solvability condition
- Numerics of acoustical 2D tomography based on the conservation laws
- Identification of the diffusion coefficient in a time fractional diffusion equation
- Linear least squares method in nonlinear parametric inverse problems