Inverse problems for stochastic partial differential equations: Some progresses and open problems

Lü, Qi; Zhang, Xu

doi:10.3934/naco.2023014

NACO

2024

DOI: 10.3934/naco.2023014

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Inverse problems for stochastic partial differential equations: Some progresses and open problems

Qi Lü¹,

Xu Zhang²

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2024

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Cited by 3 publications

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Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Wu,

Wang,

Wang

2024

Inverse Problems

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In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. We then apply these Carleman estimates to investigate two inverse problems for the space semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the H"{o}lder stability for the discrete Cauchy problem.

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Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Wu,

Wang,

Wang

2024

Inverse Problems

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Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

Dou,

Lü,

Wang

2024

Inverse Problems

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In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

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Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

Baroun,

Boulite,

Elgrou

et al. 2024

MCRF

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Inverse problems for stochastic partial differential equations: Some progresses and open problems

Cited by 3 publications

References 47 publications

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

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