An inverse potential problem for the stochastic diffusion equation with a multiplicative white noise. (English) Zbl 1529.35570
Summary: This work concerns the direct and inverse potential problems for the stochastic diffusion equation driven by a multiplicative time-dependent white noise. The direct problem is to examine the well-posedness of the stochastic diffusion equation for a given potential, while the inverse problem is to determine the potential from the expectation of the solution at a fixed observation point inside the spatial domain. The direct problem is shown to admit a unique and positive mild solution if the initial value is nonnegative. Moreover, an explicit formula is deduced to reconstruct the square of the potential, which leads to the uniqueness of the inverse problem for nonnegative potential functions. Two regularization methods are utilized to overcome the instability of the numerical differentiation in the reconstruction formula. Numerical results show that the methods are effective to reconstruct both smooth and nonsmooth potential functions.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 80A23 | Inverse problems in thermodynamics and heat transfer |
Keywords:
stochastic diffusion equation; inverse potential problem; multiplicative white noise; mild solution; uniqueness; regularizationReferences:
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