In this work the authors consider an inverse source problem in the following stochastic fractional diffusion equationThe interested inverse problem is to reconstruct f (x) and g(x) by the statistics of the final time data u(x, T ). Some direct problem results are proved at first, such as the existence, uniqueness, representation and regularity of the solution. Then the reconstruction scheme for f and g is given. To tackle the ill-posedness, the Tikhonov regularization is adopted. Finally we give a regularized reconstruction algorithm and some numerical results are displayed.
Periodic parameters are common and important in stochastic differential equations (SDEs) arising in many contemporary scientific and engineering fields involving dynamical processes. These parameters include the damping coefficient, the volatility or diffusion coefficient and possibly an external force. Identification of these periodic parameters allows a better understanding of the dynamical processes and their hidden intermittent instability. Conventional approaches usually assume that one of the parameters is known and focus on the recovery of rest parameters. By introducing the decorrelation time and calculating the standard Gaussian statistics (mean, variance) explicitly for the scalar Langevin equations with periodic parameters, we propose a parameter identification approach to simultaneously recovering all these parameters by observing a single trajectory of SDEs. Such an approach is summarized in form of regularization schemes with noisy operators and noisy right-hand sides and is further extended to parameter identification of SDEs which are indirectly observed by other random processes. Numerical examples show that our approach performs well in stable and weakly unstable regimes but may fail in strongly unstable regime which is induced by the strong intermittent instability itself.
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