Overview
This study focuses on the Bayesian recovery of the initial state within a semilinear time-fractional subdiffusion equation. The process utilizes noisy random space-time point observations. A rescaled Gaussian prior, derived from a Whittle–Matérn process, is applied to model the unknown initial condition.
Research Context
The core problem addresses the inverse problem of reconstructing an initial state in a time-fractional subdiffusion system. The equation considered is semilinear, indicating a combination of linear and nonlinear terms, and it is subjected to time-fractional derivatives. The observations are characterized as noisy and collected at discrete space-time points.
Approach
The methodology employs a Bayesian framework for initial state recovery. A specific prior distribution, a rescaled Gaussian prior based on a Whittle–Matérn process, is assigned to the initial condition. Mathematical analysis is then used to derive properties of the solution and the recovery process.
Findings
- The solution to the semilinear time-fractional subdiffusion equation possesses $\dot H^3$-regularity. This regularity holds under the specific condition that the nonlinearity satisfies a Lipschitz condition within the Sobolev norm.
- Posterior contraction rates were established for the prediction error. These rates are measured in the $L^2$-norm.
- Posterior contraction rates were also established for the parameter itself, measured in Sobolev norms.
- The derived contraction rates are polynomial in the sample size. The specific exponent of this polynomial dependence is contingent on the prior smoothness and the spatial dimension of the problem.
- A minimax lower bound was formally proven. This proof involved two primary steps: the construction of a wavelet-packing set and the controlled evaluation of Kullback–Leibler divergences.
Why This Matters
The work provides theoretical guarantees for recovering initial conditions in complex subdiffusion models, demonstrating how observation characteristics (sample size) and model properties (prior smoothness, spatial dimension) influence the accuracy of Bayesian inference. The establishment of posterior contraction rates and a minimax lower bound contributes to the foundational understanding of statistical estimation in these types of partial differential equations.