Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation
This work provides a rigorous framework for parameter identification in fractional subdiffusion equations, which is relevant for modeling anomalous diffusion in physics and biology, but the results are incremental as they extend existing regularization techniques to a specific model.
The paper develops an analytical and numerical method to identify scalar parameters (coefficients and fractional derivative orders) in a multi-term fractional subdiffusion model using a nonlocal observation. The approach yields explicit formulas, uniqueness/stability results, and effective recovery from noisy data via Tikhonov regularization with quasi-optimality, demonstrated in numerical tests.
In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, $\mathbf{D}_t$. To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation $\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=g(x,t),$ where $\mathcal{L}_{i}$ are the second order elliptic operators with time-dependent coefficients, $\mathcal{K}$ is a summable memory kernel, and $g$ is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.