Browsing by Subject "Tikhonov regularization"

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  • Niu, Pingping; Helin, Tapio; Zhang, Zhidong (2020)
    In this work the authors consider an inverse source problem the stochastic fractional diffusion equation. The interested inverse problem is to reconstruct the unknown spatial functions f and g (the latter up to the sign) in the source 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 a reconstruction scheme for f and g up to the sign is given. To tackle the ill-posedness, Tikhonov regularization is adopted and some numerical results are displayed.
  • Rundell, William; Zhang, Zhidong (2018)
    Abstract A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-independent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω and the boundary of D has the star-like shape, i.e. ∂ D = { q ( θ ) ( cos ⁡ θ , sin ⁡ θ ) ⊤ : θ ∈ [ 0 , 2 π ] } . Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [6]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.