Bayesian Calibration of Order and Diffusivity in a Space-fractional Diffusion Equation

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We focus on parameter calibration of a variable-diffusivity space-fractional diffusion model. A random, spatially-varying diffusivity field with log-normal distribution is considered. The variance and correlation length of the diffusivity field are considered uncertain parameters, and the order of the fractional sub-diffusion operator is also taken uncertain and uniformly distributed. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A non-intrusive methodology is used, and a deterministic finite-difference solver of the fractional diffusion model is utilized for this purpose. The PC surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. Different setups are considered, including distributed and localized forcing functions and data consisting of either noisy observations of the solution or its first moments.

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