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Abstract
This work focuses on parameter calibration of a variable-diffusivity 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 in the range . A Karhunen-Loève (KL) decomposition of the random diffusivity field is used, leading to a stochastic problem defined in terms of a finite number of canonical random variables. 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. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered, followed by the leading KL modes. 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. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations.
Highlights
Introduction ce an us criAnomalous diffusion occurs in a variety of natural phenomena, as well as applications involving particulate motion [1, 2, 3], viscoelastic materials [4], or subsurface transport [5].Fractional diffusion models have shown to be effective in representing and capturing such physical phenomena
This paper explores the application of a Bayesian inference approach to calibrate the fractional order parameter and infer a non-uniform diffusivity field in a space fractional diffusion equation (FDE)
We focus on the inference of the statistical characteristics of the stochastic diffusivity field jointly with the fractional order α
Summary
We focus on the steady, 1D, two-sided, fractional order diffusion equation with variable diffusivity:. In (1), κ denotes the diffusivity, f is the forcing term, whereas ∂x denotes the first-order derivative, and ∂xα is the two-sided fractional differential operator of order α ∈ (0, 1),. The variable diffusivity field, κ, is assumed to satisfy c0 ≤ κ(x) ≤ c1 for some positive constants c0 and c1 and all x ∈ [a, b]. We seek the numerical solution of problem (1) by adopting the finite difference method proposed in [17], which combines first-order forward and backward approximations of the Riemann-Liouville derivatives. Therein, the existence and uniqueness of the finite difference solution were established, and a rigorous truncation error analysis was provided. U P −1 )T denote the finite difference solution vector, and F =
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