TY - JOUR

T1 - Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation

AU - Alzahrani, Hasnaa H.

AU - Lucchesi, Marco

AU - Mustapha, Kassem

AU - Le Maître, Olivier

AU - Knio, Omar

N1 - KAUST Repository Item: Exported on 2021-07-29
Acknowledgements: Research reported in this publication was supported by research funding from King Abdullah University of Science and Technology (KAUST).

PY - 2021/7/15

Y1 - 2021/7/15

N2 - 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 (0,1). A Karhunen-Lo`eve (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.

AB - 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 (0,1). A Karhunen-Lo`eve (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.

UR - http://hdl.handle.net/10754/670334

UR - https://iopscience.iop.org/article/10.1088/2399-6528/ac1507

U2 - 10.1088/2399-6528/ac1507

DO - 10.1088/2399-6528/ac1507

M3 - Article

JO - Journal of Physics Communications

JF - Journal of Physics Communications

SN - 2399-6528

ER -