Optimal Sampling Locations for Fractional Partial Differential Equations Using D-Optimality

Researchers are becoming increasingly interested in using Fractional Partial Differential Equation (FPDE) models for physical systems such as gas flows through porous materials. These models rely on the fraction of the differentiation \(\alpha \), which needs to be estimated from empirical data. Experimentation is needed to obtain empirical data where pressures need to be measured at various times, t, from the initial pressure and distances x from the pressure source which produces an output pressure p(x, t). While sampling times are easy to choose when a sensor is in place. Typically the location of sensors from the pressure source are arbitrarily chosen. This work shows how to design experiments using a two stage design with a base design and a follow design using D-optimality to determine the location(s) of additional sensors along x which allows for the minimization of the volume of the Variance-Covariance matrix of the parameters. A Bayesian approach is utilized for parameter estimation and simulated annealing is used to search through the possible locations for sensors in the follow up design. A simple FPDE is used to illustrate the approach with a base design of six sensor locations and follow-up designs for the simultaneous addition of 1, 2, 3, 4 and 5 new sensor locations. All of the follow-up designs suggest points near where the solution to the FPDE are optimal locations as well as other locations depending the number of sensors.