Optimal Follow Up Designs for Fractional Partial Differential Equations with Application to a Convection-Advection Model

Abstract

As the mathematical properties of Fractional Partial Differential Equations are rapidly being developed, there is an increasing desire by researchers to employ these models in real world data oriented contexts. The main barrier to employing these models is the choice of the fractional order alpha. Recently, show how to both estimate and make inferences about alpha from a Bayesian perspective. However, for experimental settings one needs to be able to design experiments that will provide optimal information about alpha. This work demonstrates how to use information based criteria, namely A, D and E optimality, to choose sampling locations in follow-up designs. Specifically, the simultaneous addition of one, two and three measurement locations is considered for a simple example. The simultaneous addition of four and five measurement locations is also considered across a variety of values of alpha. The results show that each of the criteria provide different optimal measurement locations as the number of additional measurement locations is increased. Indicating that the choice of optimality criteria should not be decidedarbitrarily.