Accuracy and efficiency of adjoint state based parameter identification for fractional advection diffusion equation with space-dependent coefficients

Abstract

In many natural media, tracer dispersion deviates from the classical Advection Diffusion Equation, and in some media a fractional variant including a time derivative of fractional order was found better adapted. We describe an inverse method that achieves best adjustment to data by tuning the order of the fractional derivative and the coefficients of the fractional ADE, even when the latter depend on space. Then, the discretized fractional ADE depends on a finite number of parameters including interpolation coordinates of the coefficients, and fractional derivative order. Imagine data that are time series of a quantity measured at several points of a one-dimensional domain, and ruled by the fractional ADE equipped with coefficients which we want to determine. We present a method that finds the parameters minimizing the distance E between model solution and data along an optimization sequence in degrees of freedom space. Yet, interpolation nodes may be many (more than ten) and accurate optimization softwares need the current gradient of E as an input for to determine the step immediately following each current step of the sequence. All the components of this gradient are deduced from one adjoint state, solution of a partial differential equation adjoint to the fractional ADE, and of equivalent computational complexity. Accuracy and computing time saving are demonstrated by applying inversion to artificial data deduced from the fractional ADE equipped of parameters imposed by ourselves.