Utilizing algorithmic differentiation to efficiently compute chromatograms and parameter sensitivities

The efficiency of numerical methods for solving mass transfer equations, such as in chromatography modeling, crucially depends on the availability of specific derivatives. In particular, vector products with the system Jacobian are frequently required, i.e. derivatives of the right hand side of the spatially discretized partial differential equations with respect to the state variables. More specifically, large systems of non-linear equations are solved in each time step by Newton iterations, typically contributing more than 80% of the total compute time. We apply algorithmic differentiation (AD) combined with block and band compression for efficiently computing the required system Jacobian. Results are compared with a symbolically derived Jacobian based on non-overlapping domain decomposition. The symbolic approach turns out to be roughly two times faster, at the price of rather tedious and error-prone manual derivation of the mathematical equations. AD helps to simplify implementation of the computational code and is much more flexible when model variants are considered. In addition, AD is combined with a forward sensitivity approach for computing parameter sensitivities that are frequently applied in parameter estimation and process optimization. Results are compared with traditional finite differences. Forward sensitivity analysis is more straightforward to apply, as no perturbation factor needs to be chosen, and even computes faster than finite differences for a benchmark example with three chemical components and load, wash and elution phases.