Sensitivity Analysis for the Numerical Simulation of the Transport of Contaminants

Abstract

The questions of safety and uncertainties are central to feasibility studies for an underground waste storage site. One of the important points to be considered is the problem of the evaluation of concentration uncertainties which are due to input parameter uncertainties. These concentration uncertainties can be obtained by probabilistic methods. These methods give good results and are relatively easy to implement, but they are expensive because they require a large number of simulations. The method of deterministic sensitivities is much less demanding in computing time. This gain in computing time is accompanied a deterioration of the quality of the information obtained, but because of its much lower cost, it still deserves to be developed. The mathematical model consists of a flow equation based on Darcy's law - we assume that the Darcy velocity is stationary -, a transport equation, i. e. a mass balance equation, for each radionuclide, and a law of exchange between liquid and solid phases for each radionuclide. The deterministic sensitivity analysis is based on the first order approximation given the Jacobian matrix of the function which associates the output concentrations to the input parameters. This differentiation computation can be done either direct methods (preferable when the number of input parameters is low) or by reverse methods (preferable when the number of output concentrations is low); and since we avoid finite difference methods for their lower precision, we have two alternatives: implement by hand the derivatives or use an automatic differentiation tool. The latter uses either source code transformation or operator overloading. The main goal of this presentation is to give an evaluation of the performance of the automatic differentiation tool adolC which is able to differentiate C++ code both in direct and reverse modes. Here, we consider the simpler case of just the (3D) flow equation, and the main result is that this tool has shown both precision and efficiency comparable with those obtained from a code differentiated hand, but for a shorter development time. The major limitations to the use of adolC is the size of files automatically generated during the execution and the handling of huge outside libraries. Hence, the first experiments carried out thus far suggest differentiating by hand parts that use libraries or that contain iterative methods and using automatic differentiation for parts for which the list of operations realized during the execution depends strongly, but not iteratively, on the parameters, for example for the computation of upwinded schemes. Fortunately it is possible to combine manual and automatic differentiation, and that is what we do.