Modified QR decomposition to avoid non-uniqueness in water supply networks with extension to adjoint calculus

The dynamic simulation and optimization of water supply networks contains various difficulties. One of them is that a classical modelling may yield singularities in the form of non-unique solutions. In [1], the application of singular value decomposition (SVD) is proposed in the context of water supply networks. Since the SVD of a matrix is computationally very expensive, we introduce an approach based on the QR decomposition of a suitably modified matrix. The properties of the resulting solution are analysed and we show the applicability of our method on a real life water supply network. Further, we incorporate the presented approach in an adjoint calculus to provide sensitivity information such that gradient-based optimization methods can be applied.