Numerical Modeling of Water Distribution Systems Using the Graph p-Laplacian: Variational and Duality Methods with Applications

In this thesis we discuss how to model a Water Distribution System(WDS) by means of a surrogate graph p-Laplacian model, with 1<p<2. We exhaustively discuss both numerical and theoretical aspects, within the framework of graph based inverse problems, convex analysis and Legendre duality. The basic idea is that, starting mainly from data on pipe characteristics and measurements of water pressure and water fluxes inside pipes, it is possible to accurately determine an edge-based distribution of edges weights and p-values on the edges so that the corresponding weighted p-Poisson equation can be effectively used as a Digital Twin of the WDS under study. The successful completion of this part required the determination and subsequent numerical solution of an appropriately regularized inverse problem (a variant of Calderon’s inverse problem) defined on the WDS graph. The peculiar characteristics of a WDS, whereby neighboring pipes have typically the same edge-constant properties (read weights and p-values) required the use of a Total-Variation (TV) based regularization. Thus, on the second part of this thesis we focused on the variational characterization by duality methods of TV regularizers embedded in the solution of the weighted p-Laplace inverse problem. We discuss how to properly rewrite a convex energy functional into an equivalent saddle point formulation, to tackle the problem of finding it's minimizers from an alternative and more performing perspective. We extensively study the case of the p-Dirichlet energy for 1<p<2, and of the Total Variation energy as limit case for p=1, including it's application as a regularization term in various type of inverse problems. The derivation of these saddle point formulations is essentially based on the iteration of the Legendre transform combined with ad-hoc substitutions and transformations of the involved variables. These resulting equivalent formulations based on duality theory, leads to a class of saddle point problems that can be efficiently translated into accurate and robust numerical methods based on variant of the Dynamic-Monge-Kantorovich(DMK) equations developed earlier by the supervisors research group for the numerical solution of the L1 Optimal Transport problem. Moreover, we discuss both theoretical aspects and numerical implementation of the proposed formulation showing also the efficiency and robustness of the developed algorithms on both classical problems and real-world examples. On the third part of the thesis we focus on the development of numerical schemes for the nonlinear eigenvalue problem of p-Laplace operators on graphs with 1<p. The aim of this part is to provide a proper efficient numerical scheme in order to use eigen-information of the p-Laplace operator governing the specific WDS to develop Machine-Learning and surrogate models. A family of Energy functions, inspired again by the DMK approach, whose critical points can be proved to be variational eigenpairs of the p-Laplace operators, have been used to develop gradient-flow algorithms for the numerical calculations of p-eigenpairs. Unfortunately, only partial results have been achieved in this topic due to two main difficulties inherently related to the nonlinear eigenvalue problem. On one hand, the non-regularity of these energy functions in the presence of eigenpairs with multiplicity greater than one may cause non-convergence of the developed gradient-based method. The second important difficulty is related to the positioning of the found p-eigenpairs within the p-spectrum. Indeed, the interpretation of the DMK equations deriving from the KKT conditions of the proposed energy functions as an appropriate weighted linear Laplace eigenproblem allowed the definition of an approximate ordering of the numerically calculated p-eigenpairs. However, a complete solution of this problem is still elusive and is left of a matter of future developments.