Diffusion and ecological problems : mathematical models

Contaminant source identification using semi-supervised machine learning.

It is important to identify the source information after a sudden water contamination incident occurs in a water supply system. The accuracy of the simulation model9s parameters determines the accuracy of the source information. However, it is difficult to obtain the true value of these parameters by existing methods, so reduction of the errors caused by the uncertainty of these parameters is a crucial problem. A source identification framework which considers the uncertainty of the model9s sensitive parameters and combines Bayesian inference and Markov Chain Monte Carlo (MCMC) algorithms simulation is established, and the South-to-North Water Diversion Project is taken as the case study in this paper. Compared with a framework which does not consider the uncertainty of the model9s parameters, the proposed framework could solve the error caused by the wrong choice of model parameters and obtain more accurate results. In addition, the proposed framework based on traditional MCMC and that based on the Delayed Rejection and Adaptive Metropolis (DRAM-MCMC) are compared to prove that the DRAM-MCMC is more convergent and accurate. Lastly, the proposed framework based on DRAM-MCMC is proved to solve the problem with high practicality and generality in the studied long distance water diversion project.

Source identification of sudden contamination based on the parameter uncertainty analysis

Unsupervised machine learning based on non-negative tensor factorization for analyzing reactive-mixing

Sudden water pollution accidents in surface waters occur with increasing frequency. These accidents significantly threaten people’s health and lives. To prevent the diffusion of pollutants, identifying these pollution sources is necessary. The identification problem of pollution source, especially for multi-point source, is one of the difficulties in the inverse problem area. This study examines this issue. A new method is designed by combining differential evolution algorithm (DEA) and Metropolis–Hastings–Markov Chain Monte Carlo (MH–MCMC) based on Bayesian inference to identify multi-point sudden water pollution sources. The effectiveness and accuracy of this proposed method is verified through outdoor experiments and comparison between DEA and MH–MCMC. The average absolute error of the sources’ position and intensity, the relative error and the average standard deviations obtained using the proposed method are less than those of DEA and MH–MCMC. Moreover, the relative error and the sampling relative error under four different standard deviations of measurement error (σ = 0.01, 0.05, 0.1, 0.15) are less than 2 and 0.11 %, respectively. The proposed method (i.e., DEMH–MCMC) is effective even when the standard deviation of the measurement error increases to 0.15. Therefore, the proposed method can identify sources of multi-point sudden water pollution accidents efficiently and accurately.

Multi-point source identification of sudden water pollution accidents in surface waters based on differential evolution and Metropolis–Hastings–Markov Chain Monte Carlo

This paper deals with the identification of a time-dependent point source occurring in the right-hand side of a one-dimensional evolution linear advection–dispersion–reaction equation. The originality of this study consists in considering the general case of transport equations with spatially varying dispersion, velocity and reaction coefficients which enables to extend the applicability of the obtained results to various areas of science and engineering. We derive a main condition on the involved spatially varying coefficients that yields identifiability of the sought source, provided its time-dependent intensity function vanishes before reaching the final monitoring time, from recording the generated state at two observation points framing the source region. Then, we establish an identification method that uses those records to determine the elements defining the sought source. Some numerical experiments on a variant of the surface water pollution model are presented.

Inverse source problem in a one-dimensional evolution linear transport equation with spatially varying coefficients: application to surface water pollution

Abstract The paper deals with the nonlinear inverse source problem of identifying an unknown time-dependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The originality of this study consists in establishing a constructive identifiability theorem that leads to develop an identification method using only significant boundary observations and operating other than the classic optimization approach. To this end, we derive two dispersion-current functions that have the main property to be of orthogonal gradients which yield identifiability of the elements defining the involved unknown source from some boundary observations related to the associated state. Provided the velocity field fulfills the so-called no-slipping condition, the required boundary observations are reduced to only recording the state on the outflow boundary and its flux on the inflow boundary of the monitored domain. We establish an identification method that uses those boundary records (1) to localize the sought source position as the unique solution of a nonlinear system defined by the two dispersion-current functions, (2) to give lower and upper bounds of the total amount loaded by the unknown time-dependent source intensity function, (3) to transform the identification of this latest into solving a deconvolution problem. Some numerical experiments on a variant of the surface water BOD pollution model are presented.

Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

We address the nonlinear inverse source problem, which consists in identifying the time limit from which some unknown sources in a two-dimensional advection–dispersion-reaction equation become inactive. Under some reasonable assumptions and without requiring any a priori information about the form of the involved unknown sources, we prove the identifiability of the desired time active limit. We establish an identification method based on some boundary null controllability results that uses the records of the associated state on the outflow boundary and of its flux on the inflow boundary of the monitored domain to determine the time active limit. The application of this method to the most encountered forms of surface water pollution sources leads us to determine also lower and upper bounds that provide a significant approximation of the total amount loaded by the involved unknown sources without requiring any a priori information either on the number of active sources or on the form of their time-dependent intensity functions. Some numerical experiments on a variant of the surface water biological oxygen demand pollution model are presented.

Identification of time active limit with lower and upper bounds of total amount loaded by unknown sources in 2D transport equations

Cette these porte sur l’etude de quelques questions liees a l’identifiabilite et l’identification d’un probleme inverse non-lineaire de source. Il s’agit de l’identification d’une source ponctuelle dependante du temps constituant le second membre d’une equation de type advection-dispersion-reaction a coefficients variables. Dans le cas monodimensionnel, la souplesse du modele stationnaire nous a permis de developper des reponses theoriques concernant le nombre des capteurs necessaires et leurs emplacements permettant d’identifier la source recherchee d’une facon unique. Ces resultats nous ont beaucoup aides a definir la ligne de conduite a suivre afin d’apporter des reponses similaires pour le modele transitoire. Quant au modele bidimensionnel transitoire, en utilisant quelques resultats de nulle controlabilite frontiere et des mesures de l’etat sur la frontiere sortie et de son flux sur la frontiere entree du domaine etudie, nous avons etabli un theoreme d’identifiabilite et une methode d’identification permettant de localiser les deux coordonnees de la position de la source recherchee comme etant l’unique solution d’un systeme non-lineaire de deux equations, et de transformer l’identification de sa fonction de debit en la resolution d’un probleme de deconvolution. La derniere partie de cette these discute la difficulte principale rencontree dans ce genre de problemes inverses a savoir la non identifiabilite d’une source dans sa forme abstraite, propose une alternative permettant de surmonter cette difficulte dans le cas particulier ou le but est d’identifier le temps limite a partir duquel la source impliquee a cesse d’emettre, et donc ouvre la porte sur de nouveaux horizons.

https://tel.archives-ouvertes.fr/tel-00975168/document

Imed Mahfoudhi

Papers

Inverse source problem in a one-dimensional evolution linear transport equation with spatially varying coefficients: application to surface water pollution

Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Identification of time active limit with lower and upper bounds of total amount loaded by unknown sources in 2D transport equations

Problèmes inverses de sources dans des équations de transport à coefficients variables

Boundary null-controllability of linear diffusion–reaction equations