Identification of pollution sources in rivers using a hydrodynamic diffusion wave model and improved Bayesian-Markov chain Monte Carlo algorithm

Water quality restoration in rivers requires identification of the locations and discharges of pollution sources,and a reliable mathematical model to accomplish this identification is essential.In this paper,an innovative framework is presented to inversely estimate pollution sources for both accident preparedness and normal management of the allowable pollutant discharge.The proposed model integrates the concepts of the hydrodynamic diffusion wave equation and an improved Bayesian-Markov chain Monte Carlo method(MCMC).The methodological framework is tested using a designed case of a sudden wastewater spill incident(i.e.,source location,flow rate,and starting and ending times of the discharge)and a real case of multiple sewage inputs into a river(i.e.,locations and daily flows of sewage sources).The proposed modeling based on the improved Bayesian-MCMC method can effectively solve high-dimensional search and optimization problems according to known river water levels at pre-set monitoring sites.It can adequately provide accurate source estimation parameters using only one simulation through exploration of the full parameter space.In comparison,the inverse models based on the popular random walk Metropolis(RWM)algorithm and microbial genetic algorithm(MGA)do not produce reliable estimates for the two scenarios even after multiple simulation runs,and they fall into locally optimal solutions.Since much more water level data are available than water quality data,the proposed approach also provides a cost-effective solution for identifying pollution sources in rivers with the support of high-frequency water level data,especially for rivers receiving significant sewage discharges.

INITIAL VALUE TECHNIQUES FOR THE HELMHOLTZ AND MAXWELL EQUATIONS

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity n^2 log n on a n x n grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.

A Numerical Algorithm for Determination of a Control Parameter in Two-dimensional Parabolic Inverse Problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. In this study we present a high order scheme for determining unknown control parameter and unknown solution of two-dimensional parabolic inverse problem with overspe- cialization at a point in the spatial domain. In this approach, a compact fourth-order scheme is used to discretize spatial derivatives of equation and reduces the problem to a system of ordinary differential equations （ODEs）. Then we apply a fourth order boundary value method to the solution of resulting system of ODEs. So the proposed method has fourth order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes in the literature. Also we will investigate the effect of noise in data on the approximate solutions.