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Pollution Sources Reconstruction Based on the Topological Derivative Method
Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2020-04-30 , DOI: 10.1007/s00245-020-09685-0
L. Fernandez , A. A. Novotny , R. Prakash , J. Sokołowski

The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. The shape functional measuring the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped geometrical subdomains representing the pollution sources. The necessary conditions for optimality are derived with the help of the topological derivative method which consists in expanding the shape functional asymptotically and then truncate it up to the second order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, including their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases.



中文翻译:

基于拓扑导数法的污染源重建

拓扑导数法用于解决由稳态对流扩散方程控制的污染源重建问题。更准确地说,我们正在处理一个形状优化问题,该问题包括通过测量参考域某些子区域内污染物的浓度,在流体介质中重建一组污染源。相对于代表污染源的一组球形几何子域,最小化了测量已知数据与状态方程解之间的不匹配的形状函数。借助于拓扑导数方法,可以得出最优性的必要条件,该方法包括渐近展开形状函数,然后将其截断为二阶项。相对于所考虑的参数,最小化了所得的表达式,这导致了非迭代的二阶重构算法。考虑了两种不同的情况。首先,当给出泄漏的速度时,我们重构了对未知源的支持,包括其位置和大小。在第二种情况下,我们认为已知污染源的大小,并找出泄漏的平均速度及其位置。数值算例表明了所提算法在两种情况下重构多种污染源的能力。当给出泄漏的速度时,我们重构了对未知源的支持,包括其位置和大小。在第二种情况下,我们认为已知污染源的大小,并找出泄漏的平均速度及其位置。数值算例表明了所提算法在两种情况下重构多种污染源的能力。当给出泄漏的速度时,我们重构了对未知源的支持,包括其位置和大小。在第二种情况下,我们认为已知污染源的大小,并找出泄漏的平均速度及其位置。数值算例表明了所提算法在两种情况下重构多种污染源的能力。

更新日期:2020-04-30
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