We present a convection–diffusion inverse problem that aims to identify an unknown number of sources and their locations. We model the sources using a binary function, and we show that the inverse problem can be formulated as a large-scale mixed-integer nonlinear optimization problem. We show empirically that current state-of-the-art mixed-integer solvers cannot solve this problem and that applying simple rounding heuristics to solutions of the relaxed problem can fail to identify the correct number and location of the sources. We develop two new rounding heuristics that exploit the value and a physical interpretation of the continuous relaxation solution, and we apply a steepest-descent improvement heuristic to obtain satisfactory solutions to both two- and three-dimensional inverse problems. We also provide the code used in our numerical experiments in open-source format.

我们提出了一个对流-扩散逆问题，旨在确定未知数量的源及其位置。我们使用二元函数对源进行建模，并且证明了反问题可以表述为大规模混合整数非线性优化问题。我们凭经验表明，当前最先进的混合整数求解器无法解决此问题，将简单的舍入启发法应用于松弛问题的解决方案可能无法识别源的正确数量和位置。我们开发了两个新的舍入启发法，它们利用了连续松弛解的值和物理解释，并且我们应用了最速下降的改进启发法来对二维和三维逆问题都获得满意的解。