论文标题
图像方法的传输方法：通过继续计算所有谐波映射的零
The transport of images method: computing all zeros of harmonic mappings by continuation
论文作者
论文摘要
我们提出了一种延续方法，以计算复杂平面中的谐波映射$ f $的所有零。我们的方法在没有任何事先了解零的数量或其大约位置的情况下起作用。我们首先使用$ f（z）=η$计算所有解决方案，并用$ |η| $足够大，然后跟踪所有解决方案，因为$η$倾向于$ 0 $，以最终获得$ f $的所有零。使用谐波映射的理论结果，我们分析了$ f（z）=η$的解决方案的数量以及如何将其纳入该方法。我们证明，只要没有一个是单数的，我们的方法就可以保证计算所有零。在我们的数值示例中，该方法始终以正确数量的零终止，与通用根发现器相比非常快，并且在残差方面非常准确。易于使用的MATLAB实现可在线免费获得。
We present a continuation method to compute all zeros of a harmonic mapping $f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solution of $f(z) = η$ with $|η|$ sufficiently large and then track all solutions as $η$ tends to $0$ to finally obtain all zeros of $f$. Using theoretical results on harmonic mappings we analyze where and how the number of solutions of $f(z) = η$ changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular. In our numerical example the method always terminates with the correct number of zeros, is very fast compared to general purpose root finders and is highly accurate in terms of the residual. An easy-to-use MATLAB implementation is freely available online.
