In this paper we present a novel method for finding unknown parameters for an unknown morphogen. We postulate the existence of an unknown morphogen in a given three-dimensional domain due to the spontaneous arrangement of a downstream species on the domain boundary for which data is known. Assuming a modified Helmholtz model for the morphogen and that it is produced from a single source in the domain, our method accurately estimates the source location and other model parameters. Notably, our method does not require the forward solution of the model to be computed which can often be a challenge for three-dimensional PDE model parameter fitting. Instead, an extension is made from the problem domain to an infinite domain and the analytic nature of the fundamental solution is exploited. We explore in this manuscript strategies for best conditioning the problem and rigorously explore the accuracy of the method on two test problems. Our tests focus on the effect of source location on accuracy but also the robustness of the algorithm to experimental noise.

中文翻译：

---

点源扩散-衰减形态发生子模型的参数估计。

在本文中，我们提出了一种寻找未知形态发生素未知参数的新颖方法。我们假定给定的三维域中存在未知的形态发生子，这是由于下游物种在域边界上的自发排列而导致的，而该域的数据是已知的。假设为形态发生子修改了亥姆霍兹模型，并且该模型是从域中的单个来源生成的，则我们的方法可以准确地估计来源位置和其他模型参数。值得注意的是，我们的方法不需要计算模型的正解，这通常可能是三维PDE模型参数拟合的挑战。取而代之的是，从问题域扩展到无限域，并利用了基本解决方案的分析性质。我们在这份手稿中探索了最佳解决问题的策略，并严格探讨了该方法在两个测试问题上的准确性。我们的测试关注源位置对准确性的影响，还关注算法对实验噪声的鲁棒性。