粘性程函方程的反问题及其在电生理学中的应用
An inverse problem involving a viscous Eikonal equation with applications in electrophysiology
摘要：
在这项工作中，我们讨论了重建心脏激动时刻的基础上，粘性Eikonal方程从边界观测。该问题被制定为一个最小二乘问题，并解决了投影版本的Levenberg Marquardt方法。此外，我们分析了状态方程的适定性，并推导出最小二乘泛函关于激活时刻的梯度。在数值例子中，我们还进行了一个实验，其中的激活位点和激活时刻的位置重建共同的基础上，从https://link.springer.com/article/10.1007/s00285-019-01419-3的形状梯度法的适应版本。我们能够重建的激活时刻，以及激活的位置与高精度相对于噪声水平。
英文摘要：
In this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as an least squares problem and solved by a projected version of the Levenberg Marquardt method. Moreover, we analyze the wellposeness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from https://link.springer.com/article/10.1007/s00285-019-01419-3. We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.
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