In electrocardiography, the "classic" inverse problem consists of finding electric potentials on a surface enclosing the heart from remote recordings on the body surface and an accurate description of the anatomy. The latter being affected by noise and obtained with limited resolution due to clinical constraints, a possibly large uncertainty may be perpetuated in the inverse reconstruction. The purpose of this work is to study the effect of shape uncertainty on the forward and the inverse problem of electrocardiography. To this aim, the problem is first recast into a boundary integral formulation and then discretised with a collocation method to achieve high convergence rates and a fast time to solution. The shape uncertainty of the domain is represented by a random deformation field defined on a reference configuration. We propose a periodic-in-time covariance kernel for the random field and approximate the Karhunen-Lo\`eve expansion using low-rank techniques for fast sampling. The space-time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi-Monte Carlo method. We present several examples to illustrate the validity of the approach with parametric dimension up to 600. For the forward problem the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi-Monte Carlo methods perform as expected except with total variation regularisation, in which convergence is limited by lack of regularity. We finally investigate an $H^{1/2}$-Tikhonov regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.

中文翻译：

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心电图正逆问题中的时空形状不确定性

在心电图中，“经典”逆问题包括从体表的远程记录和解剖结构的准确描述中找到包围心脏的表面上的电势。后者受噪声影响并且由于临床限制以有限的分辨率获得，在逆重建中可能存在很大的不确定性。这项工作的目的是研究形状不确定性对心电图正反问题的影响。为此，首先将问题重铸为边界积分公式，然后使用搭配方法进行离散化，以实现高收敛速度和快速求解时间。域的形状不确定性由在参考配置上定义的随机变形场表示。我们为随机场提出了一个周期性的时间协方差内核，并使用低秩技术进行快速采样来近似 Karhunen-Lo\`eve 扩展。预期电位的时空不确定性及其方差使用各向异性稀疏正交方法进行评估，并通过准蒙特卡罗方法进行验证。我们提供了几个例子来说明参数维数高达 600 的方法的有效性。对于前向问题，稀疏正交非常有效。在逆问题中，稀疏正交和准蒙特卡罗方法按预期执行，除了全变分正则化，其中收敛受到缺乏规律性的限制。我们最终研究了 $H^{1/2}$-Tikhonov 正则化，它自然源于边界积分公式，