Space-time shape uncertainties in the forward and inverse problem of electrocardiography
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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-Loève 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.
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