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In electrocardiography, the "classic" inverse problem is the reconstructionof electric potentials at a surface enclosing the heart from remote recordingsat the body surface and an accurate description of the anatomy. The latterbeing affected by noise and obtained with limited resolution due to clinicalconstraints, a possibly large uncertainty may be perpetuated in the inversereconstruction.  The purpose of this work is to study the effect of shape uncertainty on theforward and the inverse problem of electrocardiography. To this aim, theproblem is first recast into a boundary integral formulation and thendiscretised with a collocation method to achieve high convergence rates and afast time to solution. The shape uncertainty of the domain is represented by arandom deformation field defined on a reference configuration. We propose aperiodic-in-time covariance kernel for the random field and approximate theKarhunen-Lo\`eve expansion using low-rank techniques for fast sampling. Thespace-time uncertainty in the expected potential and its variance is evaluatedwith an anisotropic sparse quadrature approach and validated by a quasi-MonteCarlo method.  We present several numerical experiments on a simplified but physiologicallygrounded 2-dimensional geometry to illustrate the validity of the approach. Thetested parametric dimension ranged from 100 up to 600. For the forward problemthe sparse quadrature is very effective. In the inverse problem, the sparsequadrature and the quasi-Monte Carlo method perform as expected, except for thetotal variation regularisation, where convergence is limited by lack ofregularity. We finally investigate an $H^{1/2}$ regularisation, which naturallystems from the boundary integral formulation, and compare it to more classicalapproaches.

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