2020/10/30 by Lia Gander, Rolf Krause, Gander, Lia +5
Computer Science · Decision Sciences · Mathematics · #Computational Engineering #FOS: Computer and information sciences #FOS: Mathematics #Finance #Image and Signal Denoising Methods #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.2010.16104
openalex publication_date 2020/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In electrocardiography, the "classic" inverse problem is the reconstruction\nof electric potentials at a surface enclosing the heart from remote recordings\nat the body surface and an accurate description of the anatomy. The latter\nbeing affected by noise and obtained with limited resolution due to clinical\nconstraints, a possibly large uncertainty may be perpetuated in the inverse\nreconstruction.\n The purpose of this work is to study the effect of shape uncertainty on the\nforward and the inverse problem of electrocardiography. To this aim, the\nproblem is first recast into a boundary integral formulation and then\ndiscretised with a collocation method to achieve high convergence rates and a\nfast time to solution. The shape uncertainty of the domain is represented by a\nrandom deformation field defined on a reference configuration. We propose a\nperiodic-in-time covariance kernel for the random field and approximate the\nKarhunen-Lo `eve expansion using low-rank techniques for fast sampling. The\nspace-time uncertainty in the expected potential and its variance is evaluated\nwith an anisotropic sparse quadrature approach and validated by a quasi-Monte\nCarlo method.\n We present several numerical experiments on a simplified but physiologically\ngrounded 2-dimensional geometry to illustrate the validity of the approach. The\ntested parametric dimension ranged from 100 up to 600. For the forward problem\nthe sparse quadrature is very effective. In the inverse problem, the sparse\nquadrature and the quasi-Monte Carlo method perform as expected, except for the\ntotal variation regularisation, where convergence is limited by lack of\nregularity. We finally investigate an H1/2 regularisation, which naturally\nstems from the boundary integral formulation, and compare it to more classical\napproaches.\n