{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T05:31:26Z","timestamp":1787376686625,"version":"build-2736575974"},"reference-count":41,"publisher":"Wiley","issue":"10","license":[{"start":{"date-parts":[[2021,9,8]],"date-time":"2021-09-08T00:00:00Z","timestamp":1631059200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Numer Methods Biomed Eng"],"published-print":{"date-parts":[[2021,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In electrocardiography, the \u201cclassic\u201d inverse problem is the reconstruction of electric potentials at a surface enclosing the heart from remote recordings at 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\u2010in\u2010time covariance kernel for the random field and approximate the Karhunen\u2013Lo\u00e8ve expansion using low\u2010rank techniques for fast sampling. The space\u2013time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi\u2010Monte Carlo method. We present several numerical experiments on a simplified but physiologically grounded two\u2010dimensional geometry to illustrate the validity of the approach. The tested parametric dimension ranged from 100 up to 600. For the forward problem, the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi\u2010Monte Carlo method perform as expected, except for the total variation regularisation, where convergence is limited by lack of regularity. We finally investigate an  regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.<\/jats:p>","DOI":"10.1002\/cnm.3522","type":"journal-article","created":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T09:20:02Z","timestamp":1629364802000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Space\u2013time shape uncertainties in the forward and inverse problem of electrocardiography"],"prefix":"10.1002","volume":"37","author":[{"given":"Lia","family":"Gander","sequence":"first","affiliation":[{"name":"Center for Computational Medicine in Cardiology Euler Institute, Universit\u00e0 della Svizzera italiana  Lugano Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rolf","family":"Krause","sequence":"additional","affiliation":[{"name":"Center for Computational Medicine in Cardiology Euler Institute, Universit\u00e0 della Svizzera italiana  Lugano Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Multerer","sequence":"additional","affiliation":[{"name":"Center for Computational Medicine in Cardiology Euler Institute, Universit\u00e0 della Svizzera italiana  Lugano Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7432-0424","authenticated-orcid":false,"given":"Simone","family":"Pezzuto","sequence":"additional","affiliation":[{"name":"Center for Computational Medicine in Cardiology Euler Institute, Universit\u00e0 della Svizzera italiana  Lugano Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2021,9,8]]},"reference":[{"key":"e_1_2_10_2_1","doi-asserted-by":"publisher","DOI":"10.1038\/nm1011"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.3389\/fphys.2018.01305"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02576202"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.3389\/fphys.2018.01708"},{"key":"e_1_2_10_6_1","first-page":"215","volume-title":"Crit Rev Biomed Eng","author":"Rudy Y.","year":"1988"},{"key":"e_1_2_10_7_1","first-page":"1142","volume-title":"SIAM J Sci Stat Comput","author":"Hansen P. 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