{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:46Z","timestamp":1787330086503,"version":"build-2736575974"},"reference-count":57,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"name":"Collaborative Research Center","award":["1324"],"award-info":[{"award-number":["1324"]}]},{"name":"Centre of Excellence of Inverse Modelling and Imaging","award":["312 125"],"award-info":[{"award-number":["312 125"]}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["2181\/1 - 390900948"],"award-info":[{"award-number":["2181\/1 - 390900948"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Pattern formation plays an important role in the development of living organisms. Since the classical work of Alan Turing, a pre-eminent way of modeling has been through reaction-diffusion mechanisms. Alternative models have been proposed that link dynamics of diffusing molecular signals with tissue mechanics. Model validation is complicated as in many experimental situations only the limiting, stationary regime of the pattern formation process is observable, without any knowledge of the transient behavior or the initial state. To overcome this problem, the initial state of the model can be randomized. But then fixed values of the model parameters correspond to a family of patterns rather than a fixed stationary solution, and standard estimation approaches, such the least squares, are not suitable. Instead, statistical characteristics of the patterns should be compared, which is difficult given the typically limited amount of available data in practical applications. To deal with this problem, we extend the recently developed statistical approach (the correlation integral likelihood method) for parameter identification by pattern data. We introduce modifications that allow increasing the accuracy of the identification process without resizing the data set. The proposed approach is tested using different classes of pattern formation models and severely limited data sets. For all considered equations, parallel, GPU-based implementations of the numerical solvers with efficient time stepping schemes are provided.<\/jats:p>","DOI":"10.1137\/22m1485553","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T04:05:01Z","timestamp":1696565101000},"page":"B673-B696","source":"Crossref","is-referenced-by-count":7,"title":["A Bayesian Approach to Modeling Biological Pattern Formation with Limited Data"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5956-2791","authenticated-orcid":true,"given":"Alexey","family":"Kazarnikov","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 69120, Heidelberg, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert","family":"Scheichl","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 69120, Heidelberg, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0538-5697","authenticated-orcid":true,"given":"Heikki","family":"Haario","sequence":"additional","affiliation":[{"name":"Department of Computational Engineering, LUT University, 53850, Lappeenranta, Finland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5831-6505","authenticated-orcid":true,"given":"Anna","family":"Marciniak-Czochra","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 69120, Heidelberg, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100292"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TCAD.1985.1270142"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-01-02819-7"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s11538-018-0518-z"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/16M1084080"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2022.08.007"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/130915005"},{"key":"ref8","volume":"6","author":"Donsker M.","year":"1951","journal-title":"Mem. 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