{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T22:47:23Z","timestamp":1785192443121,"version":"3.55.0"},"reference-count":53,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,1,30]],"date-time":"2018-01-30T00:00:00Z","timestamp":1517270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Most inverse problems in the industry (and particularly in geophysical exploration) are highly underdetermined because the number of model parameters too high to achieve accurate data predictions and because the sampling of the data space is scarce and incomplete; it is always affected by different kinds of noise. Additionally, the physics of the forward problem is a simplification of the reality. All these facts result in that the inverse problem solution is not unique; that is, there are different inverse solutions (called equivalent), compatible with the prior information that fits the observed data within similar error bounds. In the case of nonlinear inverse problems, these equivalent models are located in disconnected flat curvilinear valleys of the cost-function topography. The uncertainty analysis consists of obtaining a representation of this complex topography via different sampling methodologies. In this paper, we focus on the use of a particle swarm optimization (PSO) algorithm to sample the region of equivalence in nonlinear inverse problems. Although this methodology has a general purpose, we show its application for the uncertainty assessment of the solution of a geophysical problem concerning gravity inversion in sedimentary basins, showing that it is possible to efficiently perform this task in a sampling-while-optimizing mode. Particularly, we explain how to use and analyze the geophysical models sampled by exploratory PSO family members to infer different descriptors of nonlinear uncertainty.<\/jats:p>","DOI":"10.3390\/e20020096","type":"journal-article","created":{"date-parts":[[2018,1,30]],"date-time":"2018-01-30T12:13:37Z","timestamp":1517314417000},"page":"96","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Particle Swarm Optimization and Uncertainty Assessment in Inverse Problems"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8158-106X","authenticated-orcid":false,"given":"Jos\u00e9","family":"Pallero","sequence":"first","affiliation":[{"name":"ETSI en Topograf\u00eda, Geodesia y Cartograf\u00eda, Universidad Polit\u00e9cnica de Madrid, 28031 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6963-7507","authenticated-orcid":false,"given":"Mar\u00eda","family":"Fern\u00e1ndez-Mu\u00f1iz","sequence":"additional","affiliation":[{"name":"Grupo de Problemas Inversos, Optimizaci\u00f3n y Aprendizaje Autom\u00e1tico, Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33007 Oviedo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8890-6429","authenticated-orcid":false,"given":"Ana","family":"Cernea","sequence":"additional","affiliation":[{"name":"Grupo de Problemas Inversos, Optimizaci\u00f3n y Aprendizaje Autom\u00e1tico, Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33007 Oviedo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2244-5161","authenticated-orcid":false,"given":"\u00d3scar","family":"\u00c1lvarez-Machancoses","sequence":"additional","affiliation":[{"name":"Grupo de Problemas Inversos, Optimizaci\u00f3n y Aprendizaje Autom\u00e1tico, Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33007 Oviedo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9835-1911","authenticated-orcid":false,"given":"Luis","family":"Pedruelo-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Grupo de Problemas Inversos, Optimizaci\u00f3n y Aprendizaje Autom\u00e1tico, Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33007 Oviedo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sylvain","family":"Bonvalot","sequence":"additional","affiliation":[{"name":"Laboratoire GET, Universit\u00e9 de Toulouse, CNRS, IRD, CNES, 31400 Toulouse, France"},{"name":"Bureau Gravim\u00e9trique International (BGI), 31401 Toulouse, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4758-2832","authenticated-orcid":false,"given":"Juan","family":"Fern\u00e1ndez-Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Grupo de Problemas Inversos, Optimizaci\u00f3n y Aprendizaje Autom\u00e1tico, Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33007 Oviedo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,30]]},"reference":[{"key":"ref_1","first-page":"49","article-title":"Sur les probl\u00e8mes aux d\u00e8riv\u00e8es partielles et leur signification physique","volume":"13","author":"Hadamard","year":"1902","journal-title":"Princeton Univ. 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