{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T00:04:29Z","timestamp":1788998669663,"version":"build-2803163510"},"reference-count":53,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2026,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>For nanomaterials, the concept of heat conductivity needs to be revisited. At this fine scale, heat is propagated using the phonon transport equation, an ab initio model derived from first principles. In this problem, a material\u2019s thermal property is coded in a coefficient termed the relaxation time [Formula: see text], depending on the phonon frequency. We propose a novel inverse problem in this paper that infers the relaxation time using a material\u2019s temperature response upon heat injection. This inverse problem is formulated in a PDE-constrained optimization and is numerically solved by the stochastic gradient descent method (SGD) and its variants. In the execution of SGD, the Fr\u00e9chet derivative is computed and Lipschitz continuity is proved. This approach, in comparison to the earlier studies, enhances the ab initio concept of heat conduction in nanostructures associated to nanomaterials.<\/jats:p>","DOI":"10.1137\/25m1737341","type":"journal-article","created":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T07:00:56Z","timestamp":1777878056000},"page":"751-775","source":"Crossref","is-referenced-by-count":1,"title":["Reconstruction of the Heat Relaxation Index in the Phonon Transport Equation"],"prefix":"10.1137","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-4001-0929","authenticated-orcid":true,"given":"Peiyi","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4956-2376","authenticated-orcid":true,"given":"Irene M.","family":"Gamba","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Oden Institute, University of Texas-Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9210-8948","authenticated-orcid":true,"given":"Qin","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,5,4]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/15\/2\/022"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/12\/123010"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1364\/OE.2.000213"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/5\/053001"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2008.2.427"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2010.540608"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0743736-0"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/16M1080173"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110404"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1063\/1.1819431"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1115\/1.1454111"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.86.2297"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/aac220"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.01.034"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1080\/03605309608821207"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/12\/5\/001"},{"key":"ref17","first-page":"87","volume":"36","author":"Choulli M.","year":"1999","journal-title":"Osaka J. 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