{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T19:44:32Z","timestamp":1776109472264,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T00:00:00Z","timestamp":1769472000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T00:00:00Z","timestamp":1769472000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100008332","name":"Graz University of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100008332","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2026,2]]},"abstract":"Abstract<\/jats:title>\n \n In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator\n \n \n $$\\varvec{{\\text {\\textsf{V}}}}$$<\/jats:tex-math>\n \n \n V<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for the wave equation as a minimization problem in\n \n \n $$\\varvec{L^2(\\Sigma )}$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$\\varvec{\\Sigma := \\partial \\Omega \\times (0,T)}$$<\/jats:tex-math>\n \n \n \u03a3<\/mml:mi>\n :<\/mml:mo>\n =<\/mml:mo>\n \u2202<\/mml:mi>\n \u03a9<\/mml:mi>\n \u00d7<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the lateral boundary of the space-time domain\n \n \n $$\\varvec{Q:= \\Omega \\times (0,T)}$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n :<\/mml:mo>\n =<\/mml:mo>\n \u03a9<\/mml:mi>\n \u00d7<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.\n <\/jats:p>","DOI":"10.1007\/s10444-026-10282-y","type":"journal-article","created":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T06:22:20Z","timestamp":1769494940000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stable least-squares space-time boundary element methods for the wave equation"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-0994-6524","authenticated-orcid":false,"given":"Daniel","family":"Hoonhout","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6155-1178","authenticated-orcid":false,"given":"Richard","family":"L\u00f6scher","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2552-3022","authenticated-orcid":false,"given":"Olaf","family":"Steinbach","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0055-6175","authenticated-orcid":false,"given":"Carolina","family":"Urz\u00faa\u2013Torres","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,27]]},"reference":[{"key":"10282_CR1","doi-asserted-by":"publisher","first-page":"1746","DOI":"10.1016\/j.cam.2010.02.011","volume":"235","author":"A Aimi","year":"2011","unstructured":"Aimi, A., Diligenti, M., Guardasoni, C.: On the energetic Galerkin boundary element method applied to interior wave propagation problems. J. Comput. Appl. Math. 235, 1746\u20131754 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"10282_CR2","doi-asserted-by":"publisher","first-page":"1196","DOI":"10.1002\/nme.2660","volume":"80","author":"A Aimi","year":"2009","unstructured":"Aimi, A., Diligenti, M., Guardasoni, C., Mazzieri, I., Panizzi, S.: An energy approach to space-time Galerkin BEM for wave propagation problems. Internat. J. Numer. Methods Engrg. 80, 1196\u20131240 (2009)","journal-title":"Internat. J. Numer. Methods Engrg."},{"key":"10282_CR3","unstructured":"Aimi, A., Diligenti, M., Guardasoni, C., Panizzi, S.: A space-time energetic formulation for wave propagation analysis by BEMs. Riv. Mat. Univ. Parma, 8 (7), 171\u2013207 (2008)"},{"key":"10282_CR4","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1093\/imanum\/drs014","volume":"33","author":"R Andreev","year":"2013","unstructured":"Andreev, R.: Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33, 242\u2013260 (2013)","journal-title":"IMA J. Numer. Anal."},{"key":"10282_CR5","first-page":"673","volume":"83","author":"H Antes","year":"2004","unstructured":"Antes, H., R\u00fcberg, T., Schanz, M.: Convolution quadrature boundary element method for quasi-static visco- and poroelastic continua. Comput Struct 83, 673\u2013684 (2004)","journal-title":"Comput Struct"},{"key":"10282_CR6","doi-asserted-by":"crossref","unstructured":"Bamberger, A., Ha\u00a0Duong, T.: Formulation variationnelle pour le calcul de la diffraction d\u2019une onde acoustique par une surface rigide. Math. Meth. Appl. Sci. 8, 598\u2013608 (1986)","DOI":"10.1002\/mma.1670080139"},{"key":"10282_CR7","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1137\/070690754","volume":"47","author":"L Banjai","year":"2008","unstructured":"Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47, 227\u2013249 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"10282_CR8","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s00466-016-1281-3","volume":"58","author":"L Banz","year":"2016","unstructured":"Banz, L., Gimperlein, H., Nezhi, Z., Stephan, E.P.: Time domain BEM for sound radiation of tires. Comput. Mech. 58, 45\u201357 (2016)","journal-title":"Comput. Mech."},{"key":"10282_CR9","doi-asserted-by":"crossref","unstructured":"Bochev, P., Gunzburger, M.: Least-squares finite element methods. International Congress of Mathematicians. European Math Soc (EMS), Z\u00fcrich, Vol. III, pp. 1137\u20131162, (2006)","DOI":"10.4171\/022-3\/55"},{"key":"10282_CR10","doi-asserted-by":"crossref","unstructured":"Bochev, P., Gunzburger, M.: Least-squares methods for hyperbolic problems. Handbook of numerical methods for hyperbolic problems, Handb. Numer. Anal., 17, pp. 289\u2013317, Elsevier\/North-Holland, Amsterdam, (2016)","DOI":"10.1016\/bs.hna.2016.07.002"},{"key":"10282_CR11","doi-asserted-by":"crossref","unstructured":"Bochev, P., Gunzburger, M.: Least-squares finite element methods. Appl. Math. Sci., vol.\u00a0166, Springer, New York, (2009)","DOI":"10.1007\/b13382"},{"key":"10282_CR12","doi-asserted-by":"crossref","unstructured":"Cohen, A., Dahmen, W., Welper, G.: Adaptivity and variational stabilization for convection-diffusion equations. ESAIM: M2AN 46, 247\u20131273 (2012)","DOI":"10.1051\/m2an\/2012003"},{"key":"10282_CR13","doi-asserted-by":"crossref","unstructured":"Costabel, M., Sayas, F.-J.: Time-dependent problems with the boundary integral equation method. In: Encyclopedia of Computational Mechanics (Stein, E., Borst, R., Hughes, T.\u00a0J.\u00a0R. eds.), 2nd ed., Wiley, (2017)","DOI":"10.1002\/9781119176817.ecm2022"},{"key":"10282_CR14","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions. Numer. Methods Partial Different. Equ. 27, no. 1, 70\u2013105 (2011)","DOI":"10.1002\/num.20640"},{"key":"10282_CR15","doi-asserted-by":"publisher","first-page":"1106","DOI":"10.1137\/0733054","volume":"33","author":"W D\u00f6rfler","year":"1996","unstructured":"D\u00f6rfler, W.: A convergent adaptive algorithm for Poisson\u2019s equation. SIAM J. Numer. Anal. 33, 1106\u20131124 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"10282_CR16","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1515\/cmam-2016-0015","volume":"16","author":"W D\u00f6rfler","year":"2016","unstructured":"D\u00f6rfler, W., Findeisen, S., Wieners, C.: Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Meth. Appl. Math. 16, 409\u2013428 (2016)","journal-title":"Comput. Meth. Appl. Math."},{"key":"10282_CR17","doi-asserted-by":"crossref","unstructured":"Dohr, S.: Distributed and preconditioned space-time boundary element methods for the heat equation. PhD thesis, TU Graz, (2019)","DOI":"10.1515\/9783110548488-001"},{"key":"10282_CR18","doi-asserted-by":"crossref","unstructured":"Dohr, S., Steinbach, O., Niino, K.: Space-time boundary element methods for the heat equation. In: Space-Time Methods. Applications to Partial Differential Equations (Langer, U., Steinbach, O. eds.) pp. 1\u201360, de Gruyter, Berlin, Boston, (2019)","DOI":"10.1515\/9783110548488-001"},{"key":"10282_CR19","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1016\/j.camwa.2021.03.004","volume":"92","author":"T F\u00fchrer","year":"2021","unstructured":"F\u00fchrer, T., Karkulik, M.: Space-time least-squares finite elements for parabolic equations. Comput. Math. Appl. 92, 27\u201336 (2021)","journal-title":"Comput. Math. Appl."},{"key":"10282_CR20","doi-asserted-by":"crossref","unstructured":"F\u00fchrer, T., Gonzales, R., Karkulik, M.: Well-posedness of first-order acoustic wave equations and space-time finite element approximation. IMA J. Numer. Anal., published online, (2025)","DOI":"10.1093\/imanum\/drae104"},{"key":"10282_CR21","doi-asserted-by":"crossref","unstructured":"Gander, M.\u00a0J.: 50 years of time parallel integration. In: Multiple Shooting and Time Domain Decompositions, pp. 69\u2013114, Springer, Heidelberg, Berlin, (2015)","DOI":"10.1007\/978-3-319-23321-5_3"},{"key":"10282_CR22","doi-asserted-by":"publisher","first-page":"A2173","DOI":"10.1137\/15M1046605","volume":"38","author":"MJ Gander","year":"2016","unstructured":"Gander, M.J., Neum\u00fcller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38, A2173\u2013A2208 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"10282_CR23","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1002\/mma.3340","volume":"40","author":"H Gimperlein","year":"2017","unstructured":"Gimperlein, H., Nezhi, Z., Stephan, E.P.: A priori error estimates for a time-dependent boundary element method for the acoustic wave equation in a half-space. Math. Methods Appl. Sci. 40, 448\u2013462 (2017)","journal-title":"Math. Methods Appl. Sci."},{"key":"10282_CR24","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/j.cma.2019.07.018","volume":"356","author":"H Gimperlein","year":"2019","unstructured":"Gimperlein, H., \u00d6zdemir, C., Stark, D., Stephan, E.P.: hp-version time domain boundary elements for the wave equation on quasi-uniform meshes. Comput. Methods Appl. Mech. Engrg. 356, 145\u2013174 (2019)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"10282_CR25","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s00211-020-01142-y","volume":"146","author":"H Gimperlein","year":"2020","unstructured":"Gimperlein, H., \u00d6zdemir, C., Stark, D., Stephan, E.P.: A residual a posteriori error estimate for the time-domain boundary element method. Numer. Math. 146, 239\u2013280 (2020)","journal-title":"Numer. Math."},{"key":"10282_CR26","doi-asserted-by":"publisher","first-page":"70","DOI":"10.4208\/jcm.1610-m2016-0494","volume":"36","author":"H Gimperlein","year":"2018","unstructured":"Gimperlein, H., \u00d6zdemir, C., Stephan, E.P.: Time domain boundary element methods for the Neumann problem: error estimates and acoustic problems. J. Comput. Math. 36, 70\u201389 (2018)","journal-title":"J. Comput. Math."},{"key":"10282_CR27","unstructured":"Guardasoni, C.: Wave propogation analysis with boundary element method. PhD thesis, University of Parma, (2010)"},{"key":"10282_CR28","doi-asserted-by":"publisher","first-page":"1845","DOI":"10.1002\/nme.745","volume":"57","author":"T Ha-Duong","year":"2003","unstructured":"Ha-Duong, T., Ludwig, B., Terrasse, I.: A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Internat. J. Numer. Methods Engrg. 57, 1845\u20131882 (2003)","journal-title":"Internat. J. Numer. Methods Engrg."},{"key":"10282_CR29","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1216\/JIE-2017-29-1-107","volume":"29","author":"ME Hassell","year":"2017","unstructured":"Hassell, M.E., Qiu, T., S\u00e1nchez-Vizuet, T., Sayas, F.-J.: A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation. J. Integral Equ. Appl. 29, 107\u2013136 (2017)","journal-title":"J. Integral Equ. Appl."},{"key":"10282_CR30","doi-asserted-by":"crossref","unstructured":"John, V.: Finite element methods for incompressible flow problems. Springer Series in Comput. Math. vol.\u00a051, Springer, (2016)","DOI":"10.1007\/978-3-319-45750-5"},{"key":"10282_CR31","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1216\/JIE-2017-29-1-137","volume":"29","author":"P Joly","year":"2017","unstructured":"Joly, P., Rodr\u00edguez, J.: Mathematical aspects of variational boundary integral equations for time dependent wave propagation. J. Integral Equ. Appl. 29, 137\u2013187 (2017)","journal-title":"J. Integral Equ. Appl."},{"key":"10282_CR32","unstructured":"K\u00f6the, C., L\u00f6scher, R., Steinbach, O.: Adaptive least-squares space-time finite element methods. (2023) arXiv:2309.14300,"},{"key":"10282_CR33","doi-asserted-by":"crossref","unstructured":"Langer, U., Steinbach, O.: Space-time methods. Applications to partial differential equations. Radon Series on Comput. Appl. Math., vol. 25, de Gruyter, Berlin, (2019)","DOI":"10.1515\/9783110548488"},{"key":"10282_CR34","doi-asserted-by":"crossref","unstructured":"L\u00f6scher, R., Steinbach, O., Zank, M.: Numerical results for an unconditionally stable space-time finite element method for the wave equation. In: Domain Decomposition Methods in Science and Engineering XXVI. Lecture Notes in Computational Science and Engineering, vol. 145, pp. 625\u2013632. Springer, Cham (2022)","DOI":"10.1007\/978-3-030-95025-5_68"},{"key":"10282_CR35","doi-asserted-by":"publisher","first-page":"C320","DOI":"10.1137\/21M1430157","volume":"44","author":"M Merta","year":"2022","unstructured":"Merta, M., Of, G., Watschinger, R., Zapletal, J.: A parallel fast multipole method for a space-time boundary element method for the heat equation. SIAM J. Sci. Comput. 44, C320\u2013C345 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"10282_CR36","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s00211-017-0910-x","volume":"138","author":"A Moiola","year":"2018","unstructured":"Moiola, A., Perugia, I.: A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math. 138, 389\u2013435 (2018)","journal-title":"Numer. Math."},{"key":"10282_CR37","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.camwa.2021.08.004","volume":"99","author":"D P\u00f6lz","year":"2021","unstructured":"P\u00f6lz, D., Schanz, M.: On the space-time discretization of variational retarded potential boundary integral equations. Comput. Math. Appl. 99, 195\u2013210 (2021)","journal-title":"Comput. Math. Appl."},{"key":"10282_CR38","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s00211-012-0506-4","volume":"124","author":"F-J Sayas","year":"2013","unstructured":"Sayas, F.-J.: Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations. Numer. Math. 124, 121\u2013149 (2013)","journal-title":"Numer. Math."},{"key":"10282_CR39","doi-asserted-by":"crossref","unstructured":"Sayas, F.-J.: Retarded potentials and time domain boundary integral equations. A road map. Springer Series Comput. Math., vol. 50, Springer, Cham, (2016)","DOI":"10.1007\/978-3-319-26645-9"},{"key":"10282_CR40","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/s100920070009","volume":"37","author":"H Schulz","year":"2000","unstructured":"Schulz, H., Steinbach, O.: A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem. Calcolo 37, 79\u201396 (2000)","journal-title":"Calcolo"},{"key":"10282_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68805-3","volume-title":"Numerical Approximation Methods for Elliptic Boundary Value Problems","author":"O Steinbach","year":"2008","unstructured":"Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York (2008)"},{"key":"10282_CR42","doi-asserted-by":"publisher","unstructured":"Steinbach, O.: An adaptive least squares boundary element method for elliptic boundary value problems. Numer. Math., accepted, (2025) https:\/\/doi.org\/10.1007\/s00211-025-01520-4","DOI":"10.1007\/s00211-025-01520-4"},{"key":"10282_CR43","doi-asserted-by":"publisher","first-page":"1370","DOI":"10.1137\/21M1420034","volume":"54","author":"O Steinbach","year":"2022","unstructured":"Steinbach, O., Urz\u00faa-Torres, C.: A new approach to space-time boundary integral equations for the wave equation. SIAM J. Math. Anal. 54, 1370\u20131392 (2022)","journal-title":"SIAM J. Math. Anal."},{"key":"10282_CR44","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1216\/jie.2022.34.501","volume":"34","author":"O Steinbach","year":"2022","unstructured":"Steinbach, O., Urz\u00faa-Torres, C., Zank, M.: Towards coercive boundary element methods for the wave equation. J. Integral Equ. Appl. 34, 501\u2013515 (2022)","journal-title":"J. Integral Equ. Appl."},{"key":"10282_CR45","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1002\/pamm.201610377","volume":"16","author":"O Steinbach","year":"2016","unstructured":"Steinbach, O., Zank, M.: Adaptive space-time boundary element methods for the wave equation. PAMM. 16, 777\u2013778 (2016)","journal-title":"PAMM."},{"key":"10282_CR46","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1553\/etna_vol52s154","volume":"52","author":"O Steinbach","year":"2020","unstructured":"Steinbach, O., Zank, M.: Coercive space-time finite element methods for initial boundary value problems. Electron. Trans. Numer. Anal. 52, 154\u2013194 (2020)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"10282_CR47","doi-asserted-by":"crossref","unstructured":"Steinbach, O., Zank, M.: A generalized inf-sup stable variational formulation for the wave equation. J. Math. Anal. Appl. 505, 125457 (2022)","DOI":"10.1016\/j.jmaa.2021.125457"},{"key":"10282_CR48","first-page":"47","volume":"29","author":"O Steinbach","year":"2021","unstructured":"Steinbach, O., Zank, M.: A note on the efficient evaluation of a modified Hilbert transformation. J. Numer. Math. 29, 47\u201361 (2021)","journal-title":"J. Numer. Math."},{"key":"10282_CR49","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1093\/imanum\/drz069","volume":"41","author":"R Stevenson","year":"2021","unstructured":"Stevenson, R., Westerdiep, J.: Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic equations. IMA J. Numer. Anal. 41, 28\u201347 (2021)","journal-title":"IMA J. Numer. Anal."},{"key":"10282_CR50","unstructured":"Terrasse, I.: R\u00e9solution math\u00e9matique et num\u00e9rique des \u00e9quations de Maxwell instationnaires par une m\u00e9thode de potentiels retard\u00e9s. Doctoral dissertation, \u00c9cole Polytechnique, (1993)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10282-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-026-10282-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10282-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T10:27:05Z","timestamp":1774952825000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-026-10282-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,27]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["10282"],"URL":"https:\/\/doi.org\/10.1007\/s10444-026-10282-y","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,27]]},"assertion":[{"value":"15 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"7"}}