{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T00:07:45Z","timestamp":1778717265924,"version":"3.51.4"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071061"],"award-info":[{"award-number":["12071061"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s11075-025-02179-6","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T05:44:53Z","timestamp":1753335893000},"page":"1355-1376","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Statistical methods for the inversion of the source term and initial values of the diffusion equation"],"prefix":"10.1007","volume":"102","author":[{"given":"Yuqian","family":"Yuan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fangfang","family":"Dou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"key":"2179_CR1","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1016\/j.amc.2019.03.063","volume":"358","author":"S Ahmad","year":"2019","unstructured":"Ahmad, S., Strauss, T., Kupis, S., Khan, T.: Comparison of statistical inversion with iteratively regularized Gauss Newton method for image reconstruction in electrical impedance tomography. Appl. Math. Comput. 358, 436\u2013448 (2019)","journal-title":"Appl. Math. Comput."},{"key":"2179_CR2","doi-asserted-by":"crossref","unstructured":"Arjmand, D., Ashyraliyev, M.: Efficient low rank approximations for parabolic control problems with unknown heat source. J. Comput. Appl. Math. 450, 115959 (2024)","DOI":"10.1016\/j.cam.2024.115959"},{"key":"2179_CR3","doi-asserted-by":"crossref","unstructured":"Biccari, U., Song, Y., Yuan, X., Zuazua, E.: A two-stage numerical approach for the sparse initial source identification of a diffusion\u2013advection equation. Inverse Probl. 39(9), 095003 (2023)","DOI":"10.1088\/1361-6420\/ace548"},{"key":"2179_CR4","doi-asserted-by":"crossref","unstructured":"Bissantz, N., Hohage, T., Munk, A.: Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inverse Probl. 20(6), 1773 (2004)","DOI":"10.1088\/0266-5611\/20\/6\/005"},{"key":"2179_CR5","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/j.cam.2018.01.010","volume":"337","author":"K Cao","year":"2018","unstructured":"Cao, K., Lesnic, D.: Reconstruction of the space-dependent perfusion coefficient from final time or time-average temperature measurements. J. Comput. Appl. Math. 337, 150\u2013165 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"2179_CR6","doi-asserted-by":"crossref","unstructured":"Chen, S., Jiang, D., Wang, H.: Simultaneous identification of initial value and source strength in a transmission problem for a parabolic equation. Adv. Comput. Math. 48, 77 (2022)","DOI":"10.1007\/s10444-022-09983-x"},{"issue":"60","key":"2179_CR7","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1137\/21M1409779","volume":"2022","author":"Z Chen","year":"2022","unstructured":"Chen, Z., Zhang, W., Zou, J.: Stochastic convergence of regularized solutions and their finite element approximations to inverse source problems. SIAM J. Numer. Anal. 2022(60), 751\u2013780 (2022)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"2179_CR8","doi-asserted-by":"publisher","first-page":"5329","DOI":"10.1002\/mma.9867","volume":"47","author":"D Durdiev","year":"2024","unstructured":"Durdiev, D., Turdiev, H.: Inverse coefficient problem for a time-fractional wave equation with initial-boundary and integral type overdetermination conditions. Math. Meth. Appl. Sci. 47(6), 5329\u20135340 (2024)","journal-title":"Math. Meth. Appl. Sci."},{"key":"2179_CR9","doi-asserted-by":"crossref","unstructured":"Hussein, M.S., Dyhoum, T., Hussein, S.O., Qassim, M.: Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem. Mathematics 12(17), 2629 (2024)","DOI":"10.3390\/math12172629"},{"key":"2179_CR10","doi-asserted-by":"crossref","unstructured":"Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems, vol. 160 of Applied Mathematical Sciences. Springer New York (2005)","DOI":"10.1007\/b138659"},{"key":"2179_CR11","doi-asserted-by":"crossref","unstructured":"Kian, Y., Uhlmann, G.: Recovery of Nonlinear Terms for Reaction Diffusion Equations from Boundary Measurements. Arch. Ration. Mech. Anal. 247, 6 (2023)","DOI":"10.1007\/s00205-022-01831-y"},{"issue":"7","key":"2179_CR12","doi-asserted-by":"publisher","first-page":"1294","DOI":"10.1080\/03610926.2012.681417","volume":"42","author":"BT Knapik","year":"2013","unstructured":"Knapik, B.T., van der Vaart, A.W., van Zanten, J.H.: Bayesian recovery of the initial condition for the heat equation. Commun. Stat. Theory Methods 42(7), 1294\u20131313 (2013)","journal-title":"Commun. Stat. Theory Methods"},{"key":"2179_CR13","doi-asserted-by":"publisher","first-page":"4099","DOI":"10.1109\/TIP.2020.2969077","volume":"29","author":"G Liang","year":"2020","unstructured":"Liang, G., Ren, S., Dong, F.: A Shape-Based Statistical Inversion Method for EIT\/URT Dual-Modality Imaging. IEEE Trans. Image Process. 29, 4099\u20134113 (2020)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"2179_CR14","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1137\/S0036141092238060","volume":"25","author":"B Mair","year":"1994","unstructured":"Mair, B.: Tikhonov Regularization for Finitely and Infinitely Smoothing Operators. SIAM J. Math. Anal. 25(1), 135\u2013147 (1994)","journal-title":"SIAM J. Math. Anal."},{"issue":"5","key":"2179_CR15","doi-asserted-by":"publisher","first-page":"1424","DOI":"10.1137\/S0036139994264476","volume":"56","author":"B Mair","year":"1996","unstructured":"Mair, B., Ruymgaart, F.: Statistical Inverse Estimation in Hilbert Scales. SIAM J. Appl. Math. 56(5), 1424\u20131444 (1996)","journal-title":"SIAM J. Appl. Math."},{"issue":"3","key":"2179_CR16","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1080\/00018732.2017.1341604","volume":"66","author":"H Nguyen","year":"2017","unstructured":"Nguyen, H., Zecchina, R., Berg, J.: Inverse statistical problems: from the inverse Ising problem to data science. Adv. Phys. 66(3), 197\u2013261 (2017)","journal-title":"Adv. Phys."},{"key":"2179_CR17","doi-asserted-by":"crossref","unstructured":"Rundell, W.: Recovery of a coefficient in a diffusion equation from large time data. Inverse Probl. 40(12), 125013 (2024)","DOI":"10.1088\/1361-6420\/ad904c"},{"issue":"6","key":"2179_CR18","doi-asserted-by":"publisher","first-page":"1635","DOI":"10.1137\/0727096","volume":"27","author":"F O\u2019Sullivan","year":"1990","unstructured":"O\u2019Sullivan, F.: Convergence Characteristics of Methods of Regularization Estimators for Nonlinear Operator Equations. SIAM J. Numer. Anal. 27(6), 1635\u20131649 (1990)","journal-title":"SIAM J. Numer. Anal."},{"key":"2179_CR19","doi-asserted-by":"crossref","unstructured":"Sun, S., Nakamura, G., Wang, H.: Domain sampling methods for an inverse boundary value problem of the heat equation. Inverse Probl. 40(12), 125009 (2024)","DOI":"10.1088\/1361-6420\/ad8d78"},{"key":"2179_CR20","doi-asserted-by":"crossref","unstructured":"Umbricht, G., Rubio, D.: Regularization techniques for estimating the space-dependent source in an $$n$$-dimensional linear parabolic equation using space-dependent noisy data. Comput. Math. Appl. 172, 47\u201369 (2024)","DOI":"10.1016\/j.camwa.2024.07.029"},{"issue":"10","key":"2179_CR21","doi-asserted-by":"publisher","first-page":"6288","DOI":"10.1002\/mma.8171","volume":"45","author":"X Wang","year":"2022","unstructured":"Wang, X., Dou, F.: Statistical inverse estimation of source terms in the Helmholtz equation. Math. Methods Appl. Sci. 45(10), 6288\u20136301 (2022)","journal-title":"Math. Methods Appl. Sci."},{"issue":"17","key":"2179_CR22","doi-asserted-by":"publisher","first-page":"13221","DOI":"10.1002\/mma.10186","volume":"47","author":"J Wen","year":"2024","unstructured":"Wen, J., Liu, Y., O\u2019Regan, D., Ren, X.: Fourier truncation regularization method for simultaneous inversion in parabolic equations with convective term. Math. Methods Appl. Sci. 47(17), 13221\u201313235 (2024)","journal-title":"Math. Methods Appl. Sci."},{"issue":"3","key":"2179_CR23","doi-asserted-by":"publisher","first-page":"726","DOI":"10.1016\/j.jcp.2010.10.015","volume":"230","author":"J Wu","year":"2011","unstructured":"Wu, J., Li, T., Peng, X., Guo, H.: Statistical method for resolving the photon\u2013photoelectron-counting inversion problem. J. Comput. Phys. 230(3), 726\u2013743 (2011)","journal-title":"J. Comput. Phys."},{"key":"2179_CR24","doi-asserted-by":"crossref","unstructured":"Yu, Q., Xiong, X.: Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation. J. Comput. Appl. Math. 451, 116062 (2024)","DOI":"10.1016\/j.cam.2024.116062"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02179-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-025-02179-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02179-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T23:12:33Z","timestamp":1778713953000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-025-02179-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,24]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["2179"],"URL":"https:\/\/doi.org\/10.1007\/s11075-025-02179-6","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,24]]},"assertion":[{"value":"4 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2025","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":"Competing Interests"}}]}}