{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T09:51:55Z","timestamp":1775037115524,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10957-025-02805-2","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T11:50:06Z","timestamp":1757677806000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimating Mean and Variance of Random Coefficients in Stochastic Variational Problems Using Second-Order Methods"],"prefix":"10.1007","volume":"208","author":[{"given":"Zi-Jia","family":"Gong","sequence":"first","affiliation":[]},{"given":"Akhtar A.","family":"Khan","sequence":"additional","affiliation":[]},{"given":"Miguel","family":"Sama","sequence":"additional","affiliation":[]},{"given":"Hans-J\u00f6rg","family":"Starkloff","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"issue":"6","key":"2805_CR1","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1088\/0266-5611\/10\/6\/003","volume":"10","author":"R Acar","year":"1994","unstructured":"Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Prob. 10(6), 1217\u20131229 (1994)","journal-title":"Inverse Prob."},{"issue":"1","key":"2805_CR2","doi-asserted-by":"publisher","first-page":"1166","DOI":"10.1137\/16M106306X","volume":"5","author":"A Alexanderian","year":"2017","unstructured":"Alexanderian, A., Petra, N., Stadler, G., Ghattas, O.: Mean-variance risk-averse optimal control of systems governed by pdes with random parameter fields using quadratic approximations. SIAM\/ASA J. Uncertain. Quantif. 5(1), 1166\u20131192 (2017)","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"issue":"9","key":"2805_CR3","doi-asserted-by":"publisher","first-page":"1569","DOI":"10.1002\/nme.1015","volume":"60","author":"VA Badri Narayanan","year":"2004","unstructured":"Badri Narayanan, V.A., Zabaras, N.: Stochastic inverse heat conduction using a spectral approach. Internat. J. Numer. Methods Engrg. 60(9), 1569\u20131593 (2004)","journal-title":"Internat. J. Numer. Methods Engrg."},{"key":"2805_CR4","doi-asserted-by":"crossref","unstructured":"Banks, H.T., Reich, S., Rosen, I.G.: An approximation theory for the identification of nonlinear distributed parameter systems. SIAM J. Control. Optim. 28(3), 552\u2013569 (1990)","DOI":"10.1137\/0328033"},{"issue":"3","key":"2805_CR5","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/BF01447744","volume":"24","author":"HT Banks","year":"1991","unstructured":"Banks, H.T., Reich, S., Rosen, I.G.: Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems. Appl. Math. Optim. 24(3), 233\u2013256 (1991)","journal-title":"Appl. Math. Optim."},{"key":"2805_CR6","volume-title":"Random integral equations, Mathematics in Science and Engineering","author":"AT Bharucha-Reid","year":"1972","unstructured":"Bharucha-Reid, A.T.: Random integral equations, Mathematics in Science and Engineering, vol. 96. Academic Press, New York-London (1972)"},{"issue":"6","key":"2805_CR7","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/31\/6\/065008","volume":"31","author":"J Borggaard","year":"2015","unstructured":"Borggaard, J., van Wyk, H.-W.: Gradient-based estimation of uncertain parameters for elliptic partial differential equations. Inverse Prob. 31(6), 065008 (2015)","journal-title":"Inverse Prob."},{"issue":"3","key":"2805_CR8","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1137\/070711311","volume":"31","author":"A Borzi","year":"2009","unstructured":"Borzi, A., von Winckel, G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. 31(3), 2172\u20132192 (2009)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"2805_CR9","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1088\/0266-5611\/20\/5\/005","volume":"20","author":"M Burger","year":"2004","unstructured":"Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Prob. 20(5), 1411\u20131421 (2004)","journal-title":"Inverse Prob."},{"issue":"3","key":"2805_CR10","doi-asserted-by":"publisher","first-page":"892","DOI":"10.1137\/S0363012997318602","volume":"37","author":"Z Chen","year":"1999","unstructured":"Chen, Z., Zou, J.: An augmented lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control. Optim. 37(3), 892\u2013910 (1999)","journal-title":"SIAM J. Control. Optim."},{"issue":"1","key":"2805_CR11","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/s00211-015-0743-4","volume":"133","author":"P Chen","year":"2016","unstructured":"Chen, P., Quarteroni, A., Rozza, G.: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations. Numer. Math. 133(1), 67\u2013102 (2016)","journal-title":"Numer. Math."},{"issue":"2","key":"2805_CR12","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1137\/21M1409779","volume":"60","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. 60(2), 751\u2013780 (2022)","journal-title":"SIAM J. Numer. Anal."},{"key":"2805_CR13","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1016\/j.apnum.2018.03.002","volume":"133","author":"Y Choi","year":"2018","unstructured":"Choi, Y., Lee, H.-C.: Error analysis of finite element approximations of the optimal control problem for stochastic stokes equations with additive white noise. Appl. Numer. Math. 133, 144\u2013160 (2018)","journal-title":"Appl. Numer. Math."},{"key":"2805_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2023.103869","volume":"73","author":"J Dippon","year":"2023","unstructured":"Dippon, J., Gwinner, J., Khan, A.A., Sama, M.: A new regularized stochastic approximation framework for stochastic inverse problems. Nonlinear Anal. Real World Appl. 73, 103869 (2023)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"4","key":"2805_CR15","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1088\/0266-5611\/5\/4\/007","volume":"5","author":"HW Engl","year":"1989","unstructured":"Engl, H.W., Kunisch, K., Neubauer, A.: Convergence rates for tikhonov regularisation of nonlinear ill-posed problems. Inverse Prob. 5(4), 523\u2013540 (1989)","journal-title":"Inverse Prob."},{"issue":"2","key":"2805_CR16","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1007\/s10957-016-0888-z","volume":"171","author":"F Faraci","year":"2016","unstructured":"Faraci, F., Jadamba, B., Raciti, F.: On stochastic variational inequalities with mean value constraints. J. Optim. Theory Appl. 171(2), 675\u2013693 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"2805_CR17","doi-asserted-by":"publisher","first-page":"494","DOI":"10.1016\/j.camwa.2016.12.011","volume":"73","author":"B Faverjon","year":"2017","unstructured":"Faverjon, B., Puig, B., Baranger, T.N.: Identification of boundary conditions by solving cauchy problem in linear elasticity with material uncertainties. Comput. Math. Appl. 73(3), 494\u2013504 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"2805_CR18","doi-asserted-by":"publisher","first-page":"2982","DOI":"10.1137\/17M1123559","volume":"55","author":"B Gong","year":"2017","unstructured":"Gong, B., Liu, W., Tang, T., Zhao, W., Zhou, T.: An efficient gradient projection method for stochastic optimal control problems. SIAM J. Numer. Anal. 55(6), 2982\u20133005 (2017)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"2805_CR19","first-page":"1273","volume":"6","author":"W Grecksch","year":"2021","unstructured":"Grecksch, W., Jadamba, B., Khan, A.A., Sama, M., Tammer, C.: Inverse problem of estimating the stochastic flexural rigidity in fourth-order models. Pure Appl. Funct. Anal. 6(6), 1273\u20131301 (2021)","journal-title":"Pure Appl. Funct. Anal."},{"issue":"4","key":"2805_CR20","doi-asserted-by":"publisher","first-page":"1532","DOI":"10.1137\/100801731","volume":"49","author":"MD Gunzburger","year":"2011","unstructured":"Gunzburger, M.D., Lee, H.-C., Lee, J.: Error estimates of stochastic optimal neumann boundary control problems. SIAM J. Numer. Anal. 49(4), 1532\u20131552 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"2805_CR21","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1137\/19M1294952","volume":"9","author":"PA Guth","year":"2021","unstructured":"Guth, P.A., Kaarnioja, V., Kuo, F.Y., Schillings, C., Sloan, I.H.: A quasi-monte carlo method for optimal control under uncertainty. SIAM\/ASA J. Uncertain. Quantif. 9(2), 354\u2013383 (2021)","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"2805_CR22","volume-title":"Uncertainty quantification in variational inequalities-theory, numerics, and applications","author":"J Gwinner","year":"2022","unstructured":"Gwinner, J., Jadamba, B., Khan, A.A., Raciti, F.: Uncertainty quantification in variational inequalities-theory, numerics, and applications. CRC Press, Boca Raton, FL (2022)"},{"issue":"12","key":"2805_CR23","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/26\/12\/125014","volume":"26","author":"DN H\u00e0o","year":"2010","unstructured":"H\u00e0o, D.N., Quyen, T.N.T.: Convergence rates for tikhonov regularization of coefficient identification problems in laplace-type equations. Inverse Prob. 26(12), 125014 (2010)","journal-title":"Inverse Prob."},{"issue":"7","key":"2805_CR24","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/7\/075008","volume":"27","author":"DN H\u00e0o","year":"2011","unstructured":"H\u00e0o, D.N., Quyen, T.N.T.: Convergence rates for total variation regularization of coefficient identification problems in elliptic equations i. Inverse Prob. 27(7), 075008 (2011)","journal-title":"Inverse Prob."},{"issue":"2","key":"2805_CR25","first-page":"247","volume":"4","author":"DN Hao","year":"2019","unstructured":"Hao, D.N., Khan, A.A., Sama, M., Tammer, C.: Inverse problems in variational inequalities by minimizing energy. Pure Appl. Funct. Anal. 4(2), 247\u2013269 (2019)","journal-title":"Pure Appl. Funct. Anal."},{"issue":"1","key":"2805_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2023.127634","volume":"530","author":"F Huang","year":"2024","unstructured":"Huang, F., Chen, Y., Chen, Y., Sun, H.: Stochastic collocation for optimal control problems with stochastic pde constraints by meshless techniques. J. Math. Anal. Appl. 530(1), 127634 (2024)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"2805_CR27","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1137\/20M1323953","volume":"9","author":"B Jadamba","year":"2021","unstructured":"Jadamba, B., Khan, A.A., Sama, M., Starkloff, H.-J., Tammer, C.: A convex optimization framework for the inverse problem of identifying a random parameter in a stochastic partial differential equation. SIAM\/ASA Journal on Uncertainty Quantification 9(2), 922\u2013952 (2021)","journal-title":"SIAM\/ASA Journal on Uncertainty Quantification"},{"issue":"6","key":"2805_CR28","doi-asserted-by":"publisher","first-page":"3282","DOI":"10.1016\/j.jcp.2007.11.042","volume":"227","author":"B Jin","year":"2008","unstructured":"Jin, B., Zou, J.: Inversion of robin coefficient by a spectral stochastic finite element approach. J. Comput. Phys. 227(6), 3282\u20133306 (2008)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2805_CR29","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/26\/3\/035007","volume":"26","author":"B Kaltenbacher","year":"2010","unstructured":"Kaltenbacher, B., Hofmann, B.: Convergence rates for the iteratively regularized gauss-newton method in banach spaces. Inverse Prob. 26(3), 035007 (2010)","journal-title":"Inverse Prob."},{"issue":"2","key":"2805_CR30","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1016\/j.jmaa.2009.06.004","volume":"359","author":"P K\u00fcgler","year":"2009","unstructured":"K\u00fcgler, P., Sincich, E.: Logarithmic convergence rates for the identification of a nonlinear robin coefficient. J. Math. Anal. Appl. 359(2), 451\u2013463 (2009)","journal-title":"J. Math. Anal. Appl."},{"key":"2805_CR31","unstructured":"Reich, S.: A weak convergence theorem for the alternating method with Bregman distances, in: Theory and applications of nonlinear operators of accretive and monotone type, Vol. 178 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996, pp. 313\u2013318"},{"issue":"3","key":"2805_CR32","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1017\/S0956792514000072","volume":"25","author":"A Nouy","year":"2014","unstructured":"Nouy, A., Soize, C.: Random field representations for stochastic elliptic boundary value problems and statistical inverse problems. European J. Appl. Math. 25(3), 339\u2013373 (2014)","journal-title":"European J. Appl. Math."},{"issue":"1","key":"2805_CR33","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1080\/02331934.2018.1543295","volume":"68","author":"D Reem","year":"2019","unstructured":"Reem, D., Reich, S., De Pierro, A.: Re-examination of bregman functions and new properties of their divergences. Optimization 68(1), 279\u2013348 (2019)","journal-title":"Optimization"},{"key":"2805_CR34","doi-asserted-by":"crossref","unstructured":"Branch, M.A., Coleman, T.F., Li, Y.: A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems. SIAM J. Sci. Comput. 21(1), 1\u201323 (1999)","DOI":"10.1137\/S1064827595289108"},{"issue":"4","key":"2805_CR35","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1088\/0266-5611\/21\/4\/007","volume":"21","author":"E Resmerita","year":"2005","unstructured":"Resmerita, E.: Regularization of ill-posed problems in banach spaces: convergence rates. Inverse Prob. 21(4), 1303\u20131314 (2005)","journal-title":"Inverse Prob."},{"issue":"3","key":"2805_CR36","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1088\/0266-5611\/22\/3\/004","volume":"22","author":"E Resmerita","year":"2006","unstructured":"Resmerita, E., Scherzer, O.: Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22(3), 801\u2013814 (2006)","journal-title":"Inverse Prob."},{"issue":"11","key":"2805_CR37","doi-asserted-by":"publisher","first-page":"3021","DOI":"10.1016\/j.camwa.2018.08.005","volume":"77","author":"C Touzeau","year":"2019","unstructured":"Touzeau, C., Magnain, B., Lubineau, G., Florentin, E.: Robust method for identifying material parameters based on virtual fields in elastodynamics. Comput. Math. Appl. 77(11), 3021\u20133042 (2019)","journal-title":"Comput. Math. Appl."},{"issue":"5","key":"2805_CR38","doi-asserted-by":"publisher","first-page":"A2880","DOI":"10.1137\/18M120155X","volume":"41","author":"WIT Uy","year":"2019","unstructured":"Uy, W.I.T., Grigoriu, M.: Specification of additional information for solving stochastic inverse problems. SIAM J. Sci. Comput. 41(5), A2880\u2013A2910 (2019)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02805-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02805-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02805-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:23:31Z","timestamp":1775028211000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02805-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,12]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2805"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02805-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,12]]},"assertion":[{"value":"31 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 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":"No potential conflict of interest was reported by the author(s).","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"12"}}