Terminal value problem for nonlinear parabolic equation with Gaussian white noise

Vinh Quang Mai; Erkan Nane; Donal O'Regan; Nguyen Huy Tuan; Vinh Quang Mai; Erkan Nane; Donal O'Regan; Nguyen Huy Tuan

doi:10.3934/era.2022072

Electronic Research Archive

2022, Volume 30, Issue 4: 1374-1413. doi: 10.3934/era.2022072

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Terminal value problem for nonlinear parabolic equation with Gaussian white noise

1.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
2.
Department of Mathematics and Statistics, Auburn University, Auburn, USA
3.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
4.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
5.
Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

Received: 30 December 2021 Revised: 30 January 2022 Accepted: 15 February 2022 Published: 18 March 2022

In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane ^[1].
- Quasi-reversibility method,
- backward problem,
- parabolic equation,
- Gaussian white noise regularization
Citation: Vinh Quang Mai, Erkan Nane, Donal O'Regan, Nguyen Huy Tuan. Terminal value problem for nonlinear parabolic equation with Gaussian white noise[J]. Electronic Research Archive, 2022, 30(4): 1374-1413. doi: 10.3934/era.2022072

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Abstract

In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane ^[1].

References

[1]	N. H. Tuan, E. Nane, Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data, SIAM/ASA J. Uncertain., 6 (2018), 302–338. https://doi.org/10.1137/17M1111139 doi: 10.1137/17M1111139
[2]	H. Amann, Time-delayed Perona–Malik type problems, Acta Math. Univ. Comenian., 76 (2007), 15–38.
[3]	J. Hadamard, Lectures on the Cauchy Problems in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1923.
[4]	M. Denche, K. Bessila, A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl., 301 (2005), 419–426. https://doi.org/10.1016/j.jmaa.2004.08.001 doi: 10.1016/j.jmaa.2004.08.001
[5]	N. V. Duc, An a posteriori mollification method for the heat equation backward in time, J. Inverse Ill-Posed Probl., 25 (2017), 403–422. https://doi.org/10.1515/jiip-2016-0026 doi: 10.1515/jiip-2016-0026
[6]	B. T. Johansson, D. Lesnic, T. Reeve, A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems, Int. J. Comput. Math., 89 (2012), 1555–1568. https://doi.org/10.1080/00207160.2012.680448 doi: 10.1080/00207160.2012.680448
[7]	A. B. Mair, H. F. Ruymgaart, Statistical inverse estimation in Hilbert scales, SIAM J. Appl. Math., 56 (1996), 1424–1444. https://doi.org/10.1137/S0036139994264476 doi: 10.1137/S0036139994264476
[8]	H. Kekkonen, M. Lassas, S. Siltanen, Analysis of regularized inversion of data corrupted by white Gaussian noise, Inverse Probl., 30 (2014), 045009. https://doi.org/10.1088/0266-5611/30/4/045009 doi: 10.1088/0266-5611/30/4/045009
[9]	C. König, F. Werner, T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal., 54 (2016), 341–360. https://doi.org/10.1137/15M1022252 doi: 10.1137/15M1022252
[10]	T. Hohage, F. Weidling, Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal., 55 (2017), 598–620. https://doi.org/10.1137/16M1067445 doi: 10.1137/16M1067445
[11]	A. P. N. T. Mai, A statistical minimax approach to the Hausdorff moment problem, Inverse Probl., 24 (2008), 045018. https://doi.org/10.1088/0266-5611/24/4/045018 doi: 10.1088/0266-5611/24/4/045018
[12]	L. Cavalier, Nonparametric statistical inverse problems, Inverse Probl., 24 (2008), 034004. https://doi.org/10.1088/0266-5611/24/3/034004 doi: 10.1088/0266-5611/24/3/034004
[13]	N. Bissantz, H. Holzmann, Asymptotics for spectral regularization estimators in statistical inverse problems, Comput. Statist., 28 (2013), 435–453. https://doi.org/10.1007/s00180-012-0309-1 doi: 10.1007/s00180-012-0309-1
[14]	D. D. Cox, Approximation of method of regularization estimators, Ann. Stat., 16 (1988), 694–712. https://doi.org/10.1214/aos/1176350829 doi: 10.1214/aos/1176350829
[15]	H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Dordrecht, Boston, London, 1996. https://doi.org/10.1007/978-94-009-1740-8
[16]	B. T. Knapik, A. W. van der Vaart, J. H. van Zanten, Bayesian recovery of the initial condition for the heat equation, Comm. Statist. Theory Methods, 42 (2013), 1294–1313.
[17]	N. Bochkina, Consistency of the posterior distribution in generalized linear inverse problems, Inverse Probl., 29 (2013), 095010. https://doi.org/10.1088/0266-5611/29/9/095010 doi: 10.1088/0266-5611/29/9/095010
[18]	R. Plato, Converse results, saturation and quasi-optimality for Lavrentiev regularization of accretive problems, SIAM J. Numer. Anal., 55 (2017), 1315–1329. https://doi.org/10.1137/16M1089125 doi: 10.1137/16M1089125
[19]	L. Cavalier, Inverse problems in statistics. Inverse problems and high-dimensional estimation, In: Alquier P., Gautier E., Stoltz G. (eds) Inverse Problems and High-Dimensional Estimation. Lecture Notes in Statistics, vol 203. Springer, Berlin, Heidelberg, 3–96. https://doi.org/10.1007/978-3-642-19989-9
[20]	M. Kirane, E. Nane, N. H. Tuan, On a backward problem for multidimensional Ginzburg-Landau equation with random data, Inverse Probl., 34 (2018), 015008. https://doi.org/10.1088/1361-6420/aa9c2a doi: 10.1088/1361-6420/aa9c2a
[21]	R. Lattes, J. L. Lions, Methode de Quasi-reversibility et Applications, Dunod, Paris, 1967
[22]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier – Villars, Paris, 1969.
[23]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, Volume 19, 1997.
[24]	C. Cao, M. A. Rammaha, E. S. Titi, The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom, Z. Angew. Math. Phys., 50 (1999), 341–360. https://doi.org/10.1007/PL00001493 doi: 10.1007/PL00001493
[25]	R. Courant, D. Hilbert, Methods of mathematical physics, New York (NY): Interscience; 1953.
[26]	J. Wu, W. Wang, On backward uniqueness for the heat operator in cones, J. Differ. Equ., 258 (2015), 224–241. https://doi.org/10.1016/j.jde.2014.09.011 doi: 10.1016/j.jde.2014.09.011
[27]	A. Ruland, On the backward uniqueness property for the heat equation in two-dimensional conical domains, Manuscr. Math., 147 (2015), 415–436. https://doi.org/10.1007/s00229-015-0764-4 doi: 10.1007/s00229-015-0764-4
[28]	L. Li, V. Sverak, Backward uniqueness for the heat equation in cones, Commmun. Partial Differ. Equ., 37 (2012), 1414–1429. https://doi.org/10.1080/03605302.2011.635323 doi: 10.1080/03605302.2011.635323
[29]	N. H. Tuan, P. H. Quan, Some extended results on a nonlinear ill-posed heat equation and remarks on a general case of nonlinear terms, Nonlinear Anal. Real World Appl., 12 (2011), 2973–2984. https://doi.org/10.1016/j.nonrwa.2011.04.018 doi: 10.1016/j.nonrwa.2011.04.018
[30]	D. D. Trong, N. H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal., 71 (2009), 4167–4176. https://doi.org/10.1016/j.na.2009.02.092 doi: 10.1016/j.na.2009.02.092
[31]	P. T. Nam, An approximate solution for nonlinear backward parabolic equations, J. Math. Anal. Appl., 367 (2010), 337–349. https://doi.org/10.1016/j.jmaa.2010.01.020 doi: 10.1016/j.jmaa.2010.01.020
[32]	M. Chipot, Elements of nonlinear analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Verlag, Basel, 2000. viii+256 pp. ISBN: 3-7643-6406-8. https://doi.org/10.1007/978-3-0348-8428-0
[33]	J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196–218. https://doi.org/10.1016/S0092-8240(05)80044-8 doi: 10.1016/S0092-8240(05)80044-8
[34]	L. T. P. Ngoc, A. P. N. Dinh, N. T. Long, On a nonlinear heat equation associated with Dirichlet-Robin conditions, Numer. Funct. Anal. Optim., 33 (2012), 166–189. https://doi.org/10.1080/01630563.2011.594198 doi: 10.1080/01630563.2011.594198
[35]	N. H. Tuan, L. D. Thang, V. A. Khoa, T. Tran, On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions, J. Math. Anal. Appl., 426 (2015), 1232–1261. https://doi.org/10.1016/j.jmaa.2014.12.047 doi: 10.1016/j.jmaa.2014.12.047

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