Home A two-dimensional backward heat problem with statistical discrete data
Article
Licensed
Unlicensed Requires Authentication

A two-dimensional backward heat problem with statistical discrete data

  • Nguyen Dang Minh EMAIL logo , Khanh To Duc , Nguyen Huy Tuan and Dang Duc Trong
Published/Copyright: May 5, 2017

Abstract

We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ=θ(x,y)=u(x,y,0) such that

{ut-a(t)(uxx+uyy)=f(x,y,t),(x,y,t)Ω×(0,T),u(x,y,t)=0,(x,y)Ω×(0,T),u(x,y,T)=h(x,y),(x,y)Ω¯,

where Ω=(0,π)×(0,π). In the problem, the source f=f(x,y,t) and the final data h=h(x,y) are determined through random noise data gij(t) and dij satisfying the regression models

gij(t)=f(Xi,Yj,t)+ϑξij(t),
dij=h(Xi,Yj)+σijεij,

where (Xi,Yj) are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

MSC 2010: 35K05; 47A52; 62G08

Funding statement: This research was supported by Vietnam National University-Ho Chi Minh City (VNU-HCM) under the Grant number B2017-18-03.

Acknowledgements

We would like to thank two anonymous referees, an Associate Editor, whose constructive comments helped to improve the presentation of the paper.

References

[1] P. Alquier, E. Gautier and G. Stoltz, Inverse Problems and High-Dimensional Estimation, Springer, Berlin, 2011. 10.1007/978-3-642-19989-9Search in Google Scholar

[2] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972. Search in Google Scholar

[3] J. V. Beck, B. Blackwell and S. C. R. Clair, Inverse Heat Conduction, Ill-Posed Problems, Wiley Interscience, New York, 1985. Search in Google Scholar

[4] N. Bissantz and H. Holzmann, Statistical inference for inverse problems, Inverse Problems 24 (2008), Article ID 034009. 10.1088/0266-5611/24/3/034009Search in Google Scholar

[5] J. Burkardt, Brownian motion simulation: Simulation of Brownian motion in m dimensions, preprint, http://people.sc.fsu.edu/~jburkardt/f_src/brownian_motion_simulation/brownian_motion_simulation.html. Search in Google Scholar

[6] A. S. Carasso, J. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal. 15 (1978), no. 2, 344–367. 10.1137/0715023Search in Google Scholar

[7] L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems 24 (2008), Article ID 034004. 10.1088/0266-5611/24/3/034004Search in Google Scholar

[8] J. Cheng and J. J. Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Problems 24 (2008), no. 6, 1–18. 10.1088/0266-5611/24/6/065012Search in Google Scholar

[9] M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl. 301 (2005), 419–426. 10.1016/j.jmaa.2004.08.001Search in Google Scholar

[10] R. L. Eubank, Nonparametric Regression and Spline Smoothing, 2nd ed., Marcel Dekker, New York, 1999. 10.1201/9781482273144Search in Google Scholar

[11] D. N. Hao, A mollification method for ill-posed problems, Numer. Math. 68 (1994), 469–506. 10.1007/s002110050073Search in Google Scholar

[12] D. N. Hao and N. T. N. Oanh, Determination of the initial condition in parabolic equations from integral observations, Inverse Probl. Sci. Eng. (2016), 10.1080/17415977.2016.1229778. 10.1080/17415977.2016.1229778Search in Google Scholar

[13] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, Berlin, 1996. 10.1007/978-1-4612-5338-9Search in Google Scholar

[14] B. Mair and F. H. Ruymgaart, Statistical estimation in Hilbert scale, SIAM J. Appl. Math. 56 (1996), 1424–1444. 10.1137/S0036139994264476Search in Google Scholar

[15] G. Marsaglia and W. W. Tsang, The Ziggurat method for generating random variables, J. Stat. Softw. 5 (2000), 10.18637/jss.v005.i08. 10.18637/jss.v005.i08Search in Google Scholar

[16] P. T. Nam, An approximate solution for nonlinear backward parabolic equations, J. Math. Anal. Appl. 367 (2010), no. 2, 337–349. 10.1016/j.jmaa.2010.01.020Search in Google Scholar

[17] P. T. Nam, D. D. Trong and N. H. Tuan, The truncation method for a two-dimensional nonhomogeneous backward heat problem, Appl. Math. Comput. 216 (2010), 3423–3432. 10.1016/j.amc.2010.03.038Search in Google Scholar

[18] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, 1975. 10.1137/1.9781611970463Search in Google Scholar

[19] A. Qian and J. Mao, Quasi-reversibility regularization method for solving a backward heat conduction problem, Amer. J. Comput. Math. 1 (2011), no. 3, 159–162. 10.4236/ajcm.2011.13018Search in Google Scholar

[20] H.-H. Qin and T. Wei, Some filter regularization methods for a backward heat conduction problem, Appl. Math. Comput. 217 (2011), no. 24, 10317–10327. 10.1016/j.amc.2011.05.038Search in Google Scholar

[21] M. Renardy, W. J. Hursa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Wiley, New York, 1987. Search in Google Scholar

[22] R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, Trends in the Theory and Practice of Nonlinear Analysis (Arlington 1984), North-Holland, Amsterdam (1985), 421–425. 10.1016/S0304-0208(08)72739-7Search in Google Scholar

[23] T. H. Skaggs and Z. J. Kabala, Recovering the history of a groundwater contaminant plume: Method of quasi-reversibility, Water Resources Res. 31 (1995), no. 11, 2669–2673. 10.1029/95WR02383Search in Google Scholar

[24] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, 1977. Search in Google Scholar

[25] D. D. Trong, N. H. Tuan and P. H. Quan, A quasi-boundary value method for regularizing nonlinear ill-posed problems, Electron. J. Differential Equations 109 (2009), 1–16. Search in Google Scholar

[26] D. D. Trong, N. H. Tuan and P. H. Quan, A new version of quasi-boundary value method for a 1-D nonlinear ill-posed heat problem, J. Inverse Ill-Posed Probl. 17 (2010), no. 9, 913–932. 10.1515/JIIP.2009.053Search in Google Scholar

[27] A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer, New York, 2009. 10.1007/b13794Search in Google Scholar

Received: 2016-6-17
Revised: 2017-2-14
Accepted: 2017-2-26
Published Online: 2017-5-5
Published in Print: 2018-2-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 8.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2016-0038/html?lang=en
Scroll to top button