Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm. (English) Zbl 1530.65109

Fu, Jun-Liang; Liu, Jijun

Inverse Probl. 40, No. 1, Article ID 015010, 30 p. (2024).

Summary: We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.

© 2023 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.

MSC:

65M32	Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06	Finite difference methods for boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65Y05	Parallel numerical computation
35R25	Ill-posed problems for PDEs
35R30	Inverse problems for PDEs

Keywords:

pseudoparabolic equation; backward problem; ill-posedness; quasi-boundary value method; error estimate; parallel-in-time; numerics

Software:

ParaDiag

References:

[1]	Showalter, R. E., The final value problem for evolution equations, J. Math. Anal. Appl., 47, 563-72 (1974) · Zbl 0296.34059 · doi:10.1016/0022-247X(74)90008-0
[2]	Clark, G. W.; Oppenheimer, S. F., Quasireversibility methods for non-well-posed problems, Electron. J. Differ. Equ., 1994, 1-9 (1994) · Zbl 0811.35157
[3]	Denche, M.; Bessila, K., A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl., 301, 419-26 (2005) · Zbl 1084.34536 · doi:10.1016/j.jmaa.2004.08.001
[4]	Seidman, T. I., Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33, 162-70 (1996) · Zbl 0851.65066 · doi:10.1137/0733010
[5]	Ames, K. A.; Epperson, J. F., A kernel-based method for the approximate solution of backward parabolic problems, SIAM J. Numer. Anal., 34, 1357-90 (1997) · Zbl 0889.65100 · doi:10.1137/S0036142994276785
[6]	Muniz, W.; Ramos, F.; de Campos Velho, H., Entropy-and Tikhonov-based regularization techniques applied to the backwards heat equation, Comput. Math. Appl., 40, 1071-84 (2000) · Zbl 1016.80016 · doi:10.1016/S0898-1221(00)85017-8
[7]	Duda, P., Solution of inverse heat conduction problem using the Tikhonov regularization method, J. Therm. Sci., 26, 60-65 (2017) · doi:10.1007/s11630-017-0910-2
[8]	Wang, L.; Liu, J., Total variation regularization for a backward time-fractional diffusion problem, Inverse Problems, 29 (2013) · Zbl 1297.65116 · doi:10.1088/0266-5611/29/11/115013
[9]	Nam, P. T.; Trong, D. D.; Tuan, N. H., The truncation method for a two-dimensional nonhomogeneous backward heat problem, Appl. Math. Comput., 216, 3423-32 (2010) · Zbl 1197.65131 · doi:10.1016/j.amc.2010.03.038
[10]	Cheng, J.; Liu, J., A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Problems, 24 (2008) · Zbl 1157.35120 · doi:10.1088/0266-5611/24/6/065012
[11]	Liu, C-S, Group preserving scheme for backward heat conduction problems, Int. J. Heat Mass Transfer, 47, 2567-76 (2004) · Zbl 1100.80005 · doi:10.1016/j.ijheatmasstransfer.2003.12.019
[12]	Chang, C-W; Liu, C-S; Chang, J-R, A new shooting method for quasi-boundary regularization of multi-dimensional backward heat conduction problems, J. Chin. Inst. Eng., 32, 307-18 (2009) · doi:10.1080/02533839.2009.9671510
[13]	Ku, C-Y; Liu, C-Y; Yeih, W.; Liu, C-S; Fan, C-M, A novel space-time meshless method for solving the backward heat conduction problem, Int. J. Heat Mass Transfer, 130, 109-22 (2019) · doi:10.1016/j.ijheatmasstransfer.2018.10.083
[14]	Liu, J.; Wang, B., Solving the backward heat conduction problem by homotopy analysis method, Appl. Numer. Math., 128, 84-97 (2018) · Zbl 1393.65039 · doi:10.1016/j.apnum.2018.02.002
[15]	Lattès, R.; Lions, J. L., MéThode de Quasi-réVersibilité et Applications (1967), Dunod · Zbl 0159.20803
[16]	Liu, J.; Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89, 1769-88 (2010) · Zbl 1204.35177 · doi:10.1080/00036810903479731
[17]	Gajewski, H.; Zacharias, K., Zur regularisierung einer Klasse nichtkorrekter probleme bei evolutionsgleichungen, J. Math. Anal. Appl., 38, 784-9 (1972) · Zbl 0234.65079 · doi:10.1016/0022-247X(72)90083-2
[18]	Yosida, K., Functional Analysis (Die Grundlehren der Mathematischen Wissenschaften, Band 123) (1965), Academic, Inc., Springer · Zbl 0126.11504
[19]	Showalter, R. E., Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3, 527-43 (1972) · Zbl 0262.34047 · doi:10.1137/0503051
[20]	Showalter, R. E.; Ting, T. W., Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1, 1-26 (1970) · Zbl 0199.42102 · doi:10.1137/0501001
[21]	Maday, Y.; Rønquist, E. M., Parallelization in time through tensor-product space-time solvers, C. R. Acad. Sci., Paris, 346, 113-8 (2008) · Zbl 1133.65066 · doi:10.1016/j.crma.2007.09.012
[22]	Gander, M. J.; Liu, J.; Wu, S. L.; Yue, X.; Zhou, T., Paradiag: parallel-in-time algorithms based on the diagonalization technique (2020)
[23]	Jiang, Y.; Liu, J., Fast parallel-in-time quasi-boundary value methods for backward heat conduction problems, Appl. Numer. Math., 184, 325-39 (2023) · Zbl 1510.65229 · doi:10.1016/j.apnum.2022.10.006
[24]	Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer · Zbl 0516.47023
[25]	Brezis, H., Analyse Fonctionnelle, Théorie et Application (1983), Masson · Zbl 0511.46001
[26]	Boussetila, N.; Rebbani, F., Optimal regularization method for ill-posed Cauchy problems, Electron. J. Differ. Equ., 2006, 1-15 (2006) · Zbl 1170.35554
[27]	Hào, D. N.; Van Duc, N.; Lesnic, D., Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA J. Appl. Math., 75, 291-315 (2010) · Zbl 1194.35501 · doi:10.1093/imamat/hxp026
[28]	Stewart, G. W., On the solution of block Hessenberg systems, Numer. Linear Algebra Appl., 2, 287-96 (1995) · Zbl 0837.65014 · doi:10.1002/nla.1680020309
[29]	Favati, P.; Lotti, G.; Menchi, O., Non-recursive solution of sparse block Hessenberg systems, Numer. Linear Algebra Appl., 11, 391-409 (2004) · Zbl 1164.65335 · doi:10.1002/nla.370
[30]	Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042
[31]	Gander, M. J.; Wu, S-L, Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique, Numer. Math., 143, 489-527 (2019) · Zbl 1472.65083 · doi:10.1007/s00211-019-01060-8
[32]	Liu, J.; Xiao, M., Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework, Inverse Problems, 35 (2019) · Zbl 1425.35235 · doi:10.1088/1361-6420/ab401e