An extension of the landweber regularization for a backward time fractional wave problem

Bin Fan; Mejdi Azaïez; Chuanju Xu

doi:10.3934/dcdss.2020409

Article Contents

2021, Volume 14, Issue 8: 2893-2916. Doi: 10.3934/dcdss.2020409

This issue Previous Article Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion Next Article Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space

An extension of the landweber regularization for a backward time fractional wave problem

1.
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2.
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical, Modeling and High Performance Scientific Computing, Xiamen University, Xiamen 361005, China
3.
Bordeaux INP, Laboratoire I2M UMR 5295, Pessac 33607, France

^*Corresponding author: Chuanju Xu
^*Corresponding author: Chuanju Xu

Received: February 2020

Revised: May 2020

Early access: July 2020

Published: August 2021

This research is partially supported by NSFC grant 11971408 and NSFC/ANR joint program 51661135011/ANR-16-CE40-0026-01. The first author gratefully acknowledges the financial support from China Scholarship Council and the laboratory I2M UMR 5295 for hosting. The second author has received financial support from the French State in the frame of the "Investments for the future" Programme Idex Bordeaux, reference ANR-10-IDEX-03-02.

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Abstract

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

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Mathematics Subject Classification: Primary: 65M32, 35R11; Secondary: 47A52, 35L05.

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Figure 1. The regularized solutions with $ \alpha = 1.1$ for Examples 5.1– 5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR

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Figure 2. Regularized solutions with $\alpha = 1.6 $ for Examples 5.1–5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR

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Figure 3. (ExFLR) Regularization parameter $ m $ and relative error $ e_r $ as functions of $ \nu $, vary $ \nu $ in the range $ \{0, 0.1, \cdots, 1\} $. (a) and (d) for Example 5.1 with $ \varepsilon = 10\% $; (b) and (e) for Example 5.2 with $ \varepsilon = 0.1\% $; (c) and (f) for Example 5.3 with $ \varepsilon = 0.1\% $

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Figure 4. The computed (regularized) solutions for Example 5.1-3 (from left to right) with $ \alpha = 1.6 $: smoothness comparison of the computed solutions for three different values of $ \nu $. (a)-(c) for ExFLR; (d)-(f) for ImFLR

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Figure 5. The computed initial conditions by ImFLR and absolute errors for Example 5.4. (a)—(c) for the regularized solutions $f_4^{m, \delta}$; (d)—(f) for the absolute errors $|f_4^{m, \delta}-f_4|$

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Table 1. Examples 5.1–5.3. Relative errors and regularization parameter versus relative noise levels

		$\alpha = 1.1 $				$\alpha = 1.6 $
		ExFLR (21)		ImFLR (22)		ExFLR (21)		ImFLR (22)
$ f $	$ \varepsilon $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $
$ f_1 $	$ 1\% $	7324	0.0103	7328	0.0103	147	0.0188	152	0.0187
	$ 5\% $	4775	0.0506	4778	0.0506	96	0.0634	99	0.0634
	$ 10\% $	3677	0.1003	3680	0.1002	73	0.1209	76	0.1188
$ f_2 $	$ 0.1\% $	14913	0.0323	14915	0.0323	510	0.0647	511	0.0641
	$ 0.5\% $	3846	0.0674	3846	0.0674	8	0.1183	13	0.1187
	$ 1\% $	701	0.1068	706	0.1067	6	0.1211	11	0.1200
$ f_3 $	$ 0.1\% $	232223	0.2947	232224	0.2947	69253	0.2728	69254	0.2728
	$ 0.5\% $	32815	0.3827	32817	0.3827	7456	0.3753	7458	0.3752
	$ 1\% $	20594	0.4186	20595	0.4186	4695	0.4032	4697	0.4032

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