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

• *Corresponding author: Chuanju Xu

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

• 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.

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

 Citation: • • 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

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

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 $\varepsilon = 0.1\%$

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

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 $|f_4^{m, \delta}-f_4|$

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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