Topological sensitivity method for reconstruction of the spatial component in the source term of a time-fractional diffusion equation

Abstract

In the paper, an inverse source problem for a time-fractional diffusion equation is formulated and proved. The topological sensitivity analysis method is presented. The considered problem is formulated as a topology optimization one. The proposed process leads to a non-iterative reconstruction algorithm can be applied for large class of cost functional. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. Several numerical simulations are presented to show the efficiency and the accuracy of the algorithm.

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Apprendix A

Apprendix A

In this “Appendix”, we provide the proof of Theorem 3.5, the Propositions 3.9 and 4.6, respectively.

Proof of theorem 3.5

To start, one can refer to [24, 40] and choose \(G \in C^{2}(0,T; H^{2}(\Omega ))\) such that \(\partial _{{\mathcal {A}}} G=g\) on \(\partial \Omega \times (0,T)\). By setting \(w= u-G\), w is a solution of

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _{0^{+}}^{\alpha }{w}+ {\mathcal {A}}(w)= H &{}\quad \text{ in } &{}\Omega \times (0,T) \\ \\ \partial _{{\mathcal {A}}} w=0 &{}\quad \text{ on } &{} \partial \Omega \times (0,T) \\ \\ w =a(x) &{}\quad \text{ in } &{}\quad \Omega \times \left\{ 0\right\} \\ \\ \end{array} \right. \end{aligned}$$
(6.1)

where \(H = f+ \partial _{0^{+}}^{\alpha }{G}+ {\mathcal {A}}(G)\) and \(a(x) =-G(x, 0)\). The study of the existence and uniqueness of the solution of (1.1)–(1.3) is reduced to the existence and uniqueness of problem (6.1).

Now, let split problem (6.1) into the following two problems by taking \(w = w_1 +w_2\), where \(w_1\) is the solution of

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _{0^{+}}^{\alpha }{w_1}+ {\mathcal {A}}(w_1)= H &{}\quad \text{ in } &{}\Omega \times (0,T) \\ \\ \partial _{{\mathcal {A}}} w_1=0 &{}\quad \text{ on } &{} \partial \Omega \times (0,T) \\ \\ w_1 =0 &{}\quad \text{ in } &{}\quad \Omega \times \left\{ 0\right\} \\ \\ \end{array} \right. \end{aligned}$$
(6.2)

and \(w_2\) is solution to

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _{0^{+}}^{\alpha }{w_1}+ {\mathcal {A}}(w_1)= 0 &{}\quad \text{ in } &{}\Omega \times (0,T) \\ \\ \partial _{{\mathcal {A}}} w_1=0 &{}\quad \text{ on } &{} \partial \Omega \times (0,T) \\ \\ w_1 =a(x) &{}\quad \text{ in } &{}\quad \Omega \times \left\{ 0\right\} \\ \\ \end{array} \right. \end{aligned}$$
(6.3)
  1. 1.

    From Dongling Wang et al.(Lemma2.2 [8]) and M.Yamamoto et al.(Lemma 2.4 [20]), the problem (6.2) admits a unique weak solution \(w_1 \in { }_{0} H^{\alpha }\left( 0, T ; L^{2}(\Omega )\right) \cap C\left( [0, T] ; L^{2}(\Omega )\right) \cap L^{2}\left( 0, T ; H^{2}(\Omega )\right) \). One can find the definition of the fractional space \(_{0} H^{\alpha }\left( 0, T\right) \) in [8].

  2. 2.

    For the problem (6.3), one can consult [20], in Lemma 2.1 and Remark 2.2, and check that \(w_2 \in C\left( 0, T ; H^{2}(\Omega )\right) \)

It follows that \(w \in { }_{0} H^{\alpha }\left( 0, T ; L^{2}(\Omega )\right) \cap C\left( [0, T] ; L^{2}(\Omega )\right) \cap L^{2}\left( 0, T ; H^{2}(\Omega )\right) \cap C\left( 0, T ; H^{2}(\Omega )\right) \). Which implies that the problem (1.1)–(1.3) admits a unique solution \(u \in {\mathcal {H}}.\)

Proof of proposition 3.9

Using the weak formulation associated to the previous system (3.10), where the test function is \(u_{F}\). For almost every \(t \in (0,T)\), we get

$$\begin{aligned}&\displaystyle \int _{\Omega } \partial _{0^{+}}^{\alpha }{u_{F}}(x,t)\;u_{F}(x,t)\;dx + \int _{\Omega } p(x)\; \left| \nabla u_{F}(x,t)\right| ^{2}\;dx + \int _{\Omega } q(x)\; \left| u_{F}(x,t)\right| ^{2}\;dx\\&\quad = \displaystyle \int _{\Omega } F(x,t)\;u_{F}(x,t)\;dx. \end{aligned}$$

Now, by the lemma 3.4 and using the fact that \(p \in {\mathcal {C}}^{1}({\overline{\Omega }}),\; q \in {\mathcal {C}}({\overline{\Omega }})\), then there exists \(c_{1} > 0\) and \(c_{2} > 0\), such that

$$\begin{aligned} p(x)> c_{1}> 0 \quad \text{ and }\quad q(x)> c_{2} > 0,\quad \forall x \in \Omega , \end{aligned}$$

we get

$$\begin{aligned} \displaystyle \int _{\Omega } \partial _{0^{+}}^{\alpha }{u_{F}^{2}}(x,t)\;dx+ \left\| u_{F}(.,t)\right\| _{H^{1}(\Omega )}^{2} \le \displaystyle c \int _{\Omega } F(x,t)\;u_{F}(x,t)\;dx, \end{aligned}$$
(6.4)

By the inequality (3.1) and using the fact that \(u_{F}(.,0)\) in \(\Omega \), we get

$$\begin{aligned} \displaystyle \partial _{0^{+}}^{\alpha }{u_{F}}(.,t) = D_{0^{+}}^{\alpha }u_{F}(.,t), \end{aligned}$$

and using the fact that \(H^1(\Omega ) \hookrightarrow L^{q}(\Omega )\) for \(q \in ]2, 6[\) when \(d=2, 3\), and let denote by p the conjuguate of q, i.e. \(1/q +1/p =1\), we obtain

$$\begin{aligned} \displaystyle \frac{d}{dt}(I_{0^{+}}^{1-\alpha }\left\| u_{F}(.,t)\right\| _{2}^{2}) + \left\| u_{F}(.,t)\right\| _{H^{1}(\Omega )}^{2} \le \displaystyle c \left\| u_{F}(.,t)\right\| _{q} \left\| F(.,t)\right\| _{p}. \end{aligned}$$

by integrating this relation with respect to t from 0 to \(\tau \), we obtain for almost every \(\tau \in (0,T)\)

$$\begin{aligned} \displaystyle I_{0^{+}}^{1-\alpha }\left\| u_{F}(.,\tau )\right\| _{2}^{2} + \int _{0}^{\tau } \left\| u_{F}(.,t)\right\| _{H^{1}(\Omega )}^{2}dt \le \displaystyle c \int _{0}^{\tau }\left\| F(.,t)\right\| _{p} \left\| u_{F}(.,t)\right\| _{q} dt. \end{aligned}$$
(6.5)

In particular,

$$\begin{aligned}&\displaystyle \sup _{t \in (0,T)} I_{0^{+}}^{1-\alpha }\left\| u_{F}(.,t)\right\| _{2}^{2} + \int _{0}^{T} \left\| u_{F}(.,t)\right\| _{H^{1}(\Omega )}^{2}dt \nonumber \\&\quad \le \displaystyle c \left\| F(.,t)\right\| _{L^{2}(0,T;L^{p}(\Omega ))} \left\| u_{F}(.,t)\right\| _{L^{2}(0,T;L^{q}(\Omega ))}. \end{aligned}$$
(6.6)

Finally, we use the continuous embedding of the space \(H^{1}(\Omega )\) in \(L^{q}(\Omega )\) (see [6, 12]), we get the desired result.

6.1 Proof of proposition 4.6

One can show that J is differentiable on \({\mathcal {X}}\) and we have

$$\begin{aligned} DJ(u_{f}) w = 2 \displaystyle \int _{0}^{T}\int _{\Omega } (\nabla u-\nabla {\mathcal {U}}) \;\nabla w\; dx\;dt, \; \; \forall w \in {\mathcal {X}}, \end{aligned}$$

Using Proposition 4.1 and Remark 4.5, then the adjoint state \(v_{f} \in {\mathcal {X}}_{T}\) is solution of

$$\begin{aligned} A(w,v_{f})= - \displaystyle 2 \displaystyle \int _{0}^{T}\int _{\Omega } {\mathcal {K}}( u- {\mathcal {U}}) \;{\mathcal {K}}( w)\; dx\;dt, \; \; \forall w \in {\mathcal {X}}. \end{aligned}$$

Using the definition of \(\phi \), it’s easily to check that

$$\begin{aligned}&\Phi (f + \delta f) - \Phi (f) - \Phi ^{'}(f) (\delta f)= \displaystyle DJ(u_{f}) (u_{f+ \delta f} - u_{f})+ \left\| {\mathcal {K}} (u_{f+ \delta f} - u_{f})\right\| _{2}^{2}\\&\quad +\int _{0}^{T}{\int _{\Omega }{v_{f}\;\delta f\; dx\; dt}}. \end{aligned}$$

From the two problem (3.8) and (4.3), we get

$$\begin{aligned} \Phi (f + \delta f) - \Phi (f) - \Phi ^{'}(f) (\delta f) = \left\| {\mathcal {K}} (u_{f+ \delta f} - u_{f})\right\| _{2}^{2}. \end{aligned}$$
(6.7)

To estimate the term \(\left\| {\mathcal {K}} (u_{f+ \delta f} - u_{f})\right\| _{2}^{2}\). We apply proposition 3.9 and we take a perturbation \(\delta f\) having the same form as \(\delta f_{z,\varepsilon }\) [see (4.1)], we get

$$\begin{aligned} \left\| {\mathcal {K}} (u_{f+ \delta f_{z,\varepsilon }} - u_{f})\right\| _{2}^{2} \le c \displaystyle \left\| \delta f_{z,\varepsilon }\right\| _{L^{2}(0,T;L^{5/4}(\Omega ))}^{2}=c \displaystyle \varepsilon ^{\frac{8 d}{5}}. \end{aligned}$$

It follows,

$$\begin{aligned} \left\| {\mathcal {K}} (u_{f+ \delta f_{z,\varepsilon }} - u_{f})\right\| _{2}^{2} = o(\varepsilon ^{d}). \end{aligned}$$
(6.8)

Using (6.7) and (6.8), we deduce

$$\begin{aligned} \Phi (f + \delta f_{z,\varepsilon }) - \Phi (f) - \Phi ^{'}(f) (\delta f_{z,\varepsilon }) = o(\varepsilon ^{d}). \end{aligned}$$

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Ben Salah, M. Topological sensitivity method for reconstruction of the spatial component in the source term of a time-fractional diffusion equation. Ricerche mat (2021). https://doi.org/10.1007/s11587-020-00553-1

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Keywords

  • Time-fractional diffusion equation
  • Inverse source problem
  • Sensitivity analysis

Mathematics Subject Classification

  • 35R11
  • 35R30
  • 74P15