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# Determination of the initial density in nonlocal diffusion from final time measurements

• This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.

Mathematics Subject Classification: 35R30, 35L20, 65M32, 65F110, 65F22.

 Citation: • • Figure 1.  Exact (blue) and approximate solutions, computed with different values of the fractional Laplacian order $s$ in $\{0.1,\,0.2,\, 0.4,\, 0.8\}$

Figure 2.  The point cloud $\Big(\log(|Err(h_j)|),\,\log(h_j)\Big)_{1\leq j\leq 6}$ (blue dots) and their associated regression lines (red line) for each $s\in \{0.1,\,0.2,\,0.4,\, 0.8\}$

Figure 3.  Exact (blue dashed line) and reconstructed (red line) initial condition

Figure 4.  Exact (blue dashed line) and Reconstructed (red line) initial conditions

Figure 5.  Exact (blue dashed line) and obtained (red line) results

Figure 6.  Variation of $\gamma \mapsto Er(\gamma)$

Figure 7.  Variation of $\gamma \mapsto Er(\gamma)$

Figure 8.  Variation of $\gamma \mapsto Er(\gamma)$

Figure 9.  Variations of the $L^2$-error function when the fractional Laplacian order $s$ varies in the interval $(0,\,1)$

Figure 10.  Exact (black) and reconstructed initial conditions from noisy measured data, with different level of noise $\rho = 1\%$(blue), $\rho = 5\%$(green) and $\rho = 10\%$(red)

Figure 11.  Reconstruction results

Table 1.  Values of the error function $Err$ for each $s$ in $\{0.1,\,0.2,\,0.4,\, 0.8\}$ relative to the mesh step size variation

 Step size $h_j$ $Err(h_j),\, s=0.1$ $Err(h_j),\, s=0.2$ $Err(h_j),\, s=0.4$ $Err(h_j),\, s=0.8$ $h_1=h_0$ $3.61e-1$ $1.56e-1$ $1.24e-1$ $8.69e-2$ $h_2={h_0}/{2}$ $2.75e-1$ $1.13e-1$ $6.15e-2$ $4.41e-2$ $h_3={h_0}/{4}$ $1.70e-1$ $6.76e-2$ $3.11e-2$ $2.14e-2$ $h_4={h_0}/{8}$ $1.33e-1$ $4.28e-2$ $1.85e-2$ $1.13e-2$ $h_5={h_0}/{16}$ $9.59e-2$ $2.44e-2$ $9.51e-3$ $5.02e-3$ $h_6={h_0}/{32}$ $6.74e-2$ $1.36e-2$ $5.01e-3$ $2.49e-3$

Table 2.  Theoretical and numerical convergence rates values

 Fractional Laplacian order $s=0.1$ $s=0.2$ $s=0.4$ $s=0.8$ Expected rate $\eta(s)$ $0.4$ $0.7$ $0.9$ $1$ Obtained rate $\eta_{ap}(s)$ $0.4862$ $0.7124$ $0.9148$ $1.0242$

Table 3.  Values of the relative error function $err$ with respect to the mesh step $h$

 Mesh step size $h_j$ Example 1 Example 2 Example 3 $h=h_0$ $9.98e-3$ $1.03e-2$ $2.41e-2$ $h=h_0/2$ $7.61e-3$ $9.02e-3$ $8.32e-3$ $h=h_0/4$ $5.92e-3$ $8.23e-3$ $7.01e-3$ $h=h_0/8$ $5.01e-3$ $6.98e-3$ $5.71e-3$ $h=h_0/16$ $4.91e-3$ $6.12e-3$ $4.44e-3$

Table 4.  Values of the error function

 The parameter $\gamma$ The error $Er(\gamma)$ $10^{-1}$ $3.33e-1$ $10^{-2}$ $1.60e-1$ $10^{-3}$ $8.48e-2$ $10^{-4}$ $4.56e-2$ $10^{-5}$ $1.67e-2$ $10^{-6}$ $1.70e-3$ $10^{-7}$ $3.53e-4$ $10^{-8}$ $2.01e-4$ $0$ $2.31e-5$

Table 5.  Values of the error function

 The parameter $\gamma$ The error $Er(\gamma)$ $10^{-1}$ $5.10e-1$ $10^{-2}$ $1.50e-1$ $10^{-3}$ $8.31e-2$ $10^{-4}$ $5.95e-2$ $10^{-5}$ $4.66e-2$ $10^{-6}$ $4.53e-2$ $10^{-7}$ $4.48e-2$ $10^{-8}$ $4.51e-2$ $0$ $4.51e-2$

Table 6.  Values of the error function

 The parameter $\gamma$ The error $Er(\gamma)$ $10^{-1}$ $5.70e-1$ $10^{-2}$ $3.26e-1$ $10^{-3}$ $2.38e-1$ $10^{-4}$ $1.95e-1$ $10^{-5}$ $1.45e-1$ $10^{-6}$ $1.16e-1$ $10^{-7}$ $9.33e-2$ $10^{-8}$ $1.37e-1$ $0$ $1.73e-1$

Table 7.  Values of the $L^2-$error function related to the fractional Laplacian orders $s = 0.1,$ $s = 0.3$, $s = 0.6$ and $s = 0.9$

 Fractional order $s$ $0.1$ $0.3$ $0.6$ $0.9$ $L^2-$error norm $0.0371$ $0.0506$ $0.0736$ $0.0968$
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