makes the algorithms disconverge. Here, we replaceF by the Sobolev-gradientFsthat is the solution of the equation
−div (σ∇Fs) =F in Ω, Fs= 0 on∂Ω.
For more detail about Sobolev-gradient, we refer to [56] and the references therein. Note that by running the algorithms, we have seen that the technique of cutting the gradient used for the diffusion identification problem does not work for this problem.
We emphasize that the conditions of the convergence of the algorithms are difficult to be shown in this problem. However, the numerical example will show that the most of algorithms still work well. The replacement of the gradient also changes the converegence rates of the algorithms. We will examine the performance of the algorithms in the following example.
We setβ= 10−2, α:= 5.10−3 and assume that σ∗(x1, x2) =
6,(x1, x2)∈B0.3(0.3,0)
1, otherwise .
The exact data (j, g) :=
jN, g
and noisy data jδ, gδ
:=
jN, gδ
are illustrated in Figure4.8. Here, we have
j−jδ
L2(Γ)+g−gδ
L2(Γ)≈0.05.
Figure 4.8: Optimal currentjN, exact Dirichlet datagand noise Dirichlet datagδ withδ= 0.05.
Figure 4.9: Values of 1/sn in the algorithms; Using exact data.
We first consider the performance of the algorithms with the exact data jN, g
. Figure 4.9 shows that the stepsizes in Alg.1 are larger than those in SSQN.I and they are almost equal to s−1.
4.2. Electrical Impedance Tomography
Note that the intervals [s, s] in two algorithms have been chosen differently. Since the stepsizes in Alg.1 are larger than those inSSQN.I, Alg.1 is faster thanSSQN.I. Here, the convergence of two algorithms depend on the choice of the interval [s, s]. We have tried, for example, SSQN.I under setting [s, s] = [2.10−3,2.10] (as for Alg.1) do not converge. Similarly,Alg.1 under settings= 10−3 also disconverges.
In Figure4.10, the decrease ofM SE(σn) shows thatσn in the algorithms tend toσ∗ very slowly in the first iterations and after that they go away. The decreasing rates of the objective functional in the algorithms are very slow, too. Here,Alg.1 is faster thanAlg.2,SSQN.IandSSQN.B. However,Alg.3 is still fastest. These observations show that using the Sobolev-gradient instead of theL2−gradient not only smooth the recovered solutions but also changes the search direction. Thus,Alg.1 now turns to an accelerated version of the gradient method, whileSSQN.B using Sobolev-gradient canceled its advantage.
In electrical impedance tomography, it is very hard to obtain accurate approximations ofσ∗, especially using only one (optimal) current in our setting. With exact data, the values of M SE(σn) in the algorithms are very large and decrease very slowly in the first iterations and then they increase. The values of objective functional decrease slowly as well. From Figure 4.10, it seems that we cannot obtain an approximationσn ofσ∗ such thatM SE(σn)<4.
Figure4.12presents the physical conductivityσ∗ and the recovered solutionsσn in all algorithms. It is easy to see that the algorithms located the inhomogeneous part of the conductivityσ∗ very well, but it is more difficult to recover accurately the values ofσ∗.
Figure 4.10: Values ofD(σn)L2(Ω), M SE(σn) and Θ (σn) in the algorithms; Using exact data.
Now, we consider the case of perturbed data jN, gδ
plotted in Figure 4.8. Similar to the exact data case, the sequencesM SE(σn) decrease very slowly in the first iterations and then they increase.
Their values are still large as well. The objective functional Θ (σn) decreases monotonically, but very
Figure 4.11: Values ofD(σn)L2(Ω), M SE(σn) and Θ (σn) in the algorithms; Using data with 5%
noise.
slowly.
Figure 4.13 illustrates σ∗ and σn in the algorithms using noisy data jN, gδ
. With 5% noise, the algorithms can still recover the conductivity σ∗ very well when they are compared with the case of the exact data.
4.2. Electrical Impedance Tomography
Figure 4.12: 3D-plots and contour plots ofσ∗, σn in the algorithms; Using exact data.
Figure 4.13: 3D-plots and contour plots ofσ∗, σn in the algorithms; Using data with 5% noise.
Conclusions
We have first investigated sparsity regularization for two parameter identification problems. For these problems, sparsity regularization incorporated with the energy functional approach was analyzed.
In diffusion coefficient identification problem, the regularized problem was proven to be well-posed and convergence rates of the method was obtained under the simple source condition. An advantage of the new approach is to work with a convex and weakly lower semicontinuous functional. Therefore, the problem can be solved numerically by the fast algorithms and the proof of the well-posedness is obtained without further conditions. Another advantage is that the source condition of obtaining convergence rates is very simple. We want to emphasize that our source condition is the simplest when it is compared with the others in the least squares approach. We do not need the requirement of smallness (or generalizations of it) in the source condition.
Not as above problem, in electrical impedance tomography, although sparsity regularization incorpo-rated with the energy functional approach are applied, we can not sure about the convexity of the energy functional. In order to obtain the well-posedness of the method, we have required some regu-larity properties of the recovered parameter. For this problem, the convergence rates of the method have obtained as well. However, the source condition is not as simple as that in the previous problem.
Secondly, we have also proposed the iterative methods for the minimization problem arising from sparsity regularization of nonlinear inverse problems.
A gradient-type method has been proven to converge for the non-convex minimization problem under certain conditions. A choice of efficient heuristic step-sizes has been analyzed as well. In the special case when the minimization problem is convex, two accelerated versions have proposed and their convergence have been proved. The decreasing rates of the objective functional in two accelerated algorithms areO(n12) withnbeing the number of iterations. This order of convergence rate is known to be the best for the algorithms based on the first-order schemes.
Other iterative methods considered are the semi-smooth Newton and quasi-Newton methods. We have proposed conditions to obtain the convergence and the convergence rate of two methods. Note that the semi-smooth Newton method is difficult to implement in practice since it relates to the computation of the second derivative, which is very hard in application. To overcome, we replaced the second derivative by its approximations, which leads to the semi-smooth quasi-Newton method. Some conditions for the convergence of the semi-smooth Quasi-Newton method are given. Furthermore, two specific cases of the method have been proposed and their convergence have been proven.
All algorithms have been applied to the two parameter identification problems. The examples have showed that the algorithms work very well. It also pointed out that the conditions for the convergence of the algorithms proposed in the thesis are only sufficient, but they are not necessary. The algorithms have worked well even though some of the conditions are violated.
On the other hand, the numerical examples showed that both parameter identification problems are very ill-posed, specially electrical impedance tomography. The algorithms did not converge when the L2(Ω)−gradient is used (more exactly, the candidate of the L2(Ω)−gradient is used). For using the algorithms, we need to use a prior information of recovered parameters or use the Sobolev-gradient instead of theL2(Ω)−gradient. Note that using the Sobolev-gradient made the recovered parameters smoothed, but it made the algorithms more stable and accelerates the gradient-type method as well.
There are several possible directions for further work. For example, sparsity regularization incorpo-rated with the energy functional approach could be investigated for other problems and all algorithms in the thesis can be generalized to some other types of minimization problems. Especially, some other specific cases of the semismooth quasi-Newton method need further consideration.
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