迭代总变分正则化：非线性反问题的新方法
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本文探讨了一种新的迭代总变分正则化方法，针对的是非线性反问题。在《计算与应用数学》（Journal of Computational and Applied Mathematics）2016年第298期的40-52页中，作者李丽和鲍涵来自哈尔滨工业大学数学系，他们针对传统上已知的如Landweber迭代和总变分正则化方法可能存在的局限性，提出了创新性的解决策略。
总变分正则化因其在识别分段光滑参数和图像处理中的优越性能而广受欢迎。然而，为了进一步提升反问题求解的精度和效率，研究者们探索了结合了总变分正则化和同伦扰动技术的新方法。这种方法的关键在于利用布格曼距离（Bregman distance），这是一种度量两个函数值之间差异的距离，它在优化和机器学习中扮演着重要角色。
新方法构建了一个嵌套迭代框架，通过逐步逼近反问题的最优解，有效地控制了参数估计的误差。通过理论证明，这种方法在每次迭代中都能展现出比传统方法更显著的收敛性和稳定性。同伦扰动技术在此起到的作用可能是通过对原问题进行微小的变形或调整，使得解的搜索路径更加平滑，从而避免了局部最优的问题。
这篇论文的主要贡献在于提出了一种新颖的迭代策略，它将总变分正则化的优点与同伦扰动技术相结合，为解决非线性反问题提供了一种更高效且准确的解决方案。这对于诸如图像恢复、信号处理和科学计算等领域具有实际应用价值，特别是在处理那些本质上是不完全确定或者有噪声干扰的数据时。通过这种方式，研究者们可以期待在提高反问题求解的性能的同时，减少对初始数据的依赖，提升结果的可靠性。
42 L. Li, B. Han / Journal of Computational and Applied Mathematics 298 (2016) 40–52
Set p = 1 in (1.10), we get the approximate solution of (1.7)
u = lim
p→1
ν = u
0
+ u
1
+ u
2
+ · · ·
= u
0
− F
′
(u
0
)
∗
(F(u
0
) − y
δ
) + [I − F
′
(u
0
)
∗
F
′
(u
0
)][−F
′
(u
0
)
∗
(F(u
0
) − y
δ
)]. (1.14)
We consider the one-order approximation of this formulation (1.14), and get the classical Landweber iteration from a given
initial value u
0
[3]:
u
δ
n+1
= u
δ
n
− F
′
(u
δ
n
)
∗
(F(u
δ
n
) − y
δ
).
In the same way, a two-order approximation iterative algorithm was obtained from a given u
0
[9]:
u
δ
n+1
= u
δ
n
− [2I − F
′
(u
δ
n
)
∗
F
′
(u
δ
n
)]F
′
(u
δ
n
)
∗
[F(u
δ
n
) − y
δ
] (1.15)
and proved to be a stable regularization method according to the discrepancy principle by Li Cao and Bo Han in 2011 [10].
Compared with Landweber iteration, only half-time is needed with the same accuracy.
Subsequently, Jing Wang et al. applied this homotopy perturbation inversion method to investigate the EIT image
reconstruction problem for the first time in 2013, and testify the feasibility and effectiveness of the method in the cases
of different locations, sizes and numbers of the inclusions, as well as the robustness of the method to the noisy data [11].
However, these methods may not be effective for dealing with the discontinuous solution to the problems of image
processing and parameter identification, for example, the solution to the inverse problems consisted of large constant
regions or sharp edges. Instead, total variational regularization is one of the most effective methods for solving this kind
of problem. Total variation (TV), which was introduced by Rudin in 1987 [12], comes from real analysis. Total variation
regularization can achieve better reconstructions superior to those results obtained from the classical iteration methods.
Therefore, we can get the regularized solution by minimizing the following total variational penalized least squares
functional
1
2
∥F(u) − y
δ
∥
2
H
+ αJ(u),
where the total variational regularization functional J : X → R ∪ {+∞} is convex, and is a seminorm
J(u) =| u |
BV (Ω)
= sup
g∈C
∞
0
(Ω,R
n
),∥g∥
∞
≤1
Ω
udivg.
Here X = BV (Ω) is the space of bounded variational function. With the similar idea of the Runge–Kutta type Landweber
iteration [7], we proposed a Runge–Kutta type total variation regularization for nonlinear ill-posed problems in 2014 [13]
u
δ
k+1
= argmin
u∈D (F)
F
′
u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)
∗
F(u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)) − y
δ
,
u −
u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)
+ α
k
D
J
ξ
δ
k
u, u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)
,
ξ
δ
k+1
= ξ
δ
k
− α
−1
k
F
′
u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)
∗
F
u
δ
k
−
1
2
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
)
− y
δ
.
We considered Bregman distance D
J
ξ
(u,
¯
u) as a regularization functional, and analyzed the convergence of the proposed
method under an appropriate stopping rule. Besides, we verified this method to be more suitable for problems with
discontinuous solutions by some numerical examples.
In view of the priority of the total variational regularization for discontinuous solution, we construct a new total variation
regularization for nonlinear inverse problems based on two-order approximation iterative algorithm (1.15) and Bregman
distance in this paper,
u
δ
k+1
= arg min
u∈BV (Ω)
[2I − F
′
(u
δ
k
)
∗
F
′
(u
δ
k
)]F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
), u − u
δ
k
+ α
k
D
J
ξ
k
(u, u
δ
k
)
, (a)
ξ
δ
k+1
= ξ
δ
k
− α
−1
k
[2I − F
′
(u
δ
k
)
∗
F
′
(u
δ
k
)]F
′
(u
δ
k
)
∗
[F(u
δ
k
) − y
δ
] (b)
(1.16)
where D
J
ξ
(u,
¯
u) denotes the Bregman distance for J of u,
¯
u ∈ X, which is different from the distance in the usual norm, and
can be regarded as a generalization of the mean-square distance. In the next section, we will analyze the convergence and
stability of the new iteration (1.16)(a)–(b) (be called New TV regularization method later in this paper), and compare the
inversion results of the new method with Runge–Kutta type iteration (1.5) and Landweber type total variational method
(which is simply known as Landweber type TV), respectively
u
δ
k+1
= arg min
u∈BV (Ω)
F
′
(u
δ
k
)
∗
(F(u
δ
k
) − y
δ
), u − u
δ
k
+ α
k
D
J
ξ
k
(u, u
δ
k
)
, (a)
ξ
δ
k+1
= ξ
δ
k
− α
−1
k
F
′
(u
δ
k
)
∗
[F(u
δ
k
) − y
δ
] (b)
(1.17)
in the section of numerical examples.
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