IPI
A scaled gradient method for digital tomographic image reconstruction
Jianjun Zhang Yunyi Hu James G. Nagy
Inverse Problems & Imaging 2018, 12(1): 239-259 doi: 10.3934/ipi.2018010

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. In this paper, we develop an approach to solve the nonlinear image reconstruction problem by incorporating total variation (TV) regularization. The corresponding optimization problem is then solved by using a scaled gradient descent method. The proposed algorithm is based on KKT conditions and Nesterov's acceleration strategy. Experimental results on reconstructed polyenergetic image data illustrate the effectiveness of this proposed approach.

keywords: Tomosynthesis gradient descent ill-posed problems beam hardening artifacts total variation
DCDS-S
Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity
Hua Jin Wenbin Liu Jianjun Zhang
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 533-545 doi: 10.3934/dcdss.2018029
In this paper, we are concerned with the following fractional Kirchhoff equation
$\begin{align*}\left\{\begin{array}{ll}\left(a+b∈t_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2\right)(-Δ)^su=\lambda u+μ|u|^{q-2}u+|u|^{2_{s}^*-2}u&\ \ \mbox{in}\ \ Ω, \\ u=0&\ \ \mbox{in}\ \mathbb R^N\backslashΩ, \end{array}\right.\end{align*}$
where
$N>2s$
,
$a, b, \lambda, μ>0$
,
$s∈(0, 1)$
and
$Ω$
is a bounded open domain with continuous boundary. Here
$(-Δ)^s$
is the fractional Laplacian operator. For
$2<q≤q\min\{4, 2_s^*\}$
, we prove that if
$b$
is small or
$μ$
is large, the problem above admits multiple solutions by virtue of a linking theorem due to G. Cerami, D. Fortunato and M. Struwe [7, Theorem 2.5].
keywords: Fractional Kirchhoff equation multiple solutions critical nonlinearity
CPAA
Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
Minbo Yang Jianjun Zhang Yimin Zhang
Communications on Pure & Applied Analysis 2017, 16(2): 493-512 doi: 10.3934/cpaa.2017025

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential V. Moreover, the monotonicity of f(s)=s and the so-called Ambrosetti-Rabinowitz condition are not required.

keywords: Multi-peak solutions Choquard equation semiclassical states penalization arguments Berestycki-Lions conditions

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