从最小内部数据重构电导率的混合问题的收敛算法
A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data
AI总结：
针对仅已知电流密度幅值的混合问题，提出交替分裂Bregman算法求解加权最小梯度问题，并证明其收敛性，同时给出从幅值和边界电压恢复全矢量场的新方法。
摘要：
我们考虑混合问题，即从通过MRI测量获得的内部电流密度成像数据中重构物体$Ω$的各向同性电导率。我们仅需知道由边界$\partialΩ$上给定电压$f$产生的一个电流的幅值$|J|$。如前所述，对应的电压势$u$是加权最小梯度问题的极小化子：\[u=\hbox{argmin} \{\int_Ωa(x)| abla u|: u \in H^{1}(Ω), \ \ u|_{\partial Ω}=f\},\] 其中$a(x)= |J(x)|$。本文针对此类最小梯度问题提出了一种交替分裂Bregman算法，适用于$a\in L^2(Ω)$非负且$f\in H^{1/2}(\partial Ω)$。我们通过重点关注对偶问题给出了详细的收敛性证明，这自然引出了交替分裂Bregman算法。对偶问题还提供了一种从电流幅值和边界电压恢复全矢量场$J$的新方法。最后，我们给出了若干数值实验，展示了所提算法的收敛行为。
英文摘要：
We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $Ω$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partialΩ$. As previously shown, the corresponding voltage potential u in $Ω$ is a minimizer of the weighted least gradient problem \[u=\hbox{argmin} \{\int_Ωa(x)|\nabla u|: u \in H^{1}(Ω), \ \ u|_{\partial Ω}=f\},\] with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(Ω)$ non-negative and $f\in H^{1/2}(\partial Ω)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm.
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