关于径向Calderón问题中的非凸性问题
On the non-convexity issue in the radial Calderón problem
AI总结：
本文研究了径向Calderón问题中的非凸性问题，证明了在特定条件下不存在虚假极值点，并探讨了凸化方法在重建中的表现。
摘要：
经典的Calderón问题研究方法是通过求解非线性最小二乘问题来估计未知的导电率。这种方法导致了一个非凸优化问题，通常被认为充满了劣质局部极小值。我们重新审视了这种情况，针对分段常数径向导电率的情况，证明了在两个标量未知数且无测量噪声的情况下，不存在虚假的极值点。我们还在此一般情况下提供了一个部分证明，该结果在满足可数值验证的假设条件下成立。最后，我们调查了最近提出的一种基于凸化的做法是否能提供更好的重建效果。首次，我们提出了一种实际实施的方法，并表明它在某些最优化求解器面前表现不佳，这些求解器不仅更快，而且需要的测量更少。
英文摘要：
A classical approach to the Calderón problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.
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