论文标题
有限的许多未知数的Calderón问题等于凸半趋前优化
The Calderón problem with finitely many unknowns is equivalent to convex semidefinite optimization
论文作者
论文摘要
我们考虑了从相关的neumann-dirichlet-operator知识中确定椭圆部分微分方程中系数函数的逆边界值问题。相对于给定的像素分区，假定未知系数函数是分段常数，并且假定上限和下限是已知的A-Priori。 我们将证明，有限的许多未知数的问题可以作为最小化的问题，因为线性成本功能具有最小化的问题，并具有凸的非线性半际半径约束。我们还证明了嘈杂数据的误差估计，并将结果扩展到有限的许多测量的实际情况，其中系数应从neumann-dirichlet-operator的有限维盖金投影中重建。 我们的结果是基于先前关于Neumann-Dirichlet-operator的Loewner单调性和凸度的作品以及局部电位的技术。它连接了反系数问题的新兴领域和半决赛优化。
We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori. We will show that this Calderón problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator. Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.
