论文标题
椭圆pde中的非零约束，具有随机边界值和在混合逆问题上的应用
Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems
论文作者
论文摘要
杂交逆问题基于两种波浪的相互作用，以便以高分辨率和高对比度进行成像。反转过程通常包括两个步骤：首先，涉及未知参数和一些相关数量的内部测量值，其次，必须从内部数据中重建未知参数。第二步中的重建需要某些PDE的解决方案来满足某些非零约束，例如缺乏淋巴结或关键点，或者是非散布的Jacobian。 在这项工作中，我们考虑了二阶椭圆形PDE，并表明可以通过在次高斯分布之后随机选择边界值来满足这些约束，并以压倒性的概率来满足这些约束。该证明是基于独立利益的结果的新的定量估计值。
Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving the unknown parameters and some related quantities are obtained, and, second, the unknown parameters have to be reconstructed from the internal data. The reconstruction in the second step requires the solutions of certain PDE to satisfy some non-zero constraints, such as the absence of nodal or critical points, or a non-vanishing Jacobian. In this work, we consider a second-order elliptic PDE and show that it is possible to satisfy these constraints with overwhelming probability by choosing the boundary values randomly, following a sub-Gaussian distribution. The proof is based on a new quantitative estimate for the Runge approximation, a result of independent interest.
