论文标题
在未知介质中线性波方程的人工点源的构建
Construction of artificial point sources for a linear wave equation in unknown medium
论文作者
论文摘要
我们在$ \ Mathbb r^m $的有限域以及紧凑的Riemannian歧管$ M $上研究波动方程。我们假设波方程的系数尚不清楚，但是我们得到了与边界上物理测量相对应的双曲线neumann to-dirichlet映射$λ$。使用$λ$的知识，我们构建了一个neumann边界值的序列，因此在$ t $的时候，相应的波会收敛到零，而波的时间导数会收敛到三角洲分布。这样的波被称为人造点源。波的收敛发生在与波的能量自然相关的函数空间中。我们将结果应用于逆问题，并在1维情况下以数字为单位的焦点。
We study the wave equation on a bounded domain of $\mathbb R^m$ and on a compact Riemannian manifold $M$ with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map $Λ$ that corresponds to the physical measurements on the boundary. Using the knowledge of $Λ$ we construct a sequence of Neumann boundary values so that at a time $T$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. Such waves are called an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the 1-dimensional case.
