On the mathematical treatment of time reversal
On the mathematical treatment of time reversal
复制标题
论时间反转的数学处理
DOI：
10.1088/0266-5611/19/6/005
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发表时间：
2003
期刊：
影响因子：
2.1
通讯作者：
A. Timonov
中科院分区：
文献类型：
--
作者：
M. Klibanov;A. Timonov
我们提出了一种时间反转的数学处理方法。提出并讨论了两个近似描述时间反向场传播的数学模型。第一个模型采用零初始条件，而第二个模型采用准可逆性方法。由于时间反转的计算机模拟需要知道传播介质的材料属性，如声速或电导率，因此时间反转的一般问题是非线性的和不适定的。这种不适定性是由于不唯一性和不稳定性造成的。为了处理这一问题，提出了一个分两个阶段的程序，并证明了这一点。在第一阶段中，近似确定了传播的非均匀介质的未知材料性质。由于不假定弱散射，因此采用凸化方法来估计这些性质。在第二阶段，用拟可逆性方法从双曲型方程柯西问题的解和横向数据近似确定时间反转场。
We present a mathematical treatment of time reversal. Two mathematical models describing approximately the propagation of the time-reversed field are proposed and discussed. Zero initial conditions are exploited in the first model, whereas the method of quasi-reversibility is adopted when constructing the second model. Since computer simulation of time reversal requires knowledge of material properties of a propagating medium, such as the sound speed or electrical conductivity, the general problem of time reversal is nonlinear and ill posed. The ill-posedness is due to the nonuniqueness and instability. To treat this problem, a two-stage procedure is proposed and justified. In the first stage, the unknown material properties of a propagating inhomogeneous medium are approximately determined. Since weak scattering is not assumed, the convexification approach is adopted to estimate such properties. In the second stage, the time-reversed field is approximately determined from the solution of the Cauchy problem for a hyperbolic equation with the lateral data by the method of quasi-reversibility.