论文标题
准可逆性方法是在数值上解决双曲方程的逆源问题
The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations
论文作者
论文摘要
我们提出了一种数值方法，以解决从Cauchy数据的测量结果计算双曲线方程的初始条件的逆源问题。这个问题是在有界腔中的热和光声断层扫描中引起的，其中波的反射使广泛使用的方法（例如时间逆转方法）不适用。为了解决这个反源问题，我们通过其傅立叶级数相对于$ l^2 $的特殊正交基础近似于双曲线方程的解决方案。然后，我们为相应的傅立叶系数得出一个椭圆方程的耦合系统。我们通过准可逆性方法解决了它。所需的初始条件如下。我们严格地证明了准可逆性方法的收敛性，因为噪声水平趋向于0。提供了一些数值示例。此外，我们从数值上证明了上述特殊基本的使用非常重要。
We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo- acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthogonal basis of $L^2$. Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.
