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 orthonormal 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.

我们提出了一种数值方法来解决从柯西数据的测量中计算双曲方程初始条件的反源问题。这个问题出现在有边界的腔体中的热声和光声层析成像中，其中波的反射使广泛使用的方法（例如时间倒转方法）不适用。为了解决该逆源问题，我们针对其特殊的正交标准\（L ^ 2 \），通过其双曲方程的傅里叶级数来近似双曲型方程的解。。然后，我们为相应的傅立叶系数导出一个椭圆方程的耦合系统。我们通过准可逆性方法解决它。所需的初始条件如下。当噪声水平趋于0时，我们严格证明了准可逆性方法的收敛性。提供了一些数值示例。此外，我们通过数值证明了上述特殊基础的使用意义重大。