This work studies the inverse problem of photoacoustic tomography (more precisely, the acoustic subproblem) as the identification of a space-dependent source parameter. The model consists of a wave equation involving a time-fractional damping term to account for power law frequency dependence of the attenuation, as relevant in ultrasonics. We solve the inverse problem in a Bayesian framework using a Maximum A Posteriori (MAP) estimate, and for this purpose derive an explicit expression for the adjoint operator. On top of this, we consider optimization of the choice of the laser excitation function, which is the time-dependent part of the source in this model, to enhance the reconstruction result. The method employs the $A$-optimality criterion for Bayesian optimal experimental design with Gaussian prior and Gaussian noise. To efficiently approximate the cost functional, we introduce an approximation scheme based on projection onto finite-dimensional subspaces. Finally, we present numerical results that illustrate the theory.

翻译：本研究探讨了光声层析成像（更精确地说，其声学子问题）这一反问题，即识别空间依赖的源参数。该模型包含一个涉及时间分数阶阻尼项的波动方程，以解释超声波衰减中幂律频率依赖性的相关特性。我们在贝叶斯框架下使用最大后验概率估计求解该反问题，并为此推导了伴随算子的显式表达式。在此基础上，我们进一步优化激光激励函数（即模型中源的时间依赖部分）的选择，以提升重建结果的质量。该方法采用基于高斯先验与高斯噪声的贝叶斯最优实验设计中的$A$最优性准则。为高效逼近代价泛函，我们提出了一种基于有限维子空间投影的近似方案。最后，我们通过数值结果验证了理论的有效性。