In photoacoustic tomography (PAT), the computation of the initial pressure distribution within an object from its time-dependent boundary measurements over time is considered. This problem can be approached from two well-established points of view: deterministically using regularisation methods, or stochastically using the Bayesian framework. Both approaches frequently require the solution of a variational problem. In the paper we elaborate the connection between these approaches by establishing the equivalence between a smoothing Mat{é}rn class of covariance operators and Sobolev embedding operator $E_s: H^s \hookrightarrow L^2$. We further discuss the use of a Wavelet-based implementation of the adjoint operator $E_s^*$ which also allows for efficient evaluations for certain Mat{é}rn covariance operators, leading to efficient implementations both in terms of computational effort as well as memory requirements. The proposed methods are validated with reconstructions for the photoacoustic problem.
University of Vienna（维也纳大学）；Christian Doppler Laboratory for Mathematical Modeling and Simulation of Next Generations of Ultrasound Devices (MaMSi)（下一代超声设备数学建模与模拟克里斯蒂安·多普勒实验室）；Johann Radon Institute for Computational and Applied Mathematics (RICAM)（约翰·拉登计算与应用数学研究所）；Medical University of Vienna（维也纳医科大学）
Quantitative Photoacoustic Imaging of Two-photon Absorption
University of Texas, Austin（德克萨斯大学奥斯汀分校）；The Institute for Computational Engineering and Sciences (ICES), University of Texas, Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Department of Mathematics, University of Texas, Austin（德克萨斯大学奥斯汀分校数学系）
Fast algorithms enabling optimization and deep learning for photoacoustic tomography in a circular detection geometry
Research Unit of Mathematical Sciences, University of Oulu（奥卢大学数学科学研究所）；Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Department of Mathematics, University of Arizona（亚利桑那大学数学系）；Department of Technical Physics, University of Eastern Finland（东部芬兰大学技术物理系）
3D Uncertainty Quantification for Photoacoustic Tomography
University of Oulu（奥卢大学）；King Fahd University of Petroleum and Minerals（法赫德国王石油与矿产大学）；Technical University of Denmark（丹麦技术大学）；University of Eastern Finland（芬兰东部大学）
