Geologists and Reservoir Engineers routinely use time-domain nuclear magnetic resonance (NMR) to learn about the porous structure of rocks that hold underground fluids. In particular, two-dimensional NMR (2DNMR) technique is now gaining importance in a wide variety of applications. Crucial issue in 2DNMR analysis are the speed, robustness and accuracy of the data inversion process. This paper proposes a multi-penalty method with locally adapted regularization parameters for fast and accurate inversion of 2DNMR data. The method solves an unconstrained optimization problem whose objective function contains a data-fitting term, a single L1 penalty parameter and a multiple parameter L2 penalty. We propose an adaptation of the Fast Iterative Shrinkage and Thresholding (FISTA) method to solve the multi-penalty minimization problem, and an automatic procedure to compute all the penalty parameters. This procedure generalizes the Uniform Penalty principle introduced in [Bortolotti et al., Inverse Problems, 33(1), 2016]. The proposed approach allows us to obtain accurate 2D relaxation time distributions while keeping short the computation time. Results of numerical experiments on synthetic and real data prove that the proposed method is efficient and effective in reconstructing the peaks and the flat regions that usually characterize 2DNMR relaxation time distributions.

地质学家和油藏工程师通常使用时域核磁共振（NMR）了解容纳地下流体的岩石的多孔结构。特别是，二维NMR（2DNMR）技术现在在各种应用中变得越来越重要。2DNMR分析的关键问题是数据反演过程的速度，鲁棒性和准确性。本文提出了一种具有局部适应性正则化参数的多罚方法，用于快速，准确地反演2DNMR数据。该方法解决了目标函数包含数据拟合项，单个L 1罚分参数和多个参数L的无约束优化问题。2罚。我们提出一种快速迭代收缩和阈值（FISTA）方法的改进方案，以解决多罚分最小化问题，并提出一种自动过程来计算所有罚分参数。该过程概括了[Bortolotti等人，逆问题，33（1），2016]中引入的统一罚则原则。所提出的方法使我们能够获得准确的2D弛豫时间分布，同时保持较短的计算时间。对合成和真实数据进行数值实验的结果证明，该方法可有效地重建通常表征2DNMR弛豫时间分布的峰和平坦区域。