贝叶斯反问题的广义稀疏促进求解器：通用稀疏变换与未知噪声方差
Generalized sparsity-promoting solvers for Bayesian inverse problems: Versatile sparsifying transforms and unknown noise variances
AI总结：
本文推广贝叶斯层次反演模型，使IAS-MAP求解支持带非平凡核的通用稀疏变换并在线推断未知噪声方差，同时保持计算效率并提供收敛性与数值验证。
摘要：
贝叶斯层次模型可以为寻找不适定反问题的稀疏解提供高效算法。这类模型通常包含针对未知量的条件高斯先验模型，并辅以针对方差超参数的广义伽马超先验模型。本研究从两个方面推广了这些模型及其利用迭代交替序列（IAS）算法进行的高效最大后验（MAP）估计：（1）通用稀疏变换：与传统方法不同，我们的方法允许使用具有非平凡核的稀疏变换；（2）未知噪声方差：我们将噪声方差视为在推断过程中估计的随机变量。这在无法先验准确估计噪声的应用中非常重要。值得注意的是，这些扩展既不会显著增加算法的计算开销，也不会损害其有效性。我们给出了该方法的凸性和收敛性分析，并在若干数值实验中证明了其有效性。
英文摘要：
Bayesian hierarchical models can provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models typically comprise a conditionally Gaussian prior model for the unknown which is augmented by a generalized gamma hyper-prior model for variance hyper-parameters. This investigation generalizes these models and their efficient maximum a posterior (MAP) estimation using the iterative alternating sequential (IAS) algorithm in two ways: (1) General sparsifying transforms: Diverging from conventional methods, our approach permits the use of sparsifying transformations with nontrivial kernels; (2) Unknown noise variances: We treat the noise variance as a random variable that is estimated during the inference procedure. This is important in applications where the noise estimate cannot be accurately estimated a priori. Remarkably, these augmentations neither significantly burden the computational expense of the algorithm nor compromise its efficacy. We include convexity and convergence analysis for the method and demonstrate its efficacy in several numerical experiments.
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