Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton method (PNT), that can simultaneously update the regularization parameter and solution step by step without requiring any high-cost matrix inversions or decompositions. By reformulating the Tikhonov regularization as a constrained minimization problem and writing its Lagrangian function, a Newton-type method coupled with a Krylov subspace method, called the generalized Golub-Kahan bidiagonalization, is employed for the unconstrained Lagrangian function. The resulting PNT algorithm only needs solving a small-scale linear system to get a descent direction of a merit function at each iteration, thus significantly reducing computational overhead. Rigorous convergence results are proved, showing that PNT always converges to the unique regularized solution and the corresponding Lagrangian multiplier. Experimental results on both small and large-scale Bayesian inverse problems demonstrate its excellent convergence property, robustness and efficiency. Given that the most demanding computational tasks in PNT are primarily matrix-vector products, it is particularly well-suited for large-scale problems.

翻译：在许多应用中，计算贝叶斯线性反问题的正则化解及相应的正则化参数具有高度需求。本文提出一种新型迭代方法——投影牛顿法（PNT），该方法无需任何高成本矩阵求逆或分解，即可逐步同时更新正则化参数和解。通过将Tikhonov正则化重新表述为约束最小化问题并写出其拉格朗日函数，采用牛顿型方法与Krylov子空间方法（即广义Golub-Kahan双对角化）相结合以处理无约束拉格朗日函数。所得PNT算法在每次迭代中仅需求解一个小规模线性系统以获取目标函数的下降方向，从而显著降低计算开销。严格的收敛性证明表明，PNT始终收敛至唯一正则化解及相应的拉格朗日乘子。在小规模与大规模贝叶斯反问题上的实验结果表明，该方法具有优异的收敛特性、鲁棒性和高效性。鉴于PNT中最耗时的计算任务主要是矩阵-向量乘积，它特别适用于大规模问题。