We develop a generalized hybrid iterative approach for computing solutions to large-scale Bayesian inverse problems. We consider a hybrid algorithm based on the generalized Golub-Kahan bidiagonalization for computing Tikhonov regularized solutions to problems where explicit computation of the square root and inverse of the covariance kernel for the prior covariance matrix is not feasible. This is useful for large-scale problems where covariance kernels are defined on irregular grids or are only available via matrix-vector multiplication, e.g., those from the Mat\'{e}rn class. We show that iterates are equivalent to LSQR iterates applied to a directly regularized Tikhonov problem, after a transformation of variables, and we provide connections to a generalized singular value decomposition filtered solution. Our approach shares many benefits of standard hybrid methods such as avoiding semi-convergence and automatically estimating the regularization parameter. Numerical examples from image processing demonstrate the effectiveness of the described approaches.
翻译:我们开发了一种通用的混合迭代方法,用于计算大规模巴伊西亚反问题的解决办法。我们考虑一种基于通用的Golub-Kahan 的混合算法,用于计算Tikhonov的正规化解决办法,以解决无法明确计算平方根和前共差矩阵的共变内核反向的问题。这对于在非常规网格上定义共差内核或通过矩阵-矢量乘法(例如,Mat\'{{e}rn类)提供的混合算法,大问题很有用。我们表明,在变异变量后,对直接正规化的Tikhonov问题应用了LSQR 迭热法,我们提供了与通用单值分解过滤式解决办法的连接。我们的方法分享了标准混合方法的许多好处,例如避免半趋同和自动估计正规化参数。图像处理中的数字示例显示了所述方法的有效性。