标题： 变量投影方法求解正则化可分离反问题及其在半盲图像去模糊中的应用 显示英文标题

标题： Variable Projection Methods for Solving Regularized Separable Inverse Problems with Applications to Semi-Blind Image Deblurring

摘要： 可分离非线性最小二乘问题出现在许多反问题中，包括半盲图像去模糊问题。变量投影（VarPro）方法通过消除线性变量并将问题简化为较小的非线性问题，为解决此类问题提供了一种高效的方法。在本工作中，我们将VarPro扩展以解决包含对非线性参数的可微正则化项以及对线性变量的一般形式Tikhonov正则化项的最小化问题。此外，我们开发了一种拟牛顿方法来解决由此产生的简化问题，并在标准光滑性假设下提供了局部收敛性分析，建立了超线性或二次收敛的条件。对于大规模情况，我们引入了一种基于不精确LSQR的变体，并证明了即使在内部求解和Hessian近似的情况下，它仍具有局部收敛性。在半盲去模糊的数值实验中表明，参数正则化可以防止退化解的无模糊解，所提出的方法实现了准确的重建，不精确变体在准确性和计算成本之间提供了与理论一致的有利权衡。 显示英文摘要

摘要： Separable nonlinear least squares problems appear in many inverse problems, including semi-blind image deblurring. The variable projection (VarPro) method provides an efficient approach for solving such problems by eliminating linear variables and reducing the problem to a smaller, nonlinear one. In this work, we extend VarPro to solve minimization problems containing a differentiable regularization term on the nonlinear parameters, along with a general-form Tikhonov regularization term on the linear variables. Furthermore, we develop a quasi-Newton method for solving the resulting reduced problem, and provide a local convergence analysis under standard smoothness assumptions, establishing conditions for superlinear or quadratic convergence. For large-scale settings, we introduce an inexact LSQR-based variant and prove its local convergence despite inner-solve and Hessian approximations. Numerical experiments on semi-blind deblurring show that parameter regularization prevents degenerate no-blur solutions and that the proposed methods achieve accurate reconstructions, with the inexact variant offering a favorable accuracy-cost tradeoff consistent with the theory.