Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.

中文翻译：

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结构化L-BFGS方法及其在反问题中的应用

许多逆问题被表述为优化问题，其中目标函数是数据保真度项和正则化项的总和。通常，保真度项的 Hessian 矩阵在计算上不可用，而正则化器的 Hessian 矩阵允许廉价的矩阵向量乘积。在本文中，我们研究了利用这种结构的 L-BFGS 方法。我们证明了该方法在没有凸性假设的情况下全局收敛，并且在 Kurdyka-Łojasiewicz 型不等式下收敛是线性的。此外，我们证明了目标函数强凸的聚类点的线性收敛性。据我们所知，这是第一次在非凸设置下建立 L-BFGS 方法的线性收敛。收敛分析是在无限维希尔伯特空间中进行的，这适用于反问题，但以前从未做过。数值结果表明，在医学图像配准的非凸现实问题上，新方法优于其他结构化 L-BFGS 方法和经典 L-BFGS。在病态二次模型问题上，它与经典的 L-BFGS 相比也毫不逊色。该方法的实现是免费提供的。