[论文解读] Regularization of linear inverse problems by rational Krylov methods

Stefan Kindermann|arXiv (Cornell University)|Jan 15, 2026

Numerical methods in inverse problems被引用 0

一句话总结

论文分析了聚合（aggregation）和 RatCG 方法在病态线性反问题中的正则化性质，证明在与不一致性原则结合且正则化参数足够大时，通过有理 Krylov 空间框架形成最优阶正则化方案。

ABSTRACT

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.

研究动机与目标

• Investigate regularization properties of aggregation and RatCG methods for linear ill-posed problems.
• Understand the connection between these methods and rational Krylov spaces.
• Determine parameter choices and stopping rules that yield optimal convergence rates under Hölder-type source conditions.
• Assess whether these methods inherit saturation effects from standard Tikhonov regularization.

提出的方法

• Model aggregation and RatCG as rational Krylov space methods built from A*A and y_delta.
• Represent search spaces as K^n, R^n, and KR^n and express residuals via rational polynomials.
• Derive a decomposition of residuals into components related to iterated Tikhonov and CGNE iterations.
• Use a low-dimensional least-squares problem to compute aggregation coefficients from existing regularized solutions.
• Apply the discrepancy principle as a stopping rule to obtain regularization effects.
• Prove optimal-order convergence under Hölder-type source conditions when regularization parameters are large enough.

实验结果

研究问题

• RQ1 Do aggregation and RatCG constitute regularization methods under appropriate stopping rules?
• RQ2 How should the multiple regularization parameters alpha_i be chosen to ensure regularization without compromising convergence?
• RQ3 Do these rational Krylov-based methods exhibit the same saturation behavior as classical Tikhonov regularization?
• RQ4 Can optimal-order convergence rates be established for Hölder-type source conditions using these methods?

主要发现

• Aggregation and RatCG can be interpreted as diagonal sequences of CG-type methods in rational Krylov spaces.
• With the discrepancy principle and sufficiently large alpha_i, these methods achieve optimal-order convergence for Hölder-type source conditions.
• The residuals for the proposed methods decrease monotonically with the iteration index and reach zero only at the break-down index (in exact arithmetic).
• The analysis provides a framework to relate the rational Krylov-based methods to iterated Tikhonov regularization and CGNE, enabling convergence proofs despite nonlinearity in data dependence.