| 摘要: |
| 本文研究了线性反问题的Crank–Nicolson正则化方法.通过将连续正则化方程采用Crank–Nicolson离散，证明该迭代格式构成正则化方法并能达到最优收敛阶.理论分析表明，Crank–Nicolson方法在固定步长下具有与Landweber迭代正则化相同的正则化性能；在变步长情况下，其收敛条件比非稳态Tikhonov方法更严格，等比步长不能实现对数级迭代步数.数值实验验证了方法的正则化有效性及最优误差阶，同时揭示其在小噪声场景下相对非稳态Tikhonov方法的局限性. |
| 关键词: 线性反问题 连续正则化 Crank-Nicolson格式 |
| DOI： |
| 分类号:O177 |
| 基金项目:国家自然科学基金项目,湖北省自然科学基金 |
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| CRANK-NICOLSON REGULARIZATION METHOD FOR LINEAR INVERSE PROBLEMS |
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GAO Jian-ming, LU Xi-liang
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| Abstract: |
| This paper investigates the Crank–Nicolson regularization method for linear inverse problems.By discretizing the continuous regularization equation with the Crank–Nicolson scheme, the method is proven to be a valid regularization strategy achieving optimal convergence order.Theoretical analysis shows that, under fixed step size, it shares similar regularization properties with the Landweber method.Under variable step sizes, its convergence condition is stricter than that of the nonstationary Tikhonov method, and geometric step sequences cannot achieve logarithmic iteration order.Numerical experiments confirm its regularization effectiveness and optimal error rate, while revealing the restriction with respect to the nonstationary Tikhonov method under low-noise conditions. |
| Key words: linear inverse problems continuous regularization Crank–Nicolson scheme |
