赋相对权比的Tikhonov正则化方法及其在岭估计中的应用研究

Research on Tikhonov Regularization Method with Relative Weight Ratio and Its Application in Ridge Estimation

摘要: Tikhonov正则化准则是解决不适定问题的通用方法，当稳定泛函确定后，确定正则化参数就成为Tikhonov正则化方法应用的核心问题。针对正则化参数通常需要在一个较大正实数范围内搜索而导致效率不高的问题，提出赋相对权比的正则化方法，并推导了新准则下的正则化解的表达式。新方法将正则化参数限定在[0, 1)区间，将事先难以确定上界的搜索问题转换为一个在较小的、具有明确上下界的优化搜索问题。对岭估计岭参数确定的数值试验结果表明，新方法与原Tikhonov正则化准则具有等价性，但新方法具有更高的计算效率。

Abstract: The Tikhonov regularization criterion is a universal method for solving ill posed problems. Once the stable functional is determined, determining the regularization parameters becomes the core problem in the application of the Tikhonov regularization method. To address the problem of low efficiency caused by the need to search for regularization parameters within a large range of posi-tive real numbers, a regularization method with relative weight ratio is proposed, and the expres-sion of regularization solution under the new criterion is derived. The new method limits the regu-larization parameters to the interval of [0, 1), transforming a search problem that is difficult to de-termine an upper bound in advance into an optimization search problem with a smaller and clear upper and lower bound. Numerical experiments on the determination of ridge parameters in ridge estimation show that the new method is equivalent to the original Tikhonov regularization criteri-on, but has higher computational efficiency.