中文摘要：

TSVD通过截断参数截掉较小的奇异值来改善病态性对估计的影响,其本质是通过引入少量偏差来降低方差,以提高估值的稳定性和可靠性。截断参数是影响TSVD解算效果的关键因素,常用的广义交叉核实法（GCV法）和L曲线法未从TSVD改善模型参数估值质量的角度确定截断参数,稳定性和可靠性不足,而最小MSE法理论依据充分但受限于MSE计算的准确性。通过分析TSVD由小到大截掉奇异值后,相应的估值方差与偏差变化,本文提出了引入偏差量小于降低方差量来确定截断参数的思想,并通过估计出较大奇异值截掉后的偏差引入量建立偏差估值可信区间,利用可信区间内偏差估值与方差下降量进行比较,避免较小奇异值截掉后的方差下降量与偏差引入量的直接比较,从而解决参数真值未知截掉较小奇异值引入偏差量难以准确计算的问题。最后通过试验验证了新方法的可行性和有效性,相比于GCV法和L曲线法,新方法确定的截断参数稳定性和可靠性更高,可有效提高TSVD的解算效果。

英文摘要：

TSVD truncate small singular values by truncation parameter to improve the parameter estimation of ill-posed model.From the perspective of MSE （mean squared error）, TSVD introduce biases to reduce variances, therefore the stability and reliability of the solution can be improved.Truncation parameter is key factor of TSVD, but it is difficult to determine in case of the gently declined singular values.The parameter determined by GCV （generalized cross-validation） and L-curve often unstable and unreliable.And the minimum MSE method is limited by the accuracy of the estimated MSE.This paper compares the changes of variance and bias produced by truncating the singular values in turn and determines the truncation parameter when the reduced variance is less than the introduced bias.In order to avoid the comparison between reduced variance and introduced bias of truncating small singular values, the confidence domain of bias is established through estimating the introduced bias of truncating big singular values that are proved to be reliable.The comparisons are replaced by comparing the reduced variance with the bias in the confidence domain.Therefore, the issue of introduced bias of truncating small singular values cannot be calculated without true values of unknown parameters is solved.Numerical examples proves the feasibility and effectiveness of the new method.Truncation parameters determined by new method are more stable and reliable than GCV and L-curve and improve the TSVD solution effectively.