自适应稀疏伪谱逼近新方法

自适应稀疏伪谱逼近法是广义混沌多项式类方法的最新进展，相对于其它方法具有计算精度高、速度快的优点.但它仍存在如下缺点：1）终止判据对逼近误差的估计精度偏低；2）只适用于单输出问题.本文提出了适用于多输出问题且具有更高逼近精度的自适应稀疏伪谱逼近新方法.本文首先提出了新型终止判据及基于此新型终止判据的自适应稀疏伪谱逼近新方法，并以命题的形式证明了新型终止判据相比于现有终止判据具有更高的估计精度，从而使基于此的逼近函数精度更接近于预期精度；进而，本文基于指标集的统一策略和新型终止判据，提出了适用于多输出问题的自适应稀疏伪谱逼近新方法，该方法因能充分利用各输出变量的抽样结果，具有比将单输出方法直接推广到多输出问题更高的计算效率.多个算例验证了本文所提出新方法的有效性和正确性.

A NEW ADAPTIVE SPARSE PSEUDOSPECTRAL APPROXIMATION METHOD

The adaptive sparse pseudospectral approximation method is a highly efficient method of polynomial chaos, but there exist two defects. One defect is that its termination criterion cannot estimate its approximation error accurately, and the other is that it s only applicable to single-output problems. A new adaptive sparse pseudospectral approximation method applicable to multi-output problems is proposed in this paper. First, a new termination criterion that can estimate the approximation error more accurately than the original is put forward and its correctness is ensured by a proposition which is strictly proved in the paper. Based on the new criterion, a new adaptive sparse approximation pseudospectral method for single-output problems is thus proposed whose approximation error is closer to the required error than the original method. Then, a new adaptive sparse pseudospectral approximation method for multi-output problems is proposed by the strategy of unifying the index sets corresponding to all output variables and using the new termination criterion. The proposed method is computationally more efficient than the methods of extending the adaptive sparse pseudospectral approximation method for single-output to the multi-output problems directly. Several mathematical cases demonstrate the effectiveness and validity of the proposed method.