Numerical computation of the Karhunen--Lo\`eve expansion is computationally challenging in terms of both memory requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the K--L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed by the authors in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2]. Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth covariance kernels, while the collocation method performs slightly better for rough covariance kernels.

翻译：对Karhunen-Lo ⁇ ⁇ éeve扩展的数值计算在记忆要求和计算时间两方面都具有计算上的挑战性。我们比较了两种声称有效解决K-L扩展的先进方法:(1)使用作者在[1]和(2)中提议的基于内推的二次方位的无矩阵等离异测量加勒金方法;(2)我们采用[2]中提议的新的无矩阵同位同位法。两个三维基准问题表明,Galerkin方法在顺利共变内核方面表现得好得多,而合用法在粗共变内核方面则表现略好一些。