We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the expected value of a linear functional of the solution of the PDE. The proposed algorithm combines the multivariate decomposition method, to compute infinite-dimensional integrals, with the finite element method, to solve different instances of the PDE. The strategy of the MDFEM is to decompose the infinite-dimensional problem into multiple finite-dimensional ones which lends itself to easier parallelization than to solve a single large dimensional problem. Our first result adjusts the analysis of the multivariate decomposition method to incorporate the \((\ln (n))^d\)-factor which typically appears in error bounds for d-dimensional n-point cubature formulae and we take care of the fact that n needs to come, e.g., in powers of 2 for higher order approximations. For the further analysis we specialize the cubature methods to be two types of quasi-Monte Carlo (QMC) rules, being digitally shifted polynomial lattice rules and interlaced polynomial lattice rules. The second and main contribution then presents a bound on the error of the MDFEM and shows higher-order convergence w.r.t. the total computational cost in case of the interlaced polynomial lattice rules in combination with a higher-order finite element method. We show that the cost to achieve an error \(\epsilon \) is of order \(\epsilon ^{-a_{\mathrm {MDFEM}}}\) with \(a_{\mathrm {MDFEM}} = 1/\lambda + d'/\tau \) if the QMC cubature errors can be bounded by \(n^{-\lambda }\) and the FE approximations converge like \(h^\tau \) with cost \(h^{d'}\), where \(\lambda = \tau (1-p^*) / (p^* (1+d'/\tau ))\) and \(p^*\) is a parameter representing the “sparsity” of the random field expansion. A comparison with a dimension truncation algorithm shows that the MDFEM will perform better than the truncation algorithm if \(p^*\) is sufficiently small, i.e., the representation of the random field is sufficiently sparse.

我们引入了多元分解有限元方法(MDFEM)，用于求解具有均匀随机扩散系数的椭圆偏微分方程。我们表明，MDFEM 可用于降低估计 PDE 解的线性函数的期望值的计算复杂度。所提出的算法结合了多元分解方法，计算无限维积分，与有限元方法，解决偏微分方程的不同实例。MDFEM 的策略是将无限维问题分解为多个有限维问题，这比解决单个大维问题更容易并行化。我们的第一个结果调整了多元分解方法的分析以纳入\((\ln (n))^d\) -通常出现在d维n点体积公式的误差范围内的因子，我们会注意n需要出现的事实，例如，对于高阶近似。为了进一步分析，我们将 cuature 方法专门化为两种类型的准蒙特卡罗 (QMC) 规则，即数字移位多项式格规则和交错多项式格规则。然后，第二个也是主要的贡献提出了 MDFEM 误差的界限，并显示了在交错多项式点阵规则与高阶有限元方法相结合的情况下，与总计算成本相关的高阶收敛性。我们证明了实现错误的代价\(\epsilon \)是顺序\(\epsilon ^{-a_{\mathrm {MDFEM}}}\)与\(a_{\mathrm {MDFEM}} = 1/\lambda + d'/\tau \)如果 QMC 体积误差可以由\(n^{-\lambda }\)和有限元近似收敛\(h^\tau \)与成本\(h^{d'}\)，其中\(\lambda = \tau (1-p^*) / (p^* (1+d'/\tau ))\)和\(p^*\)是代表随机场扩展的“稀疏性”的参数。与维度截断算法的比较表明，如果\(p^*\)足够小，即随机场的表示足够稀疏，则MDFEM 的性能将优于截断算法。