We show that the expected solution operator of prototypical linear elliptic partial differential operators with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator.

中文翻译：

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预期解算子的稀疏压缩

我们表明，具有随机系数的原型线性椭圆偏微分算子的预期解算子很好地近似于可计算的稀疏矩阵。该结果基于解空间的随机局部正交多分辨率分解，该分解允许在此基础上表示的随机算子的稀疏近似反演及其随机平均。近似期望解算子可以用经典的 Haar 小波来解释。当与期望的合适采样方法结合时，这种构造导致计算期望解算子的稀疏表示的有效方法。