We present the construction of a sparse-compressed operator that approximates the solution operator of elliptic PDEs with rough coefficients. To derive the compressed operator, we construct a hierarchical basis of an approximate solution space, with superlocalized basis functions that are quasi-orthogonal across hierarchy levels with respect to the inner product induced by the energy norm. The superlocalization is achieved through a novel variant of the Super-Localized Orthogonal Decomposition method that is built upon corrections of basis functions arising from the Localized Orthogonal Decomposition method. The hierarchical basis not only induces a sparse compression of the solution space but also enables an orthogonal multiresolution decomposition of the approximate solution operator, decoupling scales and solution contributions of each level of the hierarchy. With this decomposition, the solution of the PDE reduces to the solution of a set of independent linear systems per level with mesh-independent condition numbers that can be computed simultaneously. We present an accuracy study of the compressed solution operator as well as numerical results illustrating our theoretical findings and beyond, revealing that desired optimal error rates with well-behaved superlocalized basis functions can still be attained even in the challenging case of coefficients with high-contrast channels.

翻译：本文提出了一种稀疏压缩算子的构造方法，用于逼近具有粗糙系数的椭圆型偏微分方程的解算子。为推导该压缩算子，我们构建了近似解空间的分层基，其基函数具有超局部特性，且在由能量范数诱导的内积下保持跨层级准正交性。超局部化是通过对局部正交分解方法生成的基函数进行校正而实现的新型超局部正交分解方法变体达成的。该分层基不仅诱导了解空间的稀疏压缩，还实现了近似解算子的正交多分辨率分解，从而解耦了各层级间的尺度与解贡献。通过此分解，偏微分方程的求解可简化为各层级上独立线性方程组的求解，这些方程组具有与网格无关的条件数且可并行计算。我们对压缩解算子的精度进行了分析研究，并提供了数值结果以验证理论发现及拓展结论。结果表明，即使在具有高对比度通道系数的挑战性情况下，仍能通过性质良好的超局部基函数获得所需的最优误差收敛率。