  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.
