  A novel compressed matrix format is proposed that combines an adaptive
hierarchical partitioning of the matrix with low-rank approximation. One
typical application is the approximation of discretized functions on
rectangular domains; the flexibility of the format makes it possible to deal
with functions that feature singularities in small, localized regions. To deal
with time evolution and relocation of singularities, the partitioning can be
dynamically adjusted based on features of the underlying data. Our format can
be leveraged to efficiently solve linear systems with Kronecker product
structure, as they arise from discretized partial differential equations
(PDEs). For this purpose, these linear systems are rephrased as linear matrix
equations and a recursive solver is derived from low-rank updates of such
equations. We demonstrate the effectiveness of our framework for stationary and
time-dependent, linear and nonlinear PDEs, including the Burgers' and
Allen-Cahn equations.
