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A novel compressed matrix format is proposed that combines an adaptivehierarchical partitioning of the matrix with low-rank approximation. Onetypical application is the approximation of discretized functions onrectangular domains; the flexibility of the format makes it possible to dealwith functions that feature singularities in small, localized regions. To dealwith time evolution and relocation of singularities, the partitioning can bedynamically adjusted based on features of the underlying data. Our format canbe leveraged to efficiently solve linear systems with Kronecker productstructure, as they arise from discretized partial differential equations(PDEs). For this purpose, these linear systems are rephrased as linear matrixequations and a recursive solver is derived from low-rank updates of suchequations. We demonstrate the effectiveness of our framework for stationary andtime-dependent, linear and nonlinear PDEs, including the Burgers' andAllen-Cahn equations.

Hierarchical adaptive low-rank format with applications to discretized PDEs