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Hierarchical adaptive low‐rank format with applications to discretized partial differential equations

A novel framework for hierarchical low‐rank matrices 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.