On the algebraic construction of sparse multilevel approximations of elliptic tensor product problems
For researchers solving elliptic PDEs with random inputs on complex geometries, this work enables sparse grid methods previously limited to structured grids, though the approach is incremental as it adapts existing AMG techniques.
The paper introduces an algebraic multigrid-based construction for sparse multilevel approximations of elliptic tensor product problems, enabling the sparse grid combination technique on complex geometries and unstructured grids. Numerical results demonstrate convergence behavior equivalent to geometric constructions while extending applicability to black-box PDE solvers.
We consider the solution of elliptic problems on the tensor product of two physical domains as e.g. present in the approximation of the solution covariance of elliptic partial differential equations with random input. Previous sparse approximation approaches used a geometrically constructed multilevel hierarchy. Instead, we construct this hierarchy for a given discretized problem by means of the algebraic multigrid method (AMG). Thereby, we are able to apply the sparse grid combination technique to problems given on complex geometries and for discretizations arising from unstructured grids, which was not feasible before. Numerical results show that our algebraic construction exhibits the same convergence behaviour as the geometric construction, while being applicable even in black-box type PDE solvers.