Why using Htool-DDM?¶
Htool-DDM aims to provide iterative solvers with preconditioners stemming from domain decomposition methods (DDM). It uses matrix compression via hierarchical matrices by default, and in particular, it provides
parallel matrix-vector and matrix-matrix product using MPI and OpenMP,
iterative solvers via HPDDM,
preconditioning techniques using domain decomposition methods.
- Why?
The storage and cost of assembly for dense matrices are both quadratic with respect to their size, while the cost of using linear solvers is cubic for direct solvers. For iterative solvers, the matrix-vector product has quadratic complexity, which is to be multiplied by the number of iterations. To reduce these costs, several compression techniques have been developed, which gives approximated representation of matrix-vector product and other operations. Htool-DDM provides default compression via sec_hierarchical_matrices, and emphasize has been put on
parallelisation for high-performance computing,
a black box interface, to tackle a great variety of problems.
- Applications
Hierarchical matrices are generally used to compress matrices stemming from the discretisation of asymptotically smooth kernels \(\kappa (x,y)\), i.e, for two cluster of geometric points \(X\) and \(Y\),
\[\rvert \partial_x^{\alpha} \partial_y^{\beta}\kappa (x,y)\lvert \leq C_{\mathrm{as}}\lvert x - y\rvert^{-\lvert \alpha \rvert -\lvert \beta \rvert - s}.\]with \(x\in X\), \(y\in Y\), \(x\neq y\), \(\alpha, \beta \in \mathbb{N}_0^d\) and \(\alpha+\beta \neq 0\).
Such matrices arise in the context of