The technique of hierarchical matrices

Abstract:

Matrices of large size arise in particular from elliptic partial differential equations and integral equations. In the former case one make use of the sparsity, in the latter case a standard treatment of the matrices leads already to storage problems. The technique of hierarchical matrices allows organising of the storage as well as all matrix operations (including inversion) with almost linear complexity. The hierarchical matrix operations yield only approximations, but the arising error can be made at least as small as the discretisation error. The lecture explains the matrix representation, the organisation of the operations and underlines the black-box character of the method. Furthermore, applications are described which are usually considered to be impossible for large scale matrices: computation of functions of matrices and solution of matrix equations (Lyapunov etc.)