Inverse-Based Algebraic Multilevel Factorizations Methods
Matthias Bollhöfer (TU Berlin)
We discuss an algebraic multilevel factorization
approach for the solution of large
sparse linear systems.
The objective of this approach is to detect a submatrix of the
original matrix such that its inverse is approximately sparse.
One possibility could be to construct an adapted sparse approximate
inverse of the associated leading submatrix, another approach
consists of directly
constructing an incomplete LU decomposition such that the inverse
triangular factors are bounded. From the theoretical point of view this
can be interpreted as keeping the approximate
triangular factors L and U
and their inverses close to each other.
Different strategies will be presented that address the problem of
finding a submatrix with sparse approximate inverse.
Successively applied, this leads
to an algebraic multilevel hierarchy. Numerical examples
will be shown to illustrate the effectiveness of this approach.