Approximate inverse limits and (m,n)-dimensions

Approximate Stolz Angle Limits

Refining an Approximate Inverse

Direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades. The advances in graph theoretic algorithms have not received enough attention from the iterative methods community. For example, we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate in...

Inverse Limits and Profinite Groups

Standard examples of inverse limits arise from sequences of groups, with maps between them: for instance, if we have the sequence Gn = Z/pZ for n ≥ 0, with the natural quotient maps πn+1 : Gn+1 → Gn, the inverse limit consists of tuples (g0, g1, . . . ) ∈ ∏ n≥0Gn such that πn+1(gn+1) = gn for all n ≥ 0. This is a description of the p-adic integers Zp. It is clear that more generally if the Gn a...

Multiresolution Approximate Inverse Preconditioners

We introduce a new preconditioner for elliptic PDE’s on unstructured meshes. Using a wavelet-inspired basis we compress the inverse of the matrix, allowing an effective sparse approximate inverse by solving the sparsity vs. accuracy conflict. The key issue in this compression is to use second-generation wavelets which can be adapted to the unstructured mesh, the true boundary conditions, and ev...

Parallel Approximate Inverse Preconditioners

There has been much excitement recently over the use of approximate inverses for parallel preconditioning. The preconditioning operation is simply a matrix-vector product, and in the most popular formulations, the construction of the approximate inverse seems embarassingly parallel. However, diiculties arise in practical parallel implementations. This paper will survey approximate inverse preco...