Given a sparse matrix A the SPAI Algorithm computes a sparse approximate inverse
M by minimizing || AM - I || in the Frobenius norm. The approximate inverse is
computed explicitly and can then be applied as a preconditioner to an iterative
method. The sparsity pattern of the approximate inverse is either fixed a priori
or captured automatically:

 * Fixed sparsity: The sparsity pattern of M is either banded or a subset of
 the sparsity pattern of A.
 * Adaptive sparsity: The algorithm proceeds until the 2-norm of each column of
 AM-I is less than eps. By varying eps the user controls the quality and the
 cost of computing the preconditioner. Usually the optimal eps lies between 0.5
 and 0.7.

A very sparse preconditioner is very cheap to compute but may not lead to much
improvement, while if M becomes rather dense it becomes too expensive to
compute. The optimal preconditioner lies between these two extremes and is
problem and computer architecture dependent.

The approximate inverse M can also be used as a robust (parallel) smoother for
(algebraic) multi-grid methods.

• Grigory Ustinov
• Eugeny A. Rostovtsev (REAL)

• libspai
• libspai-debuginfo
• libspai-devel
• libspai-devel-doc
• spai
• spai-debuginfo