SPAI (SParse Approximate Inverse)

Synonyms

Sparse approximate inverse matrix

Definition

For a given sparse matrix A a sparse matrix $M \approx {A}^{-1}$ is computed by minimizing $\Vert AM - {I\Vert }_{F}$ in the Frobenius norm over all matrices with a certain sparsity pattern. In the SPAI algorithm the pattern of M is updated dynamically to improve the approximation until a certain stopping criterion is reached.

Discussion

Introduction

For applying an iterative solution method like the conjugate gradient method (CG), GMRES, BiCGStab, QMR, or similar algorithms, to a system of linear equations Ax = b with sparse matrix A, it is often crucial to include an efficient preconditioner. Here, the original problem Ax = b is replaced by the preconditioned system MAx = Mb or Ax = A(My) = b. In a parallel environment a preconditioner should satisfy the following conditions:

• M can be computed efficiently in parallel.

• Mc can be computed efficiently in parallel for any given vector c.

• The iterative solver applied on AMx = b or MAx...