Modified Sparse Approximate Inverses (MSPAI) for Parallel Preconditioning

The solution of large sparse and ill-conditioned systems of linear equations is a central task in numerical linear algebra. Such systems arise from many applications like the discretization of partial differential equations or image restoration. Herefore, Gaussian elimination or other classical direct solvers can not be used since the dimension of the underlying coefficient matrices is too large and Gaussian elimination is an O ( n ) algorithm. Iterative solvers techniques are an effective remedy for this problem. They allow to exploit sparsity, bandedness, or block structures, and they can be parallelized much easier. However, due to the matrix being ill-conditioned, convergence becomes very slow or even not be guaranteed at all. Therefore, we have to employ a preconditioner. The sparse approximate inverse (SPAI) preconditioner is based on Frobenius norm minimization. It is a well-established preconditioner, since it is robust, flexible, and inherently parallel. Moreover, SPAI captures meaningful sparsity patterns automatically. The derivation of pattern update criteria for complex valued systems of equations, the reformulation and extension of a nonsingularity result, and the investigation of SPAI’s regularization qualities are our first original contributions to research. Furthermore, we investigate the effect of a fill-in-reducing graph algorithm for pattern updates in FSPAI, which is the factorized variant of SPAI. Additionally, a theoretical result for the SPAI and FSPAI of M-matrices is presented. As the main contribution to ongoing research, we develop the new modified sparse approximate inverse preconditioner MSPAI. On the one hand, this is a generalization of SPAI, because we extend SPAI to target form. This allows us also to compute explicit matrix approximations in either a factorized or unfactorized form. On the other hand, this extension enables us to add some further, possibly dense, rows to the underlying matrices, which are then also taken into account during the computation. These additional constraints for the Frobenius norm minimization generalize the idea of classical probing techniques, which are restricted to explicit approximations and very simple probing constraints. By a weighting factor, we force the resulting preconditioner to be optimal on certain probing subspaces represented by the additional rows. For instance, the vector of all ones leads to preservation of row sums, which is quite important in many applications as it reflects certain conservation laws. Therefore, MSPAI probing can also be seen as a generalization to the well-known modified preconditioners such as modified incomplete LU or modified incomplete Cholesky. Furthermore, we can improve Schur complement approximations, which are the original application area of classical probing. Given factorized preconditioners can also be improved relative to a probing subspace. For symmetric linear systems, new symmetrization techniques are introduced. The effectiveness of MSPAI probing is proven by many numerical examples such as matrices arising from domain decomposition methods and Stokes problems. Besides the theoretical development of MSPAI probing, an efficient implementation is presented. We investigate the use of a linear algebra library for sparse least squares problems in combination with QR updates and compare it to standard dense methods. Furthermore, we implement a caching strategy which helps to avoid redundant QR factorizations especially for the case of highly structured matrices. The support for maximum sparsity patterns rounds up our implementation. Various tests reveal significantly lower runtimes compared to the original implementation of SPAI.