Smoothing and regularization with modified sparse approximate inverses
Summary: Sparse approximate inverses \(M\) which satisfy \(\min_{M}\| AM-I\|_{F}\) have shown to be an attractive alternative to classical smoothers like Jacobi or Gauss-Seidel [\textit{W.-P. Tang} and \textit{W. L. Wan}, SIAM J. Matrix Anal. Appl. 21, No. 4, 1236-1252 (2000; Zbl 1049.65146)]. The static and dynamic computation of a SAI and a SPAI [\textit{M. J. Grote} and \textit{T. Huckle}, SIAM J. Sci. Comput. 18, No. 3, 838--853 (1997; Zbl 0872.65031)], respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties. Here, we are interested in developing preconditioners that can incorporate probing conditions for improving the approximation relative to high- or low-frequency subspaces. We present analytically derived optimal smoothers for the discretization of the constant-coefficient Laplace operator. On this basis, we introduce probing conditions in the generalized Modified SPAI (MSPAI) approach [\textit{T. Huckle} and \textit{A. Kallischko}, Int. J. Comput. Math. 84, No. 8, 1225--1248 (2007; Zbl 1126.65035)] which yields efficient smoothers for multigrid. In the second part, we transfer our approach to the domain of ill-posed problems to recover original information from blurred signals. Using the probing facility of MSPAI, we impose the preconditioner to act as approximately zero on the noise subspace. In combination with an iterative regularization method, it thus becomes possible to reconstruct the original information more accurately in many cases. A variety of numerical results demonstrate the usefulness of this approach.
scientific article; zbMATH DE number 5910099
scientific article