TY - JOUR A2 - Axelsson, Owe AU - Huckle, T. AU - Sedlacek, M. PY - 2010 DA - 2010/12/09 TI - Smoothing and Regularization with Modified Sparse Approximate Inverses SP - 930218 VL - 2010 AB - Sparse approximate inverses M which satisfy min M ∥ A M − I ∥ F have shown to be an attractive alternative to classicalsmoothers like Jacobi or Gauss-Seidel (Tang and Wan; 2000). The static and dynamic computation of a SAI and a SPAI(Grote and Huckle; 1997), respectively, comes along with advantages likeinherent parallelism and robustness with equal smoothingproperties (Bröker et al.; 2001). Here, we are interested indeveloping preconditioners that can incorporate probing conditionsfor improving the approximation relative to high- or low-frequencysubspaces. We present analytically derived optimal smoothers forthe discretization of the constant-coefficient Laplace operator. On this basis, we introduce probing conditions in thegeneralized Modified SPAI (MSPAI) approach (Huckle and Kallischko; 2007)which yields efficient smoothers for multigrid. In the secondpart, we transfer our approach to the domain of ill-posed problems torecover original information from blurred signals. Using the probingfacility of MSPAI, we impose the preconditioner to act asapproximately zero on the noise subspace. In combination with aniterative regularization method, it thus becomes possible toreconstruct the original information more accurately in many cases. A variety of numerical results demonstrate the usefulness of this approach. SN - 2090-0147 UR - https://doi.org/10.1155/2010/930218 DO - 10.1155/2010/930218 JF - Journal of Electrical and Computer Engineering PB - Hindawi Publishing Corporation KW - ER -