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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 (Tang and Wan; 2000). The static and dynamic computation of a SAI and a SPAI (Grote and Huckle; 1997), respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties (Bröker et al.; 2001). 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 (Huckle and Kallischko; 2007) 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. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new method for defining preconditioners for the iterative solution of a system of linear equations. By generalizing the class of modified preconditioners (e.g. MILU), the interface probing, and the class of preconditioners related to the Frobenius norm minimization (e.g. FSAI, SPAI) we develop a toolbox for computing preconditioners that are improved relative to a given small probing subspace. Furthermore, by this MSPAI (modified SPAI) probing approach we can improve any given preconditioner with respect to this probing subspace. All the computations are embarrassingly parallel. Additionally, for symmetric linear system we introduce new techniques for symmetrizing preconditioners. Many numerical examples, e.g. from PDE applications such as domain decomposition and Stokes problem, show that these new preconditioners often lead to faster convergence and smaller condition numbers. [ABSTRACT FROM AUTHOR]

This paper reviews relevant international and New Zealand literature, policy documents and reports on respite provision for children who are dying and their families. The literature describes why respite care for children is necessary, and evaluates present respite services in accord with recent reports and literature. The service provisions needed for dying children and their families are explored, including suggestions for the improvement of services and future research. The literature reveals that, although respite care for terminally ill children and their families is necessary, service provisions are uncoordinated, not always culturally sensitive and very limited for children and their families. Improvement in paediatric respite services is necessary and research needs to be conducted in this area to facilitate the development of more appropriate respite services for children and their families. [ABSTRACT FROM AUTHOR]