Smoothing and Regularization with Modified Sparse Approximate Inverses
Sparse approximate inverses 𝑀 which satisfy min𝑀‖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 lik...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
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| Series: | Journal of Electrical and Computer Engineering |
| Online Access: | http://dx.doi.org/10.1155/2010/930218 |
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| Summary: | Sparse approximate inverses 𝑀 which satisfy min𝑀‖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. |
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| ISSN: | 2090-0147 2090-0155 |