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Given a sparse matrix A the SPAI Algorithm computes a sparse approximate inverse
M by minimizing || AM - I || in the Frobenius norm. The approximate inverse is
computed explicitly and can then be applied as a preconditioner to an iterative
method.The sparsity pattern of the approximate inverse is either fixed a priori
or captured automatically:
* Fixed sparsity: The sparsity pattern of M is either banded or a subset
of the sparsity pattern of A.
* Adaptive sparsity: The algorithm proceeds until the 2-norm of each column
of AM-I is less than eps. By varying eps the user controls the quality and
the cost of computing the preconditioner. Usually the optimal eps lies between 0.5 and 0.7.
A very sparse preconditioner is very cheap to compute but may not lead to much
improvement, while if M becomes rather dense it becomes too expensive to
compute. The optimal preconditioner lies between these two extremes and is
problem and computer architecture dependent. The approximate inverse M can also
be used as a robust (parallel) smoother for (algebraic) multi-grid methods
17 lines
1.1 KiB
Text
17 lines
1.1 KiB
Text
Given a sparse matrix A the SPAI Algorithm computes a sparse approximate inverse
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M by minimizing || AM - I || in the Frobenius norm. The approximate inverse is
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computed explicitly and can then be applied as a preconditioner to an iterative
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method.The sparsity pattern of the approximate inverse is either fixed a priori
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or captured automatically:
|
|
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* Fixed sparsity: The sparsity pattern of M is either banded or a subset
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of the sparsity pattern of A.
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|
* Adaptive sparsity: The algorithm proceeds until the 2-norm of each column
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of AM-I is less than eps. By varying eps the user controls the quality and
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the cost of computing the preconditioner. Usually the optimal eps lies between 0.5 and 0.7.
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A very sparse preconditioner is very cheap to compute but may not lead to much
|
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improvement, while if M becomes rather dense it becomes too expensive to
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compute. The optimal preconditioner lies between these two extremes and is
|
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problem and computer architecture dependent. The approximate inverse M can also
|
|
be used as a robust (parallel) smoother for (algebraic) multi-grid methods
|