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- W2500593911 abstract "Previous chapter Next chapter Other Titles in Applied Mathematics Iterative Methods for Sparse Linear Systems9. Preconditioned Iterationspp.261 - 281Chapter DOI:https://doi.org/10.1137/1.9780898718003.ch9PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt Although the methods seen in previous chapters are well founded theoretically, they are all likely to suffer from slow convergence for problems that arise from typical applications such as fluid dynamics and electronic device simulation. Preconditioning is a key ingredient for the success of Krylov subspace methods in these applications. This chapter discusses the preconditioned versions of the iterative methods already seen, but without being specific about the particular preconditioners used. The standard preconditioning techniques will be covered in the next chapter. 9.1 Introduction Lack of robustness is a widely recognized weakness of iterative solvers relative to direct solvers. This drawback hampers the acceptance of iterative methods in industrial applications despite their intrinsic appeal for very large linear systems. Both the efficiency and robustness of iterative techniques can be improved by using preconditioning. A term introduced in Chapter 4, preconditioning is simply a means of transforming the original linear system into one with the same solution, but that is likely to be easier to solve with an iterative solver. In general, the reliability of iterative techniques, when dealing with various applications, depends much more on the quality of the preconditioner than on the particular Krylov subspace accelerators used. We will cover some of these preconditioners in detail in the next chapter. This chapter discusses the preconditioned versions of the Krylov subspace algorithms already seen, using a generic preconditioner. To begin with, it is worthwhile to consider the options available for preconditioning a system. The first step in preconditioning is to find a preconditioning matrix M. The matrix M can be defined in many different ways but it must satisfy a few minimal requirements. From a practical point of view, the most important requirement for M is that it be inexpensive to solve linear systems Mx = b. Previous chapter Next chapter RelatedDetails Published:2003ISBN:978-0-89871-534-7eISBN:978-0-89871-800-3 https://doi.org/10.1137/1.9780898718003Book Series Name:Other Titles in Applied MathematicsBook Code:OT82Book Pages:xvii + 520Key words:sparse matrices, iterative methods, differential equations, partial-numerical solutions" @default.
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- W2500593911 title "9. Preconditioned Iterations" @default.
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