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Iterative Methods
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
One way to do this is to simply prohibit fill-in. We allow an entry of L˜ or U˜ to be nonzero only if the corresponding entry of A is nonzero. More generally, we might specify a range of entries of L˜ and U˜ that may be filled in, e.g., up to the first three subdiagonals and up to the first three superdiagonals. Any such factorization is known as incomplete LU factorization, leading to an incomplete LU preconditionerP=L˜U˜ (often referred to simply as an ILU method). The corresponding technique for positive definite matrices is incomplete Cholesky factorization and hence an incomplete Cholesky preconditioner.
Newton projection method as applied to assembly simulation
Published in Optimization Methods and Software, 2022
S. Baklanov, M. Stefanova, S. Lupuleac
Regarding CG method it is possible to use preconditioners to improve the convergence and reduce the number of iterations . We consider two preconditioners. First one is a simple inverse diagonal preconditioner, which can be applied at no cost. Second one is incomplete Cholesky factorization (ICF) for dense matrices [18]. It requires the selection of a sparsity pattern for matrix in some way, and then to perform an incomplete Cholesky factorization for the achieved sparse matrix. The preconditioner was constructed by keeping only elements that are bigger than the threshold τ while keeping the matrix positive denite by increasing diagonal elements [15]. Namely, we keep i, j element if it satisfies with . Table 1 compares the number of iterations and computation time for CG with high accuracy for different preconditioners and for Cholesky decomposition for different matrices , emerging during NPM iterations. Both preconditioners reduce solving time, especially for the relative problem. ICF works slightly faster than inverse diagonal preconditioner for the relative problem, but a little slower for the dual problem as dual matrix usually is less diagonal dominant and requires bigger sparsity patterns to improve convergence of the CG method.