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Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

100%

Zlatev Z., Georgiev K.

Open Mathematics

2013

tom 11

nr 8

1510-1530

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.

Low rank Tucker-type tensor approximation to classical potentials

84%

Khoromskij B., Khoromskaia V.

Open Mathematics

2007

tom 5

nr 3

523-550

This paper investigates best rank-(r 1,..., r d) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r 1,..., r d) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r 1,..., r d) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example $$\tfrac{1}{{\left| {x - y} \right|}}, e^{ - \alpha \left| {x - y} \right|} , \tfrac{{e^{ - \left| {x - y} \right|} }}{{\left| {x - y} \right|}}$$ and $$\tfrac{{erf(|x|)}}{{|x|}}$$ with x, y ∈ ℝd. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.

Ograniczanie wyników

2 Open Mathematics

1 Georgiev K.

1 Khoromskaia V.

1 Khoromskij B.

1 Zlatev Z.

1 2013

1 2007

Wyniki wyszukiwania

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Low rank Tucker-type tensor approximation to classical potentials