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Gradient descent and fast artificial time integration

Uri M. Ascher, Kees van den Doel, Hui Huang, Benar F. Svaiter (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods using bolder step size selection exist. Various alternatives are investigated from both theoretical and practical...

IDR explained.

Gutknecht, Martin H. (2009)

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

Improved convergence bounds for smoothed aggregation method: linear dependence of the convergence rate on the number of levels

Jan Brousek, Pavla Fraňková, Petr Vaněk (2016)

Czechoslovak Mathematical Journal

The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear...

Improved convergence estimate for a multiply polynomially smoothed two-level method with an aggressive coarsening

Radek Tezaur, Petr Vaněk (2018)

Applications of Mathematics

A variational two-level method in the class of methods with an aggressive coarsening and a massive polynomial smoothing is proposed. The method is a modification of the method of Section 5 of Tezaur, Vaněk (2018). Compared to that method, a significantly sharper estimate is proved while requiring only slightly more computational work.

Inequality-based approximation of matrix eigenvectors

András Kocsor, József Dombi, Imre Bálint (2002)

International Journal of Applied Mathematics and Computer Science

A novel procedure is given here for constructing non-negative functions with zero-valued global minima coinciding with eigenvectors of a general real matrix A. Some of these functions are distinct because all their local minima are also global, offering a new way of determining eigenpairs by local optimization. Apart from describing the framework of the method, the error bounds given separately for the approximation of eigenvectors and eigenvalues provide a deeper insight into the fundamentally...

Influence of preconditioning and blocking on accuracy in solving Markovian models

Beata Bylina, Jarosław Bylina (2009)

International Journal of Applied Mathematics and Computer Science

The article considers the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. The paper considers some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination will be considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method...

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