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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...

Gradient-free and gradient-based methods for shape optimization of water turbine blade

Bastl, Bohumír, Brandner, Marek, Egermaier, Jiří, Horníková, Hana, Michálková, Kristýna, Turnerová, Eva (2019)

Programs and Algorithms of Numerical Mathematics

The purpose of our work is to develop an automatic shape optimization tool for runner wheel blades in reaction water turbines, especially in Kaplan turbines. The fluid flow is simulated using an in-house incompressible turbulent flow solver based on recently introduced isogeometric analysis (see e.g. J. A. Cotrell et al.: Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009). The proposed automatic shape optimization approach is based on a so-called hybrid optimization which combines...

Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

Martin Ladecký, Ivana Pultarová, Jan Zeman (2021)

Applications of Mathematics

A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite...

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.

Currently displaying 561 – 580 of 1340