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Massive parallel implementation of ODE solvers

Fischer, Cyril (2013)

Programs and Algorithms of Numerical Mathematics

The presented contribution maps the possibilities of exploitation of the massive parallel computational hardware (namely GPU) for solution of the initial value problems of ordinary differential equations. Two cases are discussed: parallel solution of a single ODE and parallel execution of scalar ODE solvers. Whereas the advantages of the special architecture in the case of a single ODE are problematic, repeated solution of a single ODE for different data can profit from the parallel...

Maximal solutions of two–sided linear systems in max–min algebra

Pavel Krbálek, Alena Pozdílková (2010)

Kybernetika

Max-min algebra and its various aspects have been intensively studied by many authors [1, 4] because of its applicability to various areas, such as fuzzy system, knowledge management and others. Binary operations of addition and multiplication of real numbers used in classical linear algebra are replaced in max-min algebra by operations of maximum and minimum. We consider two-sided systems of max-min linear equations A ⊗ x = B ⊗ x , with given coefficient matrices A and B . We present a polynomial method for...

Mixed precision GMRES-based iterative refinement with recycling

Oktay, Eda, Carson, Erin (2023)

Programs and Algorithms of Numerical Mathematics

With the emergence of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes for solving linear systems A x = b have recently been developed. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than the alternative of recomputing the LU factors in a higher precision. In this work, we incorporate the idea of Krylov subspace recycling, a well-known technique for reusing information across sequential invocations,...

Mobile sensor routing for parameter estimation of distributed systems using the parallel tunneling method

Tomasz Zięba, Dariusz Uciński (2008)

International Journal of Applied Mathematics and Computer Science

The paper deals with the problem of optimal path planning for a sensor network with mutliple mobile nodes, whose measurements are supposed to be primarily used to estimate unknown parameters of a system modelled by a partial differential equation. The adopted framework permits to consider two- or three-dimensional spatial domains and correlated observations. Since the aim is to maximize the accuracy of the estimates, a general functional defined on the relevant Fisher information matrix is used...

Modelling of multicomponent diffusive phase transformation in solids

Vala, Jiří (2008)

Programs and Algorithms of Numerical Mathematics

Physical analysis of phase transformation of materials consisting from several (both substitutional and interstitial) components, coming from the Onsager extremal thermodynamic principle, leads, from the mathematical point of view, to a system of partial differential equations of evolution type, including certain integral term, with substantial differences in particular phases ( α , γ ) and in moving interface of finite thickness ( β ), in whose center the ideal liquid material behaviour can be detected....

Modifications of the limited-memory BFGS method based on the idea of conjugate directions

Vlček, Jan, Lukšan, Ladislav (2013)

Programs and Algorithms of Numerical Mathematics

Simple modifications of the limited-memory BFGS method (L-BFGS) for large scale unconstrained optimization are considered, which consist in corrections of the used difference vectors (derived from the idea of conjugate directions), utilizing information from the preceding iteration. For quadratic objective functions, the improvement of convergence is the best one in some sense and all stored difference vectors are conjugate for unit stepsizes. The algorithm is globally convergent for convex sufficiently...

Multi-parameter asymptotic error resolution of the mixed finite element method for the Stokes problem

Aihui Zhou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a multi-parameter error resolution technique is applied into a mixed finite element method for the Stokes problem. By using this technique and establishing a multi-parameter asymptotic error expansion for the mixed finite element method, an approximation of higher accuracy is obtained by multi-processor computers in parallel.

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