Sep 23, 2014
Tomáš Gergelits, Charles University in Prague, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic Joint work with Zdenek Strakoš The method of conjugate gradients (CG) for solving linear systems of algebraic equations with a Hermitian and positive definite matrix A is computationally based on short recurrences. Assuming exact arithmetic, they ensure the global orthogonality of the residual vectors, which span at the l-th step the l-dimensional Krylov subspace Kl(A; r0) = span n r0; Ar0;A2r0; : : : ;Al1r0 o : In practical computations, however, the use of short recurrences inevitably leads to the loss of the global orthogonality and even linear independence among the computed residual vectors. Consequently, the computed Krylov subspaces can be “rank-deficient” which causes a significant delay of convergence in finite precision CG computations. In the contribution we address the question how the Krylov subspaces generated by the CG method in finite precision arithmetic differ from their exact arithmetic counterparts. After recalling the related results published previously we concentrate on the situation with a significant delay of convergence and describe the distance between Krylov subspaces of the appropriately determined equal dimension. We observe that the finite precision Krylov subspaces in this comparison do not substantially deviate from the exact Krylov subspaces.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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