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Programs and Algorithms of Numerical Mathematics 17 : Dolní Maxov, June 8-13, 2014 : Proceedings of Seminar
Chleboun, J. ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
This volume comprises peer-reviewed papers that are based on invited lectures, survey lectures, short communications, and posters presented at the 17th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Dolní Maxov, Czech Republic, June 8–13, 2014. The seminar was organized by the Institute of Mathematics of the Academy of Sciences of the Czech Republic. It continued the previous seminars on mathematical software and numerical methods held (with only one exception) biannually in\nAlšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, and Prague in the period 1983–2012. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, singleor multi-processor applications of computational methods, and new approaches to mathematical modeling.
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On Two Methods for the Parameter Estimation Problem with Spatio-Temporal FRAP Data
Papáček, Š. ; Jablonský, J. ; Matonoha, Ctirad
FRAP (Fluorescence Recovery After Photobleaching) is a measurement technique for determination of the mobility of fluorescent molecules (presumably due to the diffusion process) within the living cells. While the experimental setup and protocol are usually fixed, the method used for the model parameter estimation, i.e. the data processing step, is not well established. In order to enhance the quantitative analysis of experimental (noisy) FRAP data, we firstly formulate the inverse problem of model parameter estimation and then we focus on how the different methods of data pre- processing influence the confidence interval of the estimated parameters, namely the diffusion constant $p$. Finally, we present a preliminary study of two methods for the computation of a least-squares estimate $\hat{p}$ and its confidence interval.
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A Modified Limited-Memory BNS Method for Unconstrained Minimization Derived from the Conjugate Directions Idea
Vlček, Jan ; Lukšan, Ladislav
A modification of the limited-memory variable metric BNS method for large scale unconstrained optimization of the differentiable function $f:{\cal R}^N\to\cal R$ is considered, which consists in corrections (based on the idea of conjugate directions) of difference vectors for better satisfaction of the previous quasi-Newton conditions. In comparison with [11], more previous iterations can be utilized here. For quadratic objective functions, the improvement of convergence is the best one in some sense, all stored corrected difference vectors are conjugate and the quasi-Newton conditions with these vectors are satisfied. The algorithm is globally convergent for convex sufficiently smooth functions and our numerical experiments indicate its efficiency.
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Irregularity of turing patterns in the Thomas model with a unilateral term
Rybář, Vojtěch ; Vejchodský, Tomáš
In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This contrasts with the approximately symmetric and regular patterns of the classical Thomas model. In addition, the unilateral term yields Turing patterns even for smaller ratio of diffusion constants. These conclusions accord with the recent findings about the influence of the unilateral term in a model for mammalian coat patterns. This indicates that the observed effects of the unilateral term are general and apply to a variety of systems.
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Smooth approximation spaces based on a periodic system
Segeth, Karel
A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients obtained as the unique solution of a minimization problem. While the minimization guarantees the smoothness of the approximant and its derivatives, the constraints represent the interpolating or smoothing conditions at nodes. In the contribution, a special attention is paid to the periodic basis system $exp(-ii kx)$. A 1D numerical example is presented.
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An application of the BDDC method to the Navier-Stokes equations in 3-D cavity
Hanek, M. ; Šístek, Jakub ; Burda, P.
We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear equations is solved by Picard iteration. We explore the applicability of the Balancing Domain Decomposition by Constraints (BDDC) method to nonsymmetric problems arising from such linearisation. One step of BDDC is applied as the preconditioner for the stabilized variant of the biconjugate gradient (BiCGstab) method. We present results for a 3-D cavity problem computed on 32 cores of a parallel supercomputer.
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Different approaches to interface weights in the BDDC method in 3D
Čertíková, M. ; Šístek, Jakub ; Burda, P.
In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find relations among different choices of the interface weights and compare them numerically on model problems of the Poisson equation and linear elasticity in 3D. Problems with jumps in coefficients of material properties are considered and both regular and irregular interfaces between subdomains are tested.
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