The magic of Fibonacci codes

summary:V článku se budeme zabývat Fibonacciho kódováním, které je díky proměnlivé délce kódu vhodnější (obzvláště pro kódování posloupnosti malých čísel) než např. užití dvojkové soustavy. Ukážeme, jak lze efektivně a bez použití obřích tabulek předzpracovaných dat Fibonacciho kódy dekódovat

Global optimization using space filling curves

The existence of space filling curves opens the way to reducing multivariate optimization problems to the minimization of univariate functions. In this paper, we analyze the Hoelder continuity of space filling curves and exploit this property in the solution of global optimization problems. Subsequently, an algorithm for minimizing univariate Hoelder continuous functions is presented and analyzed. It is shown that the algorithm computes the approximate minimum with the guaranteed precision. The algorithm is tested on some types of two-dimensional functions

Finding zeros of analytic functions and local eigenvalue analysis using contour integral method in examples

A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples

Separable spherical constraints and the decrease of a quadratic function in the gradient projection step

We examine the decrease of a strictly convex quadratic function along the projected-gradient path and show that our earlier estimates obtained for the bound constraints are valid for more general feasible sets including those defined by separable spherical constraints. The result is useful for the development of in a sense optimal algorithms for the solution of some QPQC problems with separable constraints and is an important ingredient in the development of scalable algorithms for contact problems with friction.Web of Science157114013

On R-linear convergence of semi-monotonic inexact augmented Lagrangians for saddle point problems

A variant of the inexact augmented Lagrangian algorithm called SMALE (Dostál in Comput. Optim. Appl. 38:47–59, 2007) for the solution of saddle point problems with a positive definite left upper block is studied. The algorithm SMALE-M presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary unconstrained problems by a multiple of the norm of the residual of the second block equation and a constant which is updated in order to enforce increase of the Lagrangian function. A nice feature of SMALE-M inherited from SMALE is its capability to find an approximate solution in a number of iterations that is bounded in terms of the extreme eigenvalues of the left upper block and does not depend on the off-diagonal blocks. Here we prove the R-linear rate of convergence of the outer loop of SMALE-M for any regularization parameter. The theory is illustrated by numerical experiments.Web of Science5811038

Bounds on the spectra of Schur complements of large H-TFETI-DP clusters for 2D Laplacian

Bounds on the spectrum of Schur complements of subdomain stiffness matrices of the discretized Laplacian with respect to interior variables are important in the convergence analysis of finite element tearing and interconnecting (FETI)-based domain decomposition methods. Here, we are interested in bounds on the regular condition number of Schur complements of "floating" clusters, that is, of matrices comprising the Schur complements of subdomains with prescribed zero Neumann conditions that are joined on the primal level by edge averages. Using some known results, angles of subspaces, and known bounds on the spectrum of Schur complements associated with square domains, we give bounds on the regular condition number of the Schur complement of some "floating" clusters arising from the discretization and decomposition of 2D Laplacian on domains comprising square subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m x m square subdomains joined by edge averages increases proportionally to m. The estimates are compared with numerical values and used in the analysis of H-FETI-DP methods. Though the research has been motivated by an effort to extend the scope of scalability of FETI-based solvers to variational inequalities, the experiments indicate that H-TFETI-DP with large clusters can be useful for the solution of huge linear elliptic problems discretized by sufficiently regular grids.Web of Scienceart. no. e234

On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian

summary:Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities

On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.Web of Science62671869

On the solution of convex QPQC problems with elliptic and other separable constraints with strong curvature

The paper deals with an effective implementation of some algorithms for the solution of convex QPQC problems with elliptic and other separable constraints with strong curvature. Here we discuss robust quantitative refinement of the Karush–Kuhn–Tucker conditions, extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components, and plug them into the existing algorithms for the solution of the QPQC problems with R-linear rate of convergence in the bounds on the spectrum. The results are then extended to the problems with separable inequality and linear equality constraints. The performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic Tresca and Coulomb friction discretized by up to one and half million nodal variables.Web of Science24786484