Implementation of the deflated variants of the conjugate gradient method

Sdružené gradienty jsou jednou z nejpoužívanějších metod pro řešení rozsáhlých soustav lineárních rovnic se symetrickou pozitivně-semidefinitní maticí. Jeden ze způsobů urychlení konvergence metody je deflace. Principem deflace je skrývání té části spektra matice, která způsobuje zpomalení konvergence. Tato diplomová práce se zabývá efektivní implementací různých deflated verzí sdružených gradientů. Velká pozornost je také věnována teorii a volbě deflačního prostoru. Možnosti implementace jsou demonstrovány na rozsáhlém množství příkladůThe conjugate gradient algorithm is one of the most popular methods for the solution of large systems of linear equations with symmetric positive semi-definite matrix. One of the schemes accelerating the convergence of conjugate gradients is deflation which effectively hides parts of the matrix spectrum that slows down the convergence. This master's thesis deals with efficient parallel implementation of the deflated conjugate gradient method with various modifications. Detailed theoretical considerations and the crucial choice of the deflation space are also discussed. The implementation is showcased on a wide range of benchmarks9600 - IT4Innovationsvýborn

The Veneration of Saints and Their Graves in Muslim India of Premughal and Mughal Period

This thesis handles a topic of veneration of Muslim saints and their graves with emphasis on the area of Central India in Medieval period. In the first part of the thesis a process of spreading of Islam to Deccan is described and a role of sufis in it. The second part is dedicated to the concept of saints in Islam as a whole and in regard to Indian context. The third part is an introduction of the bhaktic movement of Sants in which I try to demostrate its similarities with Sufism. The shared theme of those tradition is similar concept of sanctity of persons and a their approach to the veneration of saints. Key words Saints, Veneration of Saints, Islam, Sufism, Islamic mysticismTato bakalářská práce se zabývá tématem uctívání muslimských světců a jejich hrobů s důrazem na oblast střední Indie ve středověku. Cílem této práce je nastínit problematiku uctívání světců v islámu a najít některé styčné prvky s místními hinduistickými tradicemi. V první části je zevrubně přiblížen průběh pronikání islámu do Dekkanu a popsána role, kterou v této fázi vývoje sehráli súfíjové. Druhá část je věnována pojetí světců v islámu obecně, s přihlédnutím k indickému kontextu. V třetí části bakalářské práce představuji bhaktické hnutí santů, zvláštní pozornost je věnována podobnosti tohoto hnutí se súfismem. Spojujícím tématem těchto tradic je obdobná představa svatosti osob a přístup k uctívání světců, včetně jejich hrobů. Klíčová slova Světci, uctívání světců, islám, súfismus, islámská mystika,Ústav filosofie a religionistikyInstitute of Philosophy and Religious StudiesFilozofická fakultaFaculty of Art

Parallelizations of TFETI-1 coarse problem

Import 03/11/2016Metody založené na FETI, používané pro řešení eliptických parcialních diferencialních rovnic, představují velmi úspěšnou třídu metod dekompozice oblasti, které se používají pro paralelizaci dobře známých metod konečných prvků. Původní problém ve FETI methodách je rozdělen na menší problémy definované na podoblastech. Díky tomu, že se podoblasti nepřekrývají, můžeme menší problémy nezávisle na sobě řešit paralelně. Počet podoblastí cheme zvyšovat tak, aby se menší problémy řešily rychleji. To ale zároveň vede k růstu velikosti hrubého problému. Pro složité problémy je navíc potřeba řešit hrubý problém mnohokrát. Díky tomu je potřeba najít řešení hrubého problému co nejefektivněji. Tato práce se zabývá paralelnímy strategiemi řešení hrubého problému TFETI--1 metody.The FETI based methods, used for the solution of elliptical partial differential equations, form a highly successful class of domain decomposition methods used for parallelization of well known finite element methods. In the FETI methods we partition the original problem into smaller problems defined on subdomains. Since the subdomains are non-overlapping we can naturally solve the smaller problems independently in parallel. We want to increase the number of subdomains so that the smaller problems are solved faster. This however leads to the increase in the size of the coarse problem. Moreover, for complex problems, the number of coarse problem solutions needed can be very high. Therefore, it is important to find the solution of the coarse problem efficiently. This thesis deals with parallelization strategies of the TFETI--1 coarse problem.470 - Katedra aplikované matematikyvýborn

The energy consumption optimization of the BLAS routines

The paper deals with the energy consumption evaluation of selected Sparse and Dense BLAS Level 1, 2 and 3 routines. Authors employed AXPY, Sparse Matrix-Vector, Sparse Matrix-Matrix, Dense Matrix-Vector, Dense Matrix-Matrix and Sparse Matrix-Dense Matrix multiplication routines from Intel Math Kernel Library (MKL). The measured characteristics illustrate the different energy consumption of BLAS routines, as some operations are memory-bounded and others are compute-bounded. Based on their recommendations one can explore dynamic frequency switching to achieve significant energy savings up to 23%

Non-monotone projected gradient method in linear elasticity contact problems with given friction

We are focusing on the algorithms for solving the large-scale convex optimization problem in linear elasticity contact problems discretized by Finite Element method (FEM). The unknowns of the problem are the displacements of the FEM nodes, the corresponding objective function is defined as a convex quadratic function with symmetric positive definite stiffness matrix and additional non-linear term representing the friction in contact. The feasible set constraints the displacement subject to non-penetration conditions. The dual formulation of this optimization problem is well-known as a Quadratic Programming (QP) problem and can be considered as a most basic non-linear optimization problem. Understanding these problems and the development of efficient algorithms for solving them play the crucial role in the large-scale problems in practical applications. We shortly review the theory and examine the behavior and the efficiency of Spectral Projected Gradient method modified for QP problems (SPG-QP) on the solution of a toy example in MATLAB environment.Web of Science1220art. no. 867

Суброгация

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Глобализация, региональная интеграция и экономическое развитие

The paper deals with the energy consumption evaluation of the Finite Element Tearing and Interconnect (FETI) based solvers of linear systems, which is an established method for solving real-world engineering problems. Authors evaluated the effect of the CPU frequency on the energy consumption of the FETI solver using a linear elasticity 3D cube synthetic benchmark. In this problem, the effect of frequency tuning on the energy consumption of the essential processing kernels of the FETI method was evaluated. The paper provides results for two types of frequency tuning: (1) static tuning and (2) dynamic tuning. For static tuning experiments, the frequency is set before execution and kept constant during the runtime. For dynamic tuning, the frequency is changed during the program execution to adapt the system to the actual needs of the application. The paper shows that static tuning brings up 12% energy savings when compared to default CPU settings (the highest clock rate). The dynamic tuning improves this further by up to 3%

The Veneration of Saints and Their Graves in Muslim India of Premughal and Mughal Period

This thesis handles a topic of veneration of Muslim saints and their graves with emphasis on the area of Central India in Medieval period. In the first part of the thesis a process of spreading of Islam to Deccan is described and a role of sufis in it. The second part is dedicated to the concept of saints in Islam as a whole and in regard to Indian context. The third part is an introduction of the bhaktic movement of Sants in which I try to demostrate its similarities with Sufism. The shared theme of those tradition is similar concept of sanctity of persons and a their approach to the veneration of saints. Key words Saints, Veneration of Saints, Islam, Sufism, Islamic mysticis

Comparison of selected FETI coarse space projector implementation strategies

This paper deals with scalability improvements of the FETI (Finite Element Tearing and Interconnecting) domain decomposition method solving elliptic PDEs. The main bottleneck of FETI is the solution of a coarse problem that is part of the projector onto the natural coarse space. This paper introduces and compares two strategies for the FETI coarse problem solution. The first one is a classical solution with either direct (factorization + forward/backward substitutions) or iterative solvers (conjugate gradient and deflated conjugate gradient methods). The second one is the assembly of an explicit inverse using a direct solver with the coarse problem solution realised by dense matrix-vector products. MPI subcommunicators are employed to increase arithmetic intensity and, crucially, to decrease the communication cost. PERMON library for quadratic programming implementing the Total FETI variant of FETI was used for the numerical experiments.Web of Science93art. no. UNSP 10260