Integral equations arise in many scienti c and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any eld to give rise to integral equations. Integral equations also has signi cant application in mathematical physics models, such as di rac- tion problems, scattering in quantum mechanics, conformal mapping and water waves. The Volterra's population growth model, biological species living together, propagation of stocked sh in a new lake, the heat transfer and the heat radiation are among many areas that are described by integral equations. For limited applicability of analytical techniques, the numer- ical solvers often are the only viable alternative. General computational techniques of solving integral equation involve discretization and generates equivalent system of linear equations. In most of the cases the discretization produces dense matrix. Multigrid methods are widely used to solve partial di erential equation. We discuss the multigrid algorithms to solve integral equations and propose usages of distributive relaxation and the Kaczmarz method. / Department of Mathematical Sciences

Thesis (Ph. D.)--Technische Universiteit Delft, 1989. / Summary also in Dutch. "Stellingen" (3 p.) inserted. Vita. Includes bibliographical references.

Thesis (M.S.)--Nuclear and Radiological Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Stacey, Weston M.; Committee Co-Chair: de Oliveira, Cassiano R.E.; Committee Member: Hertel, Nolan; Committee Member: van Rooijen, Wilfred F.G.

Thesis (Ph.D.)--Memorial University of Newfoundland, 2004. / Bibliography: leaves 170-183.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

abstract: From 2D planar MOSFET to 3D FinFET, the geometry of semiconductor devices is getting more and more complex. Correspondingly, the number of mesh grid points increases largely to maintain the accuracy of carrier transport and heat transfer simulations. By substituting the conventional uniform mesh with non-uniform mesh, one can reduce the number of grid points. However, the problem of how to solve governing equations on non-uniform mesh is then imposed to the numerical solver. Moreover, if a device simulator is integrated into a multi-scale simulator, the problem size will be further increased. Consequently, there exist two challenges for the current numerical solver. One is to increase the functionality to accommodate non-uniform mesh. The other is to solve governing physical equations fast and accurately on a large number of mesh grid points. This research rst discusses a 2D planar MOSFET simulator and its numerical solver, pointing out its performance limit. By analyzing the algorithm complexity, Multigrid method is proposed to replace conventional Successive-Over-Relaxation method in a numerical solver. A variety of Multigrid methods (standard Multigrid, Algebraic Multigrid, Full Approximation Scheme, and Full Multigrid) are discussed and implemented. Their properties are examined through a set of numerical experiments. Finally, Algebraic Multigrid, Full Approximation Scheme and Full Multigrid are integrated into one advanced numerical solver based on the exact requirements of a semiconductor device simulator. A 2D MOSFET device is used to benchmark the performance, showing that the advanced Multigrid method has higher speed, accuracy and robustness. / Dissertation/Thesis / Masters Thesis Materials Science and Engineering 2015

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the <i>p</i>-version of the fem. We propose several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. The proof uses interpretations of the <i>p</i>-version element stiffness matrix and mass matrix on [-1,1] as <i>h</i>-version stiffness matrix and weighted mass matrix. The analysis requires wavelet methods.

This thesis treats two essentially different subjects: V-cycle schemes are considered in Chapters 2-4, whereas the aggregation-based coarsening is analysed in Chapters 5-6. As a matter of paradox, these two multigrid ingredients, when combined together, can hardly lead to an optimal algorithm. Indeed, a V-cycle needs more accurate prolongations than the simple piecewise-constant one, associated to aggregation-based coarsening. On the other hand, aggregation-based approaches use almost exclusively piecewise constant prolongations, and therefore need more involved cycling strategies, K-cycle <a href=http://www3.interscience.wiley.com/journal/114286660/abstract?CRETRY=1&SRETRY=0>[Num.Lin.Alg.Appl. vol.15(2008), pp.473-487]</a> being an attractive alternative in this respect.<p><br><p><br><p>Chapter 2 considers more precisely the well-known V-cycle convergence theories: the approximation property based analyses by Hackbusch (see [Multi-Grid Methods and Applications, 1985, pp.164-167]) and by McCormick [SIAM J.Numer.Anal. vol.22(1985), pp.634-643] and the successive subspace correction theory, as presented in [SIAM Review, vol.34(1992), pp.581-613] by Xu and in [Acta Numerica, vol.2(1993), pp.285-326.] by Yserentant. Under the constraint that the resulting upper bound on the convergence rate must be expressed with respect to parameters involving two successive levels at a time, these theories are compared. Unlike [Acta Numerica, vol.2(1993), pp.285-326.], where the comparison is performed on the basis of underlying assumptions in a particular PDE context, we compare directly the upper bounds. We show that these analyses are equivalent from the qualitative point of view. From the quantitative point of view,<p>we show that the bound due to McCormick is always the best one.<p><br><p><br><p>When the upper bound on the V-cycle convergence factor involves only two successive levels at a time, it can further be compared with the two-level convergence factor. Such comparison is performed in Chapter 3, showing that a nice two-grid convergence (at every level) leads to an optimal McCormick's bound (the best bound from the previous chapter) if and only if a norm of a given projector is bounded on every level.<p><br><p><br><p>In Chapter 4 we consider the Fourier analysis setting for scalar PDEs and extend the comparison between two-grid and V-cycle multigrid methods to the smoothing factor. In particular, a two-sided bound involving the smoothing factor is obtained that defines an interval containing both the two-grid and V-cycle convergence rates. This interval is narrow when an additional parameter α is small enough, this latter being a simple function of Fourier components.<p><br><p><br><p>Chapter 5 provides a theoretical framework for coarsening by aggregation. An upper bound is presented that relates the two-grid convergence factor with local quantities, each being related to a particular aggregate. The bound is shown to be asymptotically sharp for a large class of elliptic boundary value problems, including problems with anisotropic and discontinuous coefficients.<p><br><p><br><p>In Chapter 6 we consider problems resulting from the discretization with edge finite elements of 3D curl-curl equation. The variables in such discretization are associated with edges. We investigate the performance of the Reitzinger and Schöberl algorithm [Num.Lin.Alg.Appl. vol.9(2002), pp.223-238], which uses aggregation techniques to construct the edge prolongation matrix. More precisely, we perform a Fourier analysis of the method in two-grid setting, showing its optimality. The analysis is supplemented with some numerical investigations. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Cette thèse se concentre sur l'étude d'un algorithme multi niveaux de régions de confiance en norme infinie, conçu pour la résolution de problèmes d'optimisation non linéaires de grande taille pouvant être soumis a des contraintes de bornes. L'étude est réalisée tant sur le plan théorique que numérique. L'algorithme RMTR8 que nous étudions ici a été élaboré a partir de l'algorithme présente par Gratton, Sartenaer et Toint (2008b), et modifie d'abord en remplaçant l'usage de la norme Euclidienne par une norme infinie, et ensuite en l'adaptant a la résolution de problèmes de minimisation soumis a des contraintes de bornes. Dans un premier temps, les spécificités du nouvel algorithme sont exposées et discutées. De plus, l'algorithme est démontré globalement convergent au sens de Conn, Gould et Toint (2000), c'est-a-dire convergent vers un minimum local au départ de tout point admissible. D'autre part, il est démontre que la propriété d'identification des contraintes actives des méthodes de régions de confiance basées sur l'utilisation d'un point de Cauchy peut être étendue a tout solveur interne respectant une décroissance suffisante. En conséquence, cette propriété d'identification est aussi respectée par une variante particulière du nouvel algorithme. Par la suite, nous étudions différents critères d'arrêt pour les algorithmes d'optimisation avec contraintes de bornes afin de déterminer le sens et les avantages de chacun, et ce pour pouvoir choisir aisément celui qui convient le mieux a certaines situations. En particulier, les critères d'arrêts sont analyses en termes d'erreur inverse (backward erreur), tant au sens classique du terme (avec l'usage d'une norme produit) que du point de vue de l'optimisation multicritères. Enfin, un algorithme pratique est mis en place, utilisant en particulier une technique similaire au lissage de Gauss-Seidel comme solveur interne. Des expérimentations numériques sont réalisées sur une version FORTRAN 95 de l'algorithme. Elles permettent d'une part de définir un panel de paramètres efficaces par défaut et, d'autre part, de comparer le nouvel algorithme a d'autres algorithmes classiques d'optimisation, comme la technique de raffinement de maillage ou la méthode du gradient conjugue, sur des problèmes avec et sans contraintes de bornes. Ces comparaisons numériques semblent donner l'avantage à l'algorithme multi niveaux, en particulier sur les cas peu non-linéaires, comportement attendu de la part d'un algorithme inspire des techniques multi grilles. En conclusion, l'algorithme de région de confiance multi niveaux présente dans cette thèse est une amélioration du précédent algorithme de cette classe d'une part par l'usage de la norme infinie et d'autre part grâce a son traitement de possibles contraintes de bornes. Il est analyse tant sur le plan de la convergence que de son comportement vis-à-vis des bornes, ou encore de la définition de son critère d'arrêt. Il montre en outre un comportement numérique prometteur. / This thesis concerns the study of a multilevel trust-region algorithm in infinity norm, designed for the solution of nonlinear optimization problems of high size, possibly submitted to bound constraints. The study looks at both theoretical and numerical sides. The multilevel algorithm RMTR8 that we study has been developed on the basis of the algorithm created by Gratton, Sartenaer and Toint (2008b), which was modified first by replacing the use of the Euclidean norm by the infinity norm and also by adapting it to solve bound-constrained problems. In a first part, the main features of the new algorithm are exposed and discussed. The algorithm is then proved globally convergent in the sense of Conn, Gould and Toint (2000), which means that it converges to a local minimum when starting from any feasible point. Moreover, it is shown that the active constraints identification property of the trust-region methods based on the use of a Cauchy step can be extended to any internal solver that satisfies a sufficient decrease property. As a consequence, this identification property also holds for a specific variant of our new algorithm. Later, we study several stopping criteria for nonlinear bound-constrained algorithms, in order to determine their meaning and their advantages from specific points of view, and such that we can choose easily the one that suits best specific situations. In particular, the stopping criteria are examined in terms of backward error analysis, which has to be understood both in the usual meaning (using a product norm) and in a multicriteria optimization framework. In the end, a practical algorithm is set on, that uses a Gauss-Seidel-like smoothing technique as an internal solver. Numerical tests are run on a FORTRAN 95 version of the algorithm in order to define a set of efficient default parameters for our method, as well as to compare the algorithm with other classical algorithms like the mesh refinement technique and the conjugate gradient method, on both unconstrained and bound-constrained problems. These comparisons seem to give the advantage to the designed multilevel algorithm, particularly on nearly quadratic problems, which is the behavior expected from an algorithm inspired by multigrid techniques. In conclusion, the multilevel trust-region algorithm presented in this thesis is an improvement of the previous algorithm of this kind because of the use of the infinity norm as well as because of its handling of bound constraints. Its convergence, its behavior concerning the bounds and the definition of its stopping criteria are studied. Moreover, it shows a promising numerical behavior.

In this thesis, techniques for efficient utilization of modern computer hardwarefor numerical simulation are considered. In particular, we study techniques for improving the performance of computations using the finite element method. One of the main difficulties in finite-element computations is how to perform the assembly of the system matrix efficiently in parallel, due to its complicated memory access pattern. The challenge lies in the fact that many entries of the matrix are being updated concurrently by several parallel threads. We consider transactional memory, an exotic hardware feature for concurrent update of shared variables, and conduct benchmarks on a prototype multicore processor supporting it. Our experiments show that transactions can both simplify programming and provide good performance for concurrent updates of floating point data. Secondly, we study a matrix-free approach to finite-element computation which avoids the matrix assembly. In addition to removing the need to store the system matrix, matrix-free methods are attractive due to their low memory footprint and therefore better match the architecture of modern processors where memory bandwidth is scarce and compute power is abundant. Motivated by this, we consider matrix-free implementations of high-order finite-element methods for execution on graphics processors, which have seen a revolutionary increase in usage for numerical computations during recent years due to their more efficient architecture. In the implementation, we exploit sum-factorization techniques for efficient evaluation of matrix-vector products, mesh coloring and atomic updates for concurrent updates, and a geometric multigrid algorithm for efficient preconditioning of iterative solvers. Our performance studies show that on the GPU, a matrix-free approach is the method of choice for elements of order two and higher, yielding both a significantly faster execution, and allowing for solution of considerably larger problems. Compared to corresponding CPU implementations executed on comparable multicore processors, the GPU implementation is about twice as fast, suggesting that graphics processors are about twice as power efficient as multicores for computations of this kind.