Relevant bibliographies by topics / Right Hand Side iteration

Academic literature on the topic 'Right Hand Side iteration'

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Published: 6 September 2023

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Journal articles on the topic "Right Hand Side iteration"

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Freitag, Melina A., Patrick Kürschner, and Jennifer Pestana. "GMRES Convergence Bounds for Eigenvalue Problems." Computational Methods in Applied Mathematics 18, no. 2 (April 1, 2018): 203–22. http://dx.doi.org/10.1515/cmam-2017-0017.

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AbstractThe convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right-hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right-hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right-hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g., tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.
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Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.
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Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.
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Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.
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Dhage, B. C., and S. Heikkilä. "On nonlinear boundary value problems with deviating arguments and discontinuous right hand side." Journal of Applied Mathematics and Stochastic Analysis 6, no. 1 (January 1, 1993): 83–91. http://dx.doi.org/10.1155/s1048953393000085.

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In this paper we shall study the existence of the extremal solutions of a nonlinear boundary value problem of a second order differential equation with general Dirichlet/Neumann form boundary conditions. The right hand side of the differential equation is assumed to contain a deviating argument, and it is allowed to possess discontinuities in all the variables. The proof is based on a generalized iteration method.
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Lampe, J., and H. Voss. "GLOBAL CONVERGENCE OF RTLSQEP: A SOLVER OF REGULARIZED TOTAL LEAST SQUARES PROBLEMS VIA QUADRATIC EIGENPROBLEMS." Mathematical Modelling and Analysis 13, no. 1 (March 31, 2008): 55–66. http://dx.doi.org/10.3846/1392-6292.2008.13.55-66.

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The total least squares (TLS) method is a successful approach for linear problems if both the matrix and the right hand side are contaminated by some noise. In a recent paper Sima, Van Huffel and Golub suggested an iterative method for solving regularized TLS problems, where in each iteration step a quadratic eigenproblem has to be solved. In this paper we prove its global convergence, and we present an efficient implementation using an iterative projection method with thick updates.
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Liu, Chein-Shan. "A Feasible Approach to Determine the Optimal Relaxation Parameters in Each Iteration for the SOR Method." Journal of Mathematics Research 13, no. 1 (January 7, 2021): 1. http://dx.doi.org/10.5539/jmr.v13n1p1.

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The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.
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Liu, Chein-Shan. "A Feasible Approach to Determine the Optimal Relaxation \\[6pt]Parameters in Each Iteration for the SOR Method." Journal of Mathematics Research 13, no. 1 (January 7, 2021): 1. http://dx.doi.org/10.5539/jmr.v13n1p1.

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The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.
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Laguta, V. V. "THE ITERATION METHOD OF THE FIRST BOUNDARY-VALUE PROBLEM SOLUTION WITH NONLINEAR RIGHT-HAND SIDE." Science and Transport Progress, no. 18 (October 25, 2007): 95–99. http://dx.doi.org/10.15802/stp2007/17452.

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The numerical method of the decision of the first boundery problem with the nonlinear ordinary differential equation of the second order is offered. The initial problem is reduced to the decision of the two integrated equations with use Green's function. The system is solved Picard's method.
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Aghamiry, Hossein S., Ali Gholami, and Stéphane Operto. "Accurate and efficient data-assimilated wavefield reconstruction in the time domain." GEOPHYSICS 85, no. 2 (January 30, 2020): A7—A12. http://dx.doi.org/10.1190/geo2019-0535.1.

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Wavefield reconstruction inversion (WRI) mitigates cycle skipping in full-waveform inversion by computing wavefields that do not exactly satisfy the wave equation to match data with inaccurate velocity models. We refer to these wavefields as data assimilated wavefields because they are obtained by combining the physics of wave propagation and the observations. Then, the velocity model is updated by minimizing the wave-equation errors, namely, the source residuals. Computing these data-assimilated wavefields in the time domain with explicit time stepping is challenging. This is because the right-hand side of the wave equation to be solved depends on the back-propagated residuals between the data and the unknown wavefields. To bypass this issue, a previously proposed approximation replaces these residuals by those between the data and the exact solution of the wave equation. This approximation is questionable during the early WRI iterations when the wavefields computed with and without data assimilation differ significantly. We have developed a simple backward-forward time-stepping recursion to refine the accuracy of the data-assimilated wavefields. Each iteration requires us to solve one backward and one forward problem, the former being used to update the right side of the latter. An application to the BP salt model indicates that a few iterations are enough to reconstruct data-assimilated wavefields accurately with a crude velocity model. Although this backward-forward recursion leads to increased computational overheads during one WRI iteration, it preserves its capability to extend the search space.
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Dissertations / Theses on the topic "Right Hand Side iteration"

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Balsa, Carlos Jorge da Rocha. "Inexact Subspace Iteration to Accelerate the Solution of Linear Systems with Multiple Right-Hand Sides." Phd thesis, Toulouse, INPT, 2006. http://oatao.univ-toulouse.fr/7470/1/balsa.pdf.

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We propose a two-phase acceleration technique for the solution of Symmetric and Positive Definite (SPD) linear systems with multiple right-hand sides. In the first phase we compute some partial spectral information related to the ill conditioned part of the given coefficient matrix and, in the second phase, we use this information to improve the convergence of the Conjugate Gradient (CG) algorithm. This approach is adequate for large scale problems, like the simulation of time dependent differential equations, where it is necessary to solve consecutively several linear systems with the same coefficient matrix (or with matrices that present very close spectral properties) but with changing right-hand sides. To compute the spectral information, we combine the block Conjugate Gradient algorithm with the Subspace Iteration to build a purely iterative algorithm, that we call Block-CGSI. We analyze the convergence of the blockCG algorithm and exploit the possibility of reducing the total amount of computational work by controlling in a same appropriate manner the accuracy during the solution of linear systems at each subspace iteration. We also improve the global convergence of this algorithm by using Chebyshev polynomials as a spectral filtering tool when building the starting vectors. The concept of “sliding window” was also introduced as an algorithmic feature that enables the computation of a near-invariant subspace of any dimension. The spectral information computed by the BlockCGSI algorithm is then used to remove the effect of the smallest eigenvalues in two different ways: either by building a Spectral Low Rank Update (SLRU) preconditioner that basically adds the value 1 to the approximated eigenvalues, or by performing a deflation of the initial residual in order to remove part of the solution corresponding to the smallest eigenvalues. Both techniques yield important reductions of the total number of iterations and computational work in each subsequent runs of the Conjugate Gradient algorithm. We report on experiments on a 2D diffusion equation as well as on two applications coming from Computational Fluid Dynamics (CFD). The analysis of costs and benefits, in terms of floating point operations, helps to validate the strategy as a good way to speed up the solution of symmetric and positive definite linear systems
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Balsa, Carlos Jorge da Rocha Dayde Michel Palma José Laginha. "Inexact subspace iteration to accelerate the solution of linear systems with multiple right-hand sides." Toulouse : INP Toulouse, 2008. http://ethesis.inp-toulouse.fr/archive/00000315.

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Reproduction de : Thèse de doctorat : Informatique et télécommunications : Toulouse, INPT : 2006. Reproduction de : Thèse de doctorat : Informatique et télécommunications : Porto, FEUP : 2006.
Thèse soutenue en co-tutelle. Titre provenant de l'écran-titre. Bibliogr. 47 réf.
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SCAGLIA, MICHELE. "NONTRIVIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2015. http://hdl.handle.net/10281/77845.

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The main subject of this thesis is the study, by a variational approach, of semilinear elliptic problems with measure data. Starting with a semilinear problem with unique solution, we introduce a parametrized perturbation and study the bifurcation phenomena giving rise to further solutions. The main feature is that we are able to use a direct variational approach, even when the semilinearity has no growth assumptions. In this setting, we prove bifurcation results in the line of classical results of Boehme-Marino and Rabinowitz and also global existence results.
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Nilsson, Martin. "Fast Numerical Techniques for Electromagnetic Problems in Frequency Domain." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3884.

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Slavova, Tzvetomila. "Résolution triangulaire de systèmes linéaires creux de grande taille dans un contexte parallèle multifrontal et hors-mémoire." Thesis, Toulouse, INPT, 2009. http://www.theses.fr/2009INPT016H/document.

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Nous nous intéressons à la résolution de systèmes linéaires creux de très grande taille par des méthodes directes de factorisation. Dans ce contexte, la taille de la matrice des facteurs constitue un des facteurs limitants principaux pour l'utilisation de méthodes directes de résolution. Nous supposons donc que la matrice des facteurs est de trop grande taille pour être rangée dans la mémoire principale du multiprocesseur et qu'elle a donc été écrite sur les disques locaux (hors-mémoire : OOC) d'une machine multiprocesseurs durant l'étape de factorisation. Nous nous intéressons à l'étude et au développement de techniques efficaces pour la phase de résolution après une factorization multifrontale creuse. La phase de résolution, souvent négligée dans les travaux sur les méthodes directes de résolution directe creuse, constitue alors un point critique de la performance de nombreuses applications scientifiques, souvent même plus critique que l'étape de factorisation. Cette thèse se compose de deux parties. Dans la première partie nous nous proposons des algorithmes pour améliorer la performance de la résolution hors-mémoire. Dans la deuxième partie nous pousuivons ce travail en montrant comment exploiter la nature creuse des seconds membres pour réduire le volume de données accédées en mémoire. Dans la première partie de cette thèse nous introduisons deux approches de lecture des données sur le disque dur. Nous montrons ensuite que dans un environnement parallèle le séquencement des tâches peut fortement influencer la performance. Nous prouvons qu'un ordonnancement contraint des tâches peut être introduit; qu'il n'introduit pas d'interblocage entre processus et qu'il permet d'améliorer les performances. Nous conduisons nos expériences sur des problèmes industriels de grande taille (plus de 8 Millions d'inconnues) et utilisons une version hors-mémoire d'un code multifrontal creux appelé MUMPS (solveur multifrontal parallèle). Dans la deuxième partie de ce travail nous nous intéressons au cas de seconds membres creux multiples. Ce problème apparaît dans des applications en electromagnétisme et en assimilation de données et résulte du besoin de calculer l'espace propre d'une matrice fortement déficiente, du calcul d'éléments de l'inverse de la matrice associée aux équations normales pour les moindres carrés linéaires ou encore du traitement de matrices fortement réductibles en programmation linéaire. Nous décrivons un algorithme efficace de réduction du volume d'Entrées/Sorties sur le disque lors d'une résolution hors-mémoire. Plus généralement nous montrons comment le caractère creux des seconds -membres peut être exploité pour réduire le nombre d'opérations et le nombre d'accès à la mémoire lors de l'étape de résolution. Le travail présenté dans cette thèse a été partiellement financé par le projet SOLSTICE de l'ANR (ANR-06-CIS6-010)
We consider the solution of very large systems of linear equations with direct multifrontal methods. In this context the size of the factors is an important limitation for the use of sparse direct solvers. We will thus assume that the factors have been written on the local disks of our target multiprocessor machine during parallel factorization. Our main focus is the study and the design of efficient approaches for the forward and backward substitution phases after a sparse multifrontal factorization. These phases involve sparse triangular solution and have often been neglected in previous works on sparse direct factorization. In many applications, however, the time for the solution can be the main bottleneck for the performance. This thesis consists of two parts. The focus of the first part is on optimizing the out-of-core performance of the solution phase. The focus of the second part is to further improve the performance by exploiting the sparsity of the right-hand side vectors. In the first part, we describe and compare two approaches to access data from the hard disk. We then show that in a parallel environment the task scheduling can strongly influence the performance. We prove that a constraint ordering of the tasks is possible; it does not introduce any deadlock and it improves the performance. Experiments on large real test problems (more than 8 million unknowns) using an out-of-core version of a sparse multifrontal code called MUMPS (MUltifrontal Massively Parallel Solver) are used to analyse the behaviour of our algorithms. In the second part, we are interested in applications with sparse multiple right-hand sides, particularly those with single nonzero entries. The motivating applications arise in electromagnetism and data assimilation. In such applications, we need either to compute the null space of a highly rank deficient matrix or to compute entries in the inverse of a matrix associated with the normal equations of linear least-squares problems. We cast both of these problems as linear systems with multiple right-hand side vectors, each containing a single nonzero entry. We describe, implement and comment on efficient algorithms to reduce the input-output cost during an outof- core execution. We show how the sparsity of the right-hand side can be exploited to limit both the number of operations and the amount of data accessed. The work presented in this thesis has been partially supported by SOLSTICE ANR project (ANR-06-CIS6-010)
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Druet, Pierre-Etienne. "Analysis of a coupled system of partial differential equations modeling the interaction between melt flow, global heat transfer and applied magnetic fields in crystal growth." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2009. http://dx.doi.org/10.18452/15893.

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Hauptthema der Dissertation ist die Analysis eines nichtlinearen, gekoppelten Systems partieller Differentialgleichungen (PDG), das in der Modellierung der Kristallzüchtung aus der Schmelze mit Magnetfeldern vorkommt. Die zu beschreibenden Phenomäne sind einerseits der im elektromagnetisch geheizten Schmelzofen erfolgende Wärmetransport (Wärmeleitung, -konvektion und -strahlung), und andererseits die Bewegung der Halbleiterschmelze unter dem Einfluss der thermischen Konvektion und der angewendeten elektromagnetischen Kräfte. Das Modell besteht aus den Navier-Stokeschen Gleichungen für eine inkompressible Newtonsche Flüssigkeit, aus der Wärmeleitungsgleichung und aus der elektrotechnischen Näherung des Maxwellschen Systems. Wir erörtern die schwache Formulierung dieses PDG Systems, und wir stellen ein Anfang-Randwertproblem auf, das die Komplexität der Anwendung widerspiegelt. Die Hauptfrage unserer Untersuchung ist die Wohlgestelltheit dieses Problems, sowohl im stationären als auch im zeitabhängigen Fall. Wir zeigen die Existenz schwacher Lösungen in geometrischen Situationen, in welchen unstetige Materialeigenschaften und nichtglatte Trennfläche auftreten dürfen, und für allgemeine Daten. In der Lösung zum zeitabhängigen Problem tritt ein Defektmaß auf, das ausser der Flüssigkeit im Rand der elektrisch leitenden Materialien konzentriert bleibt. Da eine globale Abschätzung der im Strahlungshohlraum ausgestrahlten Wärme auch fehlt, rührt ein Teil dieses Defektmaßes von der nichtlokalen Strahlung her. Die Eindeutigkeit der schwachen Lösung erhalten wir nur unter verstärkten Annahmen: die Kleinheit der gegebenen elektrischen Leistung im stationären Fall, und die Regularität der Lösung im zeitabhängigen Fall. Regularitätseigenschaften wie die Beschränktheit der Temperatur werden, wenn auch nur in vereinfachten Situationen, hergeleitet: glatte Materialtrennfläche und Temperaturunabhängige Koeffiziente im Fall einer stationären Analysis, und entkoppeltes, zeitharmonisches Maxwell für das transiente Problem.
The present PhD thesis is devoted to the analysis of a coupled system of nonlinear partial differential equations (PDE), that arises in the modeling of crystal growth from the melt in magnetic fields. The phenomena described by the model are mainly the heat-transfer processes (by conduction, convection and radiation) taking place in a high-temperatures furnace heated electromagnetically, and the motion of a semiconducting melted material subject to buoyancy and applied electromagnetic forces. The model consists of the Navier-Stokes equations for a newtonian incompressible liquid, coupled to the heat equation and the low-frequency approximation of Maxwell''s equations. We propose a mathematical setting for this PDE system, we derive its weak formulation, and we formulate an (initial) boundary value problem that in the mean reflects the complexity of the real-life application. The well-posedness of this (initial) boundary value problem is the mainmatter of the investigation. We prove the existence of weak solutions allowing for general geometrical situations (discontinuous coefficients, nonsmooth material interfaces) and data, the most important requirement being only that the injected electrical power remains finite. For the time-dependent problem, a defect measure appears in the solution, which apart from the fluid remains concentrated in the boundary of the electrical conductors. In the absence of a global estimate on the radiation emitted in the cavity, a part of the defect measure is due to the nonlocal radiation effects. The uniqueness of the weak solution is obtained only under reinforced assumptions: smallness of the input power in the stationary case, and regularity of the solution in the time-dependent case. Regularity properties, such as the boundedness of temperature are also derived, but only in simplified settings: smooth interfaces and temperature-independent coefficients in the case of a stationary analysis, and, additionally for the transient problem, decoupled time-harmonic Maxwell.
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Shao, Yuanyuan. "Beiträge zur Regularisierung inverser Probleme und zur bedingten Stabilität bei partiellen Differentialgleichungen." Doctoral thesis, Universitätsbibliothek Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-102801.

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Wir betrachten die lineare inverse Probleme mit gestörter rechter Seite und gestörtem Operator in Hilberträumen, die inkorrekt sind. Um die Auswirkung der Inkorrektheit zu verringen, müssen spezielle Lösungsmethode angewendet werden, hier nutzen wir die sogenannte Tikhonov Regularisierungsmethode. Die Regularisierungsparameter wählen wir aus das verallgemeinerte Defektprinzip. Eine typische numerische Methode zur Lösen der nichtlinearen äquivalenten Defektgleichung ist Newtonverfahren. Wir schreiben einen Algorithmus, die global und monoton konvergent für beliebige Startwerte garantiert. Um die Stabilität zu garantieren, benutzen wir die Glattheit der Lösung, dann erhalten wir eine sogenannte bedingte Stabilität. Wir demonstrieren die sogenannte Interpolationsmethode zur Herleitung von bedingten Stabilitätsabschätzungen bei inversen Problemen für partielle Differentialgleichungen.
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Negi, Yoginder Kumar. "Fast Solvers and Preconditioning Methods in Computational Electromagnetics." Thesis, 2018. https://etd.iisc.ac.in/handle/2005/4509.

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Method of Moments (MoM) is an integral equation based solver and is one of the most popular computational techniques to solve complex 3D Electromagnetic problems efficiently and accurately. Compared to the conventional differential equation solvers, MoM does not require a volumetric discretization of the entire bounding box containing the structure or imposes absorbing boundary condition or perfect match layer. However, due to Green's function interactions, the MoM matrix is dense leading to quadratic matrix fill time and cubic solve time complexity. As the scale and the complexity of the problem increases, matrix size increases with high storage memory and solve time requirement. Thus, to further expedite the solve time and decrease the storage requirement, fast solvers are extremely important. This thesis addresses the above problem by developing fast solvers and proposing a couple of novel preconditioning approaches for improving the solution speed of the fast solvers. The recent development of kernel-independent fast solvers has gained more popularity among the CEM community because of ease of their implementation. In this thesis, first, a brief overview of kernel dependent and kernel independent fast solvers is presented along with their parallelization. A new compression method is introduced based on the Adaptive Cross Approximation (ACA) sampling, where the pivot rows and columns can give a QR factorized orthogonal compressed matrices. These orthogonal matrices can be compressed further by using Interpolative Decomposition (ID) or can further be compressed by using Singular Value Decomposition (SVD) compression. The entire spectrum of fast solvers is highly dependent on the condition number of the matrices, specifically the spread of their eigenvalues. Hence, an ill-conditioned matrix jeopardizes the solution time and accuracy. This ill-conditioning mostly due to either geometry, meshing or the frequency of operation. Preconditioning the system of equations is the most efficient way to overcome ill-condition. Conventional preconditioners are either difficult to parallelize or lack linear scaling with problem size. In this thesis, we propose new preconditioners, which overcome the lacuna of the existing preconditioners. In the null-field preconditioner, near-field blocks are scaled to the diagonal blocks using the null-field method. The preconditioner is computed from all near-field blocks and selected far-field blocks in accordance with the null-field procedure. The final form of the preconditioner is block diagonal, therefore generates no additional fill-ins in its inverse and is also amenable to parallelization. A complexity analysis is presented to show the near-linear cost of preconditioner construction and usage in terms of computation time and memory. Numerical experiments with a sequential implementation demonstrate on an average 1.5-3x speed-up in the iterative solution time over Incomplete LU (ILUT) based preconditioners. Thus, giving a robust and stable preconditioner better than ILUT. The main drawback of the null-field method is the improper scaling of the near-field blocks. The next preconditioning method proposed in the thesis is based on the Schur’s complement method. The Schur’s complement method diagonalizes the near-field blocks to a block-diagonal form. The fill-in blocks generated in the process can be compressed efficiently and used for completely scaling the near-field blocks. Further fill-in reduction can be achieved by arranging the near-field blocks by using graph algorithms. Both these processes lead to a reduction in final matrix-vector product time in the iterative solver. Numerical experiments demonstrate a significant advantage over ILUT or recently published null-field based methods. In the case of multiport system or the problem with multiple Right Hand Side (RHS) iteration, the fast solver may turn out to be costly since each RHS may take different iteration thus taking more time to solve all RHSs so in such cases a direct solver is highly desirable. The complexity of the conventional direct solver is cubic so in this thesis, a fast direct solver based on the extended sparsification of FMM is proposed.
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Lai, Kuan-Yin, and 賴冠吟. "Biomechanical Analysis of Hand during Piano Playing:Concerns of Hand Size and Right-Left Side Difference." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/51272473187529684878.

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碩士
國立成功大學
職能治療學系碩博士班
96
Background & purpose: Pianists are an occupational group. To understand the injury mechanism associated with pianists, it is important to investigate the biomechanics of the hand in depth while playing the piano. The purposes of the study are: 1) to compare the kinematic and kinetic performance of the small hand span with those of large hand span; 2) to compare the kinematic and kinetic performance of the right-hand side with those of the left-hand side by the objective and quantitative biomechanical methods. The results could offer further insights to clinical application. Methods: Twenty female pianists were recruited from the southern area of Taiwan. All subjects passed through the hand span screening and assigned to play a chord and an octave. The kinematic parameters, such as the ratio of maximal digit-to-digit abduction angle, the range of motion (ROM) of finger joint flexion-extension (F/E) angle, the ROM of wrist joint flexion-extension (F/E) as well as radial-ulnar deviation (R/U) angle, and the movement unit of flexion-extension (F/E) of each finger joint were measured using a video-based passive marker detection system. The fingertip force was measured using the MIDI interface and calculated by a kinetic equation. Results: There is significant difference of the ratio of maximal digit-to-digit abduction angle between large and small hand span pianists while playing the chord and the octave. Significant difference of the ROM of wrist joint F/E angle between large and small hand span pianists while playing the chord is also presented. The ROM of finger joint F/E angle , the ROM of wrist joint R/U angle, and the fingertip force do not show significant differences between these two groups. When playing the chord and the octave, both the movement unit of F/E of each finger joint and the fingertip force of the right hand are significantly larger than those of the left hand. Conclusions: The small hand span pianists use the wide-extent of finger abduction to play the chord and octave. Similarly, the pianists of this group use larger wrist F/E movement to play the chord. The left hand also represents smoother motions and less fingertip force as compared with the performance of the right hand. Accordingly we may claim that the small hand span and the right-hand side might potentially be the risk factors which could induce the injuries in piano playing.
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Hsieh, Shang-Jen, and 謝尚仁. "Sensitivity Analysis under a Perturbation of Right-Hand-Side Parameter in the Transportation Problem." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/85078863913934647427.

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碩士
國立清華大學
工業工程與工程管理學系
99
An optimum solution is usually the result of the mathematical model, but what decision makers require are the options in some ranges for them to choose, make decisions and further take actions instead of a definitely exact answer. Accordingly, it is important to perform sensitivity analysis to investigate the effects on the optimum solution. In fact, sensitivity analysis is one of the most important areas in postoptimality analysis. On the other hand, the transportation problem (TP) is an important concern that arises in several contexts and has received much attention in the literature. Therefore, this study investigates the sensitivity analysis of the TP and concentrates on the so-called one-change-at-a-time sensitivity ranges of the right-hand-side elements. Due to the special structure of the TP, i.e., the balanced condition, the sensitivity range is derived by the perturbation between one original shipment and one dummy shipment. Thus, the revised auxiliary perturbed problems are demonstrated to apply on the TP, and we further develop two algorithms based on the Labeling procedure for the sensitivity ranges. The proposed algorithms are divided into two parts for finding the lower bound and the upper bound respectively. Two numerical examples are presented in order to illustrate the two algorithms respectively which are the effective ways for determining the one-change-at-a-time sensitivity range of the right-hand-side elements in the TP.
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Books on the topic "Right Hand Side iteration"

1

Fischer, Paul F. Projection techniques for iterative solution of Ax = b with successive right-hand sides. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1993.

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Center, Langley Research, ed. Projection techniques for iterative solution of Ax̲=b̲ with successive right-hand sides. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1993.

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Fischer, P. F. Projection techniques for iterative solution of Ax̲=b̲ with successive right-hand sides. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1993.

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Fists, wits, and a wicked right: Surviving on the wild side of the street. Boulder, Colo: Paladin Press, 1991.

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Capacchione, Lucia. Power of Your Other Hand: Unlock Creativity and Inner Wisdom Through the Right Side of Your Brain. Red Wheel/Weiser, 2019.

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The Power of Your Other Hand: Unlock Creativity and Inner Wisdom Through the Right Side of Your Brain. Red Wheel, 2019.

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Grant, Darren. Weekly Journal: Beautifully Simple 53 Week 108 Page 5. 5 X8. 5 Journal with Jobs to Do and Notes on the Left Hand Side and Day of the Week on the Right Hand Side. Independently Published, 2021.

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Grant, Darren. Fitness Planner Gym Workout Body Building Exercise Log: Beautifully Simple 53 Week 108 Page 5. 5 X8. 5 Journal with Exercises and Notes on the Left Hand Side and Days of the Week on the Right Hand Side. Independently Published, 2021.

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Tourneau, Thierry Le, Luis Caballero, and Tsai Wei-Chuan. Right atrium. Oxford University Press, 2016. http://dx.doi.org/10.1093/med/9780198726012.003.0024.

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The right atrium (RA) is located on the upper right-hand side of the heart and has relatively thin walls. From an anatomical point of view, the RA comprises three basic parts, the appendage, the vestibule of the tricuspid valve, and the venous component (superior and inferior vena cava, and the coronary sinus) receiving the deoxygenated blood. The RA is a dynamic structure dedicated to receive blood and to assist right ventricular (RV) filling. The three components of atrial function are the reservoir function during ventricular systole, the conduit function which consists in passive blood transfer from veins to the RV in diastole, and the booster pump function in relation to atrial contraction in late diastole to complete ventricular filling. Right atrial function depends on cardiac rhythm (sinus or atrial fibrillation), pericardial integrity, RV load and function, and tricuspid function. Right atrial dimension assessment is limited in two-dimensional (2D) echocardiography. Right atrial planimetry in the apical four-chamber view is commonly used with an upper normal value of 18-20 cm2. Minor and major diameters can also be measured. Three-dimensional (3D) echocardiography could overcome the limitation of conventional echocardiography in assessing RA size. Right atrial function has been poorly explored by echocardiography both in physiological and pathological contexts. Although tricuspid inflow and tissue Doppler imaging of tricuspid annulus can be used in the exploration of RA function, 2D speckle tracking and 3D echocardiography appear promising tools to dissect RA function and to overcome the limitations of standard echocardiography.
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616SRM, Aionic Star. Black Master Manifest-O: The Black Master Manifest-O Seeks to Re-manifest the Black Tradition, the Darker Side of the Various Right Hand Paths, to Re-establish a Dynamic Equilibrium. Independently Published, 2017.

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Book chapters on the topic "Right Hand Side iteration"

1

Balsa, Carlos, Michel Daydé, José M. L. M. Palma, and Daniel Ruiz. "Inexact Subspace Iteration for the Consecutive Solution of Linear Systems with Changing Right-Hand Sides." In Springer Proceedings in Mathematics & Statistics, 49–78. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-74086-7_3.

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Gass, Saul I., and Carl M. Harris. "Right-hand side." In Encyclopedia of Operations Research and Management Science, 720. New York, NY: Springer US, 2001. http://dx.doi.org/10.1007/1-4020-0611-x_895.

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Weik, Martin H. "right-hand side." In Computer Science and Communications Dictionary, 1494. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_16392.

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Gass, Saul I., and Carl M. Harris. "Right-hand-side ranging." In Encyclopedia of Operations Research and Management Science, 720. New York, NY: Springer US, 2001. http://dx.doi.org/10.1007/1-4020-0611-x_896.

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Tolstonogov, Alexander. "Inclusions with Non-Compact Right Hand Side." In Differential Inclusions in a Banach Space, 235–54. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-015-9490-5_5.

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Candito, Pasquale, and Roberto Livrea. "Nonlinear Difference Equations with Discontinuous Right-Hand Side." In Differential and Difference Equations with Applications, 331–39. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7333-6_26.

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Schilling, Thorsten Ernst, and Håvard Raddum. "Analysis of Trivium Using Compressed Right Hand Side Equations." In Information Security and Cryptology - ICISC 2011, 18–32. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31912-9_2.

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Schilling, Thorsten Ernst, and Håvard Raddum. "Solving Compressed Right Hand Side Equation Systems with Linear Absorption." In Lecture Notes in Computer Science, 291–302. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-30615-0_27.

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Pestana, Jennifer. "Right-Hand Side Dependent Bounds for GMRES Applied to Ill-Posed Problems." In IFIP Advances in Information and Communication Technology, 230–36. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-45504-3_22.

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Walden, Aaron, Sabbir Khan, Bálint Joó, Desh Ranjan, and Mohammad Zubair. "Optimizing a Multiple Right-Hand Side Dslash Kernel for Intel Knights Corner." In Lecture Notes in Computer Science, 390–401. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-46079-6_28.

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Conference papers on the topic "Right Hand Side iteration"

1

Killough, J. E., Yogeshwar Sharma, Alain Dupuy, Robert Bissell, and John Wallis. "A Multiple Right Hand Side Iterative Solver for History Matching." In SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, 1995. http://dx.doi.org/10.2118/29119-ms.

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Yang, Shao-Chong, and Qing-Sheng Yang. "Geometrically Nonlinear Transient Response of Laminated Plates With Nonlinear Viscoelastic Restraints." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59252.

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In this work, geometrically nonlinear transient response of laminated plates with nonlinear viscoelastic restraints is investigated by a numerical method, in the computational platform - MSC.Nastran. The variable constraining forces as functions of dynamic displacements and velocities are added to the right-hand side of equation of motion as additional applied loads. These loads are represented by an independent set of functions that satisfy the various constraint conditions for specific cases. The nonlinear equations are solved by using the load increments scheme in conjunction with Newton-Raphson iteration. The time history of transverse displacement at a typical point is given through a series of transient analysis. Then the comparisons of the responses for different parameters of the Kelvin-Voigt model and boundary conditions are made. The numerical results show that the present method is validated to be effective for treating geometrically nonlinear transient problems with nonlinear viscoelastic restraints.
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Seibold, Andreas S., Daniel J. Rixen, and Javier Del Fresno Zarza. "On Total Reuse of Krylov Subspaces for an iterative FETI-solver in multirate integration." In VI ECCOMAS Young Investigators Conference. València: Editorial Universitat Politècnica de València, 2021. http://dx.doi.org/10.4995/yic2021.2021.12306.

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Recycling techniques for Krylov subspaces in parallel FETI-solvers are able to increase efficiency of solution processes with repeated right-hand-sides. One simple technique consists in a Total Reuse of the Krylov Subspace (TRKS) provided by conjugate directions generated during the solution of previous problems. This applies especially for linear structural dynamics and recycling also reduced global iterations for nonlinear structural dynamics. The structure of the interface-operator's eigenvalues governs the possible efficiency-gain. Only if high clustered eigenvalues are captured early enough, global FETI iterations will be reduced accordingly. Besides these advances, multirate integrators have been proposed, that enable adapted time-step-sizes on each substructure and are expected to accelerate the parallel simulation of nonlinear dynamic simulations with local highly nonlinear processes, e.g. damaging. Based on the linear BGC-macro and nonlinear PH-method, a nonlinear BGC-macro method has been proposed and on the other hand a more flexible and accurate multirate-integrator has been derived from the variational principle. These multirate-integration schemes require several local integration-steps in each global iteration, leading to a non-symmetric structure of the linearized local problems and the local boundary conditions are altered compared to FETI for standard singlerate integration. So, we have to solve the global interface-problem with a GMRes-solver. In this contribution, we show our recent results on the eigenvalues and application of a TRKS to these new problems.
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Dragojlovic, Zoran, Deborah A. Kaminski, and Juntaek Ryoo. "Tuning of Membership Functions in a Fuzzy Rule Set for Controlling Convergence of Laminar CFD Solutions." In ASME 2001 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/imece2001/htd-24288.

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Abstract Under-relaxation in an iterative CFD solver is guided by fuzzy logic to achieve automatic convergence with minimum CPU time. The fuzzy rule set uses information from a Fourier transform of a set of characteristic values. The control algorithm adjusts the relaxation factors for the system variables on each iteration and restarts the solver if divergence occurs. Four problems of laminar fluid flow and heat transfer are solved. They include buoyancy driven flow in a square cavity, lid driven flow in a square enclosure, mixed convection over a backward facing step and Dean flow. The consideration of buoyancy driven flow in a square cavity took into account a case with isothermal vertical walls, as well as a case with heat conduction in the solid wall on the right hand side of the cavity. The incompressible Newtonian conservation equations are solved by the SIMPLER algorithm with simple substitution. In order to achieve the best performance of the fuzzy controller, a set of triangular membership functions was tuned by using a gradient method. The fuzzy control algorithm with the optimal membership functions significantly reduced the CPU time needed for solving the problem, compared to performance of triangular membership functions before optimization. The improvement in speed of convergence varied from 10% in the case of Dean flow to 70% in the case of buoyancy driven flow in a square enclosure with isothermal walls. For all the problems considered, the fuzzy rule set provided convergence comparable in speed to that obtained with the best choice of constant relaxation factors. In the case of buoyancy driven cavity flow with conjugate heat transfer, the fuzzy controller outperformed any choice of constant relaxation factors.
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Cromieres, Fabien, and Sadao Kurohashi. "Translation Rules with Right-Hand Side Lattices." In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). Stroudsburg, PA, USA: Association for Computational Linguistics, 2014. http://dx.doi.org/10.3115/v1/d14-1063.

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Jackson, Michael Anthony. "The Right-Hand Side Problem: Research Topics in RE." In 2017 IEEE 25th International Requirements Engineering Conference (RE). IEEE, 2017. http://dx.doi.org/10.1109/re.2017.16.

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Li, Wei, and Gui-xiang Wang. "General Solutions for Linear Programming with Interval Right Hand Side." In 2006 International Conference on Machine Learning and Cybernetics. IEEE, 2006. http://dx.doi.org/10.1109/icmlc.2006.259046.

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Richtmann, Daniel, Simon Heybrock, and Tilo Wettig. "Multiple right-hand-side setup for the DD-$\alpha$AMG." In The 33rd International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2016. http://dx.doi.org/10.22323/1.251.0035.

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Garcia, Juan Carlos Figueroa. "Linear programming with Interval Type-2 Fuzzy Right Hand Side parameters." In NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society. IEEE, 2008. http://dx.doi.org/10.1109/nafips.2008.4531280.

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Michel, Anthony N., and Ling Hou. "Floquet theory for linear periodic differential equations with discontinuous right-hand side." In 2009 IEEE International Conference on Control and Automation (ICCA). IEEE, 2009. http://dx.doi.org/10.1109/icca.2009.5410523.

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Reports on the topic "Right Hand Side iteration"

1

Nijhuis, Chris, Bertjan Verbeek, and Andrej Zaslove. Disagreement among populists in the Netherlands: The diverging rhetorical and policy positions of Dutch populist Radical Right parties following Russia’s invasion of Ukraine. European Center for Populism Studies (ECPS), March 2023. http://dx.doi.org/10.55271/rp0032.

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The Netherlands boasts a wide array of populist Radical Right parties, from the Freedom Party (PVV) and Forum for Democracy (FvD) to Correct Alternative 2021 (JA21). To complicate matters further, the left-wing Socialist Party (SP) is also considered a populist party. Mirroring the diversity of responses to the Russian invasion of Ukraine in the rest of Europe, Dutch populist parties have reacted in myriad ways. Whereas the PVV condemned the Russian invasion, the FvD remained highly supportive of Putin. Interestingly, while many European populist Radical Right leaders, following public opinion, pivoted away from support for Russia, the FvD has maintained its support for Putin’s regime throughout 2022. JA21, on the other hand, has followed the non-populist parties, calling for tougher sanctions on Russia. This report maps the diverse positions of the three populist Radical Right parties regarding Russia. However, we also seek to explain why they have chosen such diverse paths. The report focuses on supply-side considerations, such as the impact of the parties’ relative degree of populism, their attaching ideology, and their position in the highly fragmented party system.
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