87 research outputs found

    Drazin inverse based numerical methods for singular linear differential systems

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    [EN] In this paper, numerical methods for the solution of linear singular differential system are analyzed. The numerical solution of initial value problem by means of a corresponding finite difference approach and a possible implementation of the product Drazin inverse by vector is discussed. Examples of index-1 and index-2 DAEs have been studied numerically.This paper was partially supported by Grant GS1 DGI MTM2010-18228, by Ministry of Education of Argentina (PPUA, grant Resol. 228, SPU, 14-15-222) and by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. No. 049/11).Coll Aliaga, PDC.; Ginestar Peiro, D.; Sánchez Juan, E.; Thome Coppo, NJ. (2012). Drazin inverse based numerical methods for singular linear differential systems. ADVANCES IN ENGINEERING SOFTWARE. (50):37-43. https://doi.org/10.1016/j.advengsoft.2012.04.001S37435

    Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

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    Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt

    A Note on Computing the Generalized Inverse A^(2)_{T,S} of a Matrix A

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    The generalized inverse A T,S (2) of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T,S (2) has been recently developed with the condition σ (GA| T)⊂(0,∞), where G is a matrix with R(G)=T andN(G)=S. In this note, we remove the above condition. Three types of iterative methods for A T,S (2) are presented if σ(GA|T) is a subset of the open right half-plane and they are extensions of existing computational procedures of A T,S (2), including special cases such as the weighted Moore-Penrose inverse A M,N † and the Drazin inverse AD. Numerical examples are given to illustrate our results

    Author index to volumes 301–400

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    Author index for volumes 101–200

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    주기경계조건을 갖는 P1-비순응유한요소공간과 멀티스케일 문제에 대한 응용

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    학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 협동과정 계산과학전공, 2018. 2. 신동우.We consider the P1-nonconforming quadrilateral finite space with periodic boundary condition, and investigate characteristics of the finite space and discrete Laplace operators in the first part of this dissertation. We analyze dimension of the finite element spaces in help of concept of minimally essential discrete boundary conditions. Based on the analysis, we classify functions in a basis for the finite space with periodic boundary condition into two types. And we introduce several Krylov iterative schemes to solve second-order elliptic problems, and compare their solutions. Some of the schemes are based on the Drazin inverse, one of generalized inverse operators, since the periodic nature may derive a singular linear system of equations. An application to the Stokes equations with periodic boundary condition is considered. Lastly, we extend our results for elliptic problems to 3-D case. Some numerical results are provided in our discussion. In the second part, we introduce a nonconforming heterogeneous multiscale method for multiscale problems. Its formulation is based on the P1-nonconforming quadrilateral finite element, mainly with periodic boundary condition. We analyze a priori error estimates of the proposed scheme by following general framework for the finite element heterogeneous multiscale method. For numerical implementations, we use one of the proposed iterative schemes for singular linear systems in the previous part. Several numerical examples and results are given.I P1-Nonconforming Quadrilateral Finite Space with Periodic Boundary Condition 1 Chapter 1 Introduction 3 Chapter 2 Preliminaries 7 2.1 P1-nonconforming quadrilateral finite element 7 2.2 Drazin inverse 8 2.3 Notations 9 Chapter 3 Dimension of the Finite Spaces 13 3.1 Induced relation between boundary DoF values 13 3.2 Minimally essential discrete boundary conditions 16 Chapter 4 Deeper Look on the Finite Space with Periodic B.C. 19 4.1 Linear dependence of B 19 4.2 A Basis for V^h_per 21 4.3 Stiffness matrix associated with B 22 4.4 Numerical schemes for elliptic problems with periodic boundary condition 24 4.4.1 Option 1: S = E^♭ for a nonsingular nonsymmetric system 27 4.4.2 Option 2: S = E^♭ for a symmetric positive semi-definite system with rank deficiency 1 28 4.4.3 Option 3: S = E for a symmetric positive semi-definite system with rank deficiency 2 31 4.4.4 Option 4: S = B for a symmetric positive semi-definite system with rank deficiency 2 33 4.5 Numerical results 34 Chapter 5 Application to Stokes Equations 37 5.1 Discrete inf-sup stability 38 5.2 Numerical scheme: Uzawa variant with a semi-definite block 41 5.3 Numerical results 49 Chapter 6 3-D Case 51 6.1 Dimension of finite spaces in 3-D 51 6.2 Linear dependence of B in 3-D 56 6.3 A basis for V^h_per in 3-D 64 6.4 Stiffness matrix associated with B in 3-D 66 6.5 Numerical schemes in 3-D 67 6.6 Numerical results 73 II Nonconforming Heterogeneous Multiscale Method 75 Chapter 1 Introduction 77 Chapter 2 Preliminaries 81 2.1 Homogenization 81 2.2 Notations 83 Chapter 3 FEHMM Based on Nonconforming Spaces 85 Chapter 4 Fundamental Properties of Nonconforming HMM 91 4.1 Existence and uniqueness 91 4.2 Recovered homogenized tensors 93 4.3 The case of periodic coupling 95 4.4 The case of Dirichlet coupling 101 4.5 A priori error estimate 102 4.5.1 Macro error 102 4.5.2 Modeling error 102 4.5.3 Micro error 104 4.6 Main theorem for error estimates 105 Chapter 5 Numerical Results 107 5.1 Periodic diagonal example 108 5.1.1 Comparison between approaches to solve micro problem 110 5.2 Periodic example with off-diagonal terms 112 5.3 Example with noninteger-ε-multiple sampling domain and Dirichlet coupling 112 5.4 Example on mixed domain 115 국문초록 127Docto

    Laurent expansion of the inverse of perturbed, singular matrices

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    In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix. The matrix coefficients of the Laurent expansion of the inverse are computed using recursive analytical formulae. We show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix, but exponentially with the order of the singularity. We apply this algorithm to several matrices that arise in applications. We make special emphasis to interpolation problems with radial basis functions.This work has been supported by Spanish MICINN Grants FIS2013-41802-R and CSD2010-00011
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