Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I

The molded block fuel element (FE) also called monolith is a molded body, consisting of a substantially isotropic highly crystalline graphite matrix, fuel regions within the same matrix and cooling channels. The fuel regions contain the fuel in the form of coated particles which are well bonded to the remaining graphite matrix, so that both parts of the block form a monolithic structure. The monolith meets the requirements for the very high temperature reactors attaining helium outlet temperatures above 1000°C. To fabricate the molded blocks FE demonstration plant was erected and put into operation. The equipment worked without malfunction. The produced block FEs meet the specifications of GA machined block FEs. All specimens and block segments irradiated at temperature up to 1600°C and max. fast fluence E > 0, 1 MeV of 11×1021 n/cm2 show perfect behaviour without any damage.

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Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

Numerische Mathematik ◽

10.1007/s00211-020-01143-x ◽

2020 ◽

Vol 146 (2) ◽

pp. 335-368

Author(s):

Owe Axelsson ◽

János Karátson

Keyword(s):

Optimal Control ◽

Rate Of Convergence ◽

Large Scale ◽

Optimal Control Problems ◽

Iterative Solution ◽

Control Problems ◽

Block Form ◽

Solution Methods ◽

Preconditioned Matrix ◽

Iterative Solution Methods

Abstract Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

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An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

Journal of Applied Mathematics ◽

10.1155/2014/549597 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 10

Author(s):

Lee Ken Yap ◽

Fudziah Ismail ◽

Norazak Senu

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Collocation Method ◽

Film Flow ◽

Direct Solution ◽

Thin Film Flow ◽

Block Method ◽

Third Order ◽

Block Form ◽

The Stability

The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.

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Block-form control of linear time-invariant discrete-time systems defined over commutative rings

Linear Algebra and its Applications ◽

10.1016/0024-3795(94)90370-0 ◽

1994 ◽

Vol 205-206 ◽

pp. 805-829 ◽

Cited By ~ 3

Author(s):

E.W. Kamen

Keyword(s):

Discrete Time ◽

Linear Time ◽

Commutative Rings ◽

Linear Time Invariant ◽

Discrete Time Systems ◽

Block Form ◽

Time Invariant ◽

Time Systems

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Transformations preserving the Lyapunov exponents

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500274 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750027 ◽

Cited By ~ 5

Author(s):

Luis Barreira ◽

Claudia Valls

Keyword(s):

Lyapunov Exponents ◽

Asymptotic Properties ◽

Regular Sequence ◽

Complete Characterization ◽

Block Form ◽

The Family ◽

Coordinate Changes ◽

Invertible Matrices ◽

Finite Exponent

We give a complete characterization of the existence of Lyapunov coordinate changes bringing an invertible sequence of matrices to one in block form. In other words, we give a criterion for the block-trivialization of a nonautonomous dynamics with discrete time while preserving the asymptotic properties of the dynamics. We provide two nontrivial applications of this criterion: we show that any Lyapunov regular sequence of invertible matrices can be transformed by a Lyapunov coordinate change into a constant diagonal sequence; and we show that the family of all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices (with finite exponent) coincides with the family of Lyapunov coordinate changes.

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