Kind Of Linear Systems Research Articles

Initial Results of H-Matrix Reconstruction Method for 1-D Mirrored Aperture Synthesis Radiometers

The mirrored aperture synthesis radiometer (MASR) has been proposed for high-resolution observation in recent years. Since the MASR is a kind of linear system, the matrix reconstruction method should be an alternative for image reconstruction of a practical MASR. However, the matrix reconstruction method is neither theoretically analyzed nor experimentally validated at present, and the performance of the matrix reconstruction method for a MASR in practical applications is still unknown. In this letter, the H-matrix reconstruction method including the truncated singular value decomposition (SVD) and Tikhonov regularization, not yet applied to a MASR, is introduced. Then, the numerical simulations and imaging experiments are performed to validate the presented method, and the performance of the presented method is also shown. Finally, the comparisons between the presented method and the cosine-transform-based iterative method are made in the view of imaging experiments. The H-matrix reconstruction method is more convenient in the situation of a small MASR with a small alias-free field of view (FOV).

The Schwarz alternating method in solid mechanics

We advance the Schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics. We prove that the Schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well-posed, i.e. their corresponding energy density functions are quasi-convex. It is shown that the use of a Newton-type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems, depending on the treatment of the Dirichlet boundary conditions. The first kind is a symmetric block-diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain, i.e. the off-diagonal blocks are all zero and the coupling terms appear only on the right-hand side. The second kind is a nonsymmetric block system with off-diagonal coupling terms. Several variants of the Schwarz alternating method are proposed for the first kind of linear system, including one in which the Schwarz alternating iterations and the Newton iterations are combined into a single scheme. This version of the method is particularly attractive, as it lends itself to a minimally intrusive implementation into existing finite element codes. Finally, we demonstrate the performance of the proposed variants of the Schwarz alternating method on several one-dimensional and three-dimensional examples.

Should Pruning be a Pre-Processor of any Linear System?

There are many real-world problems whose mathematical models turn out to be linear systems Ax = b, where A is an m x n matrix. Each equation of the linear system is an information. An information, in a physical problem, such as 4 mangoes, 6 bananas, and 5 oranges cost $10, is mathematically modeled as an equation 4x(sub 1) + 6x(sub 2) + 5x(sub 3) = 10 , where x(sub 1), x(sub 2), x(sub 3) are each cost of one mango, that of one banana, and that of one orange, respectively. All the information put together in a specified context, constitutes the physical problem and need not be all distinct. Some of these could be redundant, which cannot be readily identified by inspection. The resulting mathematical model will thus have equations corresponding to this redundant information and hence are linearly dependent and thus superfluous. Consequently, these equations once identified should be better pruned in the process of solving the system. The benefits are (i) less computation and hence less error and consequently a better quality of solution and (ii) reduced storage requirements. In literature, the pruning concept is not in vogue so far although it is most desirable. It is assumed that at least one information, i.e. one equation is known to be correct and which will be our first equation. In a numerical linear system, the system could be slightly inconsistent or inconsistent of varying degree. If the system is too inconsistent, then we should fall back on to the physical problem (PP), check the correctness of the PP derived from the material universe, modify it, if necessary, and then check the corresponding mathematical model (MM) and correct it. In nature/material universe, inconsistency is completely nonexistent. If the MM becomes inconsistent, it could be due to error introduced by the concerned measuring device and/or due to assumptions made on the PP to obtain an MM which is relatively easily solvable or simply due to human error. No measuring device can usually measure a quantity with an accuracy greater that 0.005% or, equivalently with a relative error less than 0.005%. Hence measurement error is unavoidable in a numerical linear system when the quantities are continuous (or even discrete with extremely large number). Assumptions, though not desirable, are usually made when we find the problem sufficiently difficult to be solved within the available means/tools/resources and hence distort the PP and the corresponding MM. The . error thus introduced in the system could (not always necessarily though) make the system somewhat inconsistent. If the inconsistency (contradiction) is too much then one should definitely not proceed to solve the system in terms of getting a least-squares solution or the minimum-norm least-squares solution. All these solutions will be invariably of no real-world use. If, on the other hand, inconsistency is reasonably low, i.e. the system is near-consistent or, equivalently, has near-linearly-dependent rows, then the foregoing solutions are useful. Pruning in such a near-consistent system should be performed based on the desired accuracy and on the definition of near-linear dependence. In this article, we discuss pruning over various kinds of linear systems and strongly suggest its use as a pre-processor or as a part of an algorithm. Ideally pruning should (i) be a part of the solution process (algorithm) of the system, (ii) reduce both computational error and complexity of the process, and (iii) take into account the numerical zero defined in the context. These are precisely what we achieve through our proposed O(mn2) algorithm presented in Matlab, that uses a subprogram of solving a single linear equation and that has embedded in it the pruning.

Coarse grid correction by aggregation / disaggregation with application in image reconstruction

AbstractIn algebraic reconstruction of images in computerized tomography we are dealing with rectangular, large, sparse and ill‐conditioned linear systems of equations. In this paper we describe a two‐grid algorithm for solving such kind of linear systems, which uses Kaczmarz's projection method as relaxation. The correction step is performed with a special “local” aggregation / disaggregation type procedure. In this respect, we have to solve a small sized minimization problem associated to each coarse grid pixel. The information so obtained is then “distributed” to the neighbour fine grid pixels. Some image reconstruction experiments are also presented. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Statistical applications for equivariant matrices

Solving linear system of equations Ax = b enters into many scientific applications. In this paper, we consider a special kind of linear systems, the matrix A is an equivariant matrix with respect to a finite group of permutations. Examples of this kind are special Toeplitz matrices, circulant matrices, and others. The equivariance property of A may be used to reduce the cost of computation for solving linear systems. We will show that the quadratic form is invariant with respect to a permutation matrix. This helps to know the multiplicity of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. Applications for such systems from the area of statistics will be presented. These include Fourier transforms on a symmetric group as part of statistical analysis of rankings in an election, spectral analysis in stationary processes, prediction of stationary processes and Yule‐Walker equations and parameter estimation for autoregressive processes.

Bounds for the solutions of a class of tridiagonal linear systems

We derive bounds for the solution of an irreducible tridiagonal linear system of dimension N which arises in many areas of numerical analysis. The conditions for which the linear system is invertible and its solution bounded by a constant independent of N or dependent weakly on N are established. The results derived are strictly related to the study of the condition number for the coefficient matrix of the system. Two examples which lead to this kind of linear systems are given, the second of which derives from the solution of boundary value problems by means of difference methods.

Computer Aided Analysis Software for Discrete-event Dynamic System

A discrete-event system is a system whose behaviour can be discribed by a set of time-consuming activities, performed according to a prescribed ordering. The events concerned correspond to starting or ending time of some activities. This kind of system is not linear in the conventional arithmetic, but is, however, linear in max-algebra. A new kind of linear system, also called psuedo-linear discrete-event system, has been set up. Referring to usual linear system theory, a series of results for modelling and analysis of discrete-event dynamic system (DEDS) are developed, such as state equation, stability, characteristic equation, eigenvalue, periodicity, controllability, observability, etc. The Psuedo-linear Discrete-event System Analysis Package (PDSAP) is to implement the matrix calculations in max-algebra sense, model DEDS and analize performances of the system. Some fundamental contents of max-algebra and DEDS, functions and some algorithms of PDSAP are presented in this paper.

Control problem for linear systems with input derivatives control

Abstract Descriptor variable system theory is applied to one kind of linear system with input derivatives control, focusing on the problems of controllability and compensation. Necessary and sufficient conditions for the existence and design methods of its dynamic compensator are given. Examples are also considered to illustrate the results.

The relationship between the stability and the optimality of linear systems (II)

Different from the inverse problem put forward by R.E.Kalman, another kind of inverse problem of linear optimal control is proposed and discussed in [1] as follows: Given an asymptotically stable linear constant system and a nonnegative quadratic performance index, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended. Similar conclusions are obtained for linear discrete systems and linear time-variable systems. According to these conclusions we can say that the correspondence between the asymptotically stable system and the optimal feedback system is the inherent character of all kinds of linear systems.