Showing papers on "Linear map published in 1990"
TL;DR: In this paper, a general algebraic framework for characterizing the set of possible effective tensors of composites is developed for characterising the class of possible tensors for non-self-adjoint problems.
Abstract: A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.
214 citations
TL;DR: In this article, an inversion method for the solution of ill-posed linear problems is presented, based on the idea of computing a mollified version of the searched-for solution and the approximate inverse operator is computed with exactly given quantities.
Abstract: An inversion method for the solution of ill-posed linear problems is presented. It is based on the idea of computing a mollified version of the searched-for solution and the approximate inverse operator is computed with exactly given quantities. The method is compared with known methods such as the Tikhonov-Phillips and Backus-Gilbert methods. Numerical tests verify the advantages, which are: no additional or artificial discretisation of the solution is needed, locally varying point-spread functions are easily realised, a simple change of the regularisation parameter with regard of a posteriori parameter strategies is implemented and a straightforward interpretation of the regularised solution is possible. When the approximation inversion operator is computed the solution can be computed by parallel processing.
160 citations
TL;DR: A linear map A : C(T) → C(S) is called separating if f • g ≡ 0 implies Af • Ag = 0 as discussed by the authors, and any separating isomorphism is continuous.
Abstract: A linear map A : C(T) → C(S) is called separating if f • g ≡ 0 implies Af • Ag = 0. We describe the general form of such maps and prove that any separating isomorphism is continuous.
146 citations
01 Jan 1990
TL;DR: In this article, a generalized controller canonical form is proposed for linear and nonlinear dynamics, obtained using the theorem of the primitive element from differential algebra, which does not apply to multivariable constant linear systems.
Abstract: Abstmct- A generalized controller canonical form is proposed for linear and nonlinear dynamics. It is obtained using the theorem of the primitive element from differential algebra. This area of mathematics, which was introduced in system theory in order to provide an answer to multivariable nonlinear input-output inversion, has also permitted a renewed understanding of synthesis problems, especially those involving control-dependent state transformations. Our derivation of the controller form does not apply to multivariable constant linear systems. This means, that in a certain sense, the structure of these systems is more complex than the structure of time-varying linear systems and of general nonlinear systems, unless we allow time-varying control-dependent linear transformations in the study of constant linear systems. Under our transformations, all nonlinear dynamics can he exactly linearized via dynamic feedback. The main departure from standard theory is that transformations may depend on input derivatives. Once differential-algebraic tools are introduced, the proofs of the results are easy.
102 citations
TL;DR: In this paper, the authors considered the optimal control of a system whose states are governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval and showed that catching-up optimal solutions asymptotically approach a unique optimal steady state.
Abstract: The optimal control of a system whose states are governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. The results obtained extend those of Brock and Haurie [Math. Oper. Res., 1 (1976), pp. 337–346] and Leizarowitz [Math. Oper. Res., 10 (1985), pp. 450–461]. In particular, it is shown that (under appropriate hypotheses) catching-up optimal solutions asymptotically approach a unique optimal steady state, and thus enjoy the so-called “turnpike” property found in the economics literature. By combining this result with an associated optimal control problem, the desired existence result is obtained. Furthermore, it is remarked that, in addition to extending these earlier works to the time-delay case, the results presented below utilize convexity, seminormality, and growth hypotheses that in some cases are weaker than those encountered in the above-mentioned papers.
41 citations
TL;DR: In this article, a class of continuous-time generalized fractional programming problems with nonlinear operator inequality and linear operator equality constraints is considered. But the primal and dual problems considered in this paper contain, as special cases, the continuous time analogues of various primal-dual pairs of similar problems previously studied in the areas of finite-dimensional linear, quadratic, and nonlinear programming.
36 citations
TL;DR: In this paper, the necessary and sufficient conditions for the existence of an optimal solution of a vector-valued, n-set functions optimization problem were obtained for the case of n = 2.
28 citations
TL;DR: In this paper, the condition number of the Bernstein coefficients of a polynomial P(t) on a given interval [a, b] was shown to have the simple form K. This suggests a practical rule-of-thumb in assessing how far Bezier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points.
Abstract: The linear map M that takes the Bernstein coefficients of a polynomial P(t) on a given interval [a, b] into those on any subinterval [a, b] is specified by a stochastic matrix which depends only on the degree n of P(t) and the size and location of [a, T] relative to [a, b]. We show that in the 11 * 110j-norm, the condition number of M has the simple form K. (M) = [2f max(u7, VO)], where u5 = (m a)/(b a) and v, = (b m)/(b a) are the barycentric coordinates of the subinterval midpoint m = 2 (a + b) , and f denotes the "zoom" factor (b a)/(b a) of the subdivision map. This suggests a practical rule-of-thumb in assessing how far Bezier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points. The exponential growth of Koo (M) with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.
21 citations
19 citations
TL;DR: Both error analysis and graphical experiments given prove the importance of those combinations to computer vision, image processing, graphs and pattern recognition.
16 citations
TL;DR: In this article, it was shown that such approximations can yield strictly smaller error than even optimal linear ones in Hubert space (cf. [KW], [TWW], [PW]).
Abstract: In this note we answer two open questions on the finite-dimensional approximation of linear operators in Banach spaces. The first result establishes bounds on the ratio a of the error of adaptive approximations to the error of nonadaptive approximations of linear operators (see [PW], open problem 1); terms are defined more precisely below. This result is of interest partly because of its connection to questions regarding the error arising in parallel computational solutions of linear problems in infinite dimension (see [TWW], [PW]). The second result concerns (possibly nonlinear) continuous finite-dimensional approximations of infinite-dimensional linear operators in Banach space (see [KW]). It is shown that such approximations can yield strictly smaller error than even optimal linear ones. This statement has been shown to be false in Hubert space (cf. [KW]). We defer discussion of related results to give some precise definitions. Let S: F —• G be a bounded linear operator from a linear space F to a Banach space G. We wish to evaluate S at an element f G F (the "problem element"), restricted to lie in a bounded balanced convex subset B of F. The element ƒ is uncertain to the extent that it is specified only by the value of its image N(f) under a finite rank operator N (to be defined below). We induce a norm on F whose unit ball is B. Let N: F —• Y be a linear operator (information operator), with Y = R finite dimensional. Decompose N into component linear functional, N = (lx, l2, /3, . . . , ln). The image Nf represents the (finite-dimensional) information available about the (high or infinite-dimensional) problem element ƒ .
17 Sep 1990
TL;DR: A suite of modules which has been used extensively in practice for this purpose and which forms the core of a number of computer packages including the National Physical Laboratory’s Data Approximation Subroutine Library is described.
Abstract: Many problems in least-squares data approximation, modelling of discrete processes and other fields require suitable linear algebra modules for their solution. This paper describes a suite of modules which has been used extensively in practice for this purpose and which forms the core of a number of computer packages including the National Physical Laboratory’s Data Approximation Subroutine Library. The suite consists of routines that treat three types of structured linear system, viz those with banded, bordered and block-angular matrices, in addition to conventional systems with full matrices. The design of the suite is discussed, and the coverage of the suite given. Particular attention is paid to the use of the suite in solving rank-deficient least-squares problems.
TL;DR: In this paper, conditions are obtained under which R(T+S) is topologically complemented whenever S belongs to the class of precompact operators, or to some wider class.
TL;DR: In this article, a linear map on the space of n × n real symmetric matrices which maps the set of matrices having inertia (n 2, n 2, 0) into itself is presented.
TL;DR: In this paper, the notion of a symmetrizable finite difference oper- ator was introduced and sufficient conditions for its stability were presented. But the conditions for the existence of such operators are still open.
Abstract: We introduce the notion of a symmetrizable finite difference oper- ator and prove that such operators are stable. We then present some sufficient conditions for symmetrizability. One of these extends H.-O. Kreiss' theorem on dissipative difference schemes for hyperbolic equations to a more general case with full (jc , invariable coefficients.
Journal Article•
TL;DR: In this paper, the spectral assignability properties of the linear map A+BF are exploited under the structural constraints that a given controlled invariant complement of the imaginary part of B are both (A+BF) invariants.
Abstract: The spectral assignability properties of the linear map A+BF are exploited under the structural constraints that a given controlled invariant complement of the imaginary part of B are both (A+BF) invariants. Correlations with previous results are established, and the duality is performed with conditioned invariants. Direct applications are found in the linear geometric theory of many reduced-order syntheses, e.g. those of observers, regulators, etc. >
TL;DR: In this paper, it was shown that every surjective linear mapping from the mth tensor space of U to mth symmetric product space over U is induced by m isomorphisms on U if dim U ≥ 3.
Abstract: Let U be a vector space over a field of characteristic 0. We show in this paper that every surjective linear mapping from the mth tensor space of U to the mth symmetric product space over U that takes nonzero decomposable elements to nonzero decomposable elements is induced by m isomorphisms on U if dim U ≥ 3. The result is applied to symmetry classes of tensors over U associated with permutation groups and the character identically 1.
TL;DR: The use of the linear transformation method to systolize the Warshall algorithm for computing the transitive closure of a graph on a mesh-connected array (without wraparound connections) is discussed and the technique is extended to design linear systolic arrays.
Abstract: The use of the linear transformation method to systolize the Warshall algorithm for computing the transitive closure of a graph on a mesh-connected array (without wraparound connections) is discussed. The technique is extended to design linear systolic arrays. The advantage of this approach is easy verification of correctness, as well as synthesis of a family of arrays with tradeoffs between I/O bandwidth, number of processing elements, and local storage. The technique can be further refined to cope with problems that entail nonconstant dependency vectors. >
TL;DR: In this paper, the convergence speed of successive approximations to solutions of linear operator equations is characterized by the growth of the (Fredholm) resolvent when approaching the peripheral spectrum.
Abstract: The aim of this paper is to describe some relations between the convergence speed of successive approximations to solutions of linear operator equations, on the one hand, and various spectral properties of the corresponding operators, on the other. We shall show, in particular, that the estimates for the convergence speed of successive approximations is basically determined by certain properties of the pheripheral spectrum of the operator involved (recall that the peripheral spectrum is that part of the spectrum which lies on the boundary, i.e. consists of numbers with absolute values equal to the spectral radius). Equivalently, the convergence speed is characterized by the growth of the (Fredholm) resolvent when approaching the peripheral spectrum. Interestingly, these properties are essentially different for Volterra and non-Volterra operators, where by Volterra operator we mean, as usual, an operator whose spectrum consists only of zero.
TL;DR: This work presents a neural convolution network with shift invariant coupling that nevertheless exhibits autoassociative restoration of distorted images and has one more advantage: associative recall does not depend on object position.
Abstract: Processing images by a neural network means performing a repeated sequence of operations on the images. The sequence consists of a general linear transformation and a nonlinear mapping of pixel intensities. The general (shift variant) linear transformation is time consuming for large images if done with a serial computer. A shift invariant linear transformation can be implemented much easier by fast Fourier transform or optically, but the shift invariant transform has fewer degrees of freedom because the coupling matrix is Toeplitz. We present a neural convolution network with shift invariant coupling that nevertheless exhibits autoassociative restoration of distorted images. Besides the simple implementation, the network has one more advantage: associative recall does not depend on object position.
TL;DR: In this article, the surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined.
Abstract: The surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.
Journal Article•
TL;DR: Problem: Find all occurrences of the pattern in the text where the pattern may appear subject to any one of these transformations.
Abstract: Suppose we are given two strings of real numbers. The longer string is called text and the other is called pattern. We consider problems within the following framework. Suppose each symbol of the pattern was modified by any transformation which is a member in some family of transformations. Find all occurrences of the pattern in the text where the pattern may appear subject to any one of these transformations. Problems are introduced and efficient algorithms are given.
TL;DR: In this article, iso-spectral perturbations of operators of the formHA=A*A withA being a densely defined closed linear operator from a Hilbert space to another Hilbert space were considered.
Abstract: We consider (essentially) iso-spectral perturbations of operators of the formHA=A*A withA being a densely defined closed linear operator from a Hilbert space ℋ to another Hilbert space ℋ We perturbHA by perturbingA asA+B withB being a linear operator from ℋ to ℋ Two classes ofB are defined so as to obtain (essentially) iso-spectral perturbations ofHA The abstract results are applied to Schrodinger operators Our approach gives also a mathematical unification for the so-called ‘factorization method’ in quantum mechanics
01 Jan 1990
TL;DR: Techniques to find optimal solutions to three related problems to find time optimal and conflict-free mappings are presented and both outperform previously proposed approaches in terms of computational complexity and/or optimality.
Abstract: Three related problems, among others, are faced when trying to execute an algorithm on a parallel machine. The scheduling problem deals with minimizing the time taken to execute all computations of the algorithm without violating data dependencies. The independent partitioning problem deals with partitioning the algorithm into blocks so that no data communications take place between computations in different blocks, and the conflict-free mapping problem involves mapping n-dimensional algorithms (algorithms with n nested loops) into lower dimensional processor arrays without computation conflicts. In this thesis, techniques to find optimal solutions to these problems are presented. Optimality is guaranteed when the techniques are applied to algorithms with uniform dependencies and linear schedules are used. These algorithms occur frequently in scientific computing and digital signal processing applications and can often be coded as nested-loop programs. The proposed solutions can be used in optimizing compilers and to map algorithms into processor arrays, especially to program bit-level processor arrays.
A uniform dependence algorithm consists of a set of indexed computations and a set of uniform dependence vectors. If one computation uses data generated by another computation, then this data dependence is represented by the difference of their indices (called dependence vector). A dependence vector is uniform if its value is independent of the indices of computations. Linear schedules are a special class of schedules described by a linear mapping of computation indices into time. The complexity of the proposed method to identify optimal linear schedules is independent of the algorithm size. Also, linear schedules are compared with free schedules, the best schedules possible. The comparison indicates that optimal linear schedules can be as efficient as free schedules and identifies a class of algorithms for which this is always true. Two methods are presented which find independent algorithm partitions, and both outperform previously proposed approaches in terms of computational complexity and/or optimality. To map n-dimensional algorithms into lower dimensional processor arrays, necessary and sufficient conditions are derived for a mapping to be conflict-free, that is no two or more computations are mapped into the same processor and the same execution time. By these conditions and other optimization techniques, procedures are proposed to find time optimal and conflict-free mappings.
TL;DR: In this article, dispersion orderings are considered, including dispersion-diminishing linear transformations, concentration properties of which are shown, and their properties studied when σ is positive definite.
Abstract: Let D(σ) consist of matrices congruent to and dominated by a given matrix σ , and let T(σ) be the corresponding congruent transformations. These classes are characterized and their properties studied when σ is positive definite. Dispersion orderings are considered, including dispersion-diminishing linear transformations, concentration properties of which are shown. Arbitrary linear transformations are decomposed into contractions, isometries and dilations on subspaces relative to Mahalanobis norms. Applications are noted in statistical process control and linear inference
TL;DR: A method for finding a linear transformation of an initial pattern space into a one-dimensional new space, which optimizes the L2 distance between density functions, using an orthogonal expansion with Hermite functions to compute the criterion.