Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Polynomial representations of the Lucas logarithm

We deduce exact formulas for polynomials representing the Lucas logarithm and prove lower bounds on the degree of interpolation polynomials of the Lucas logarithm for subsets of given data.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

Existence theorems for perturbed abstract measure differential equations

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11-23] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503-510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497-506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53-54] as special cases under weaker continuity condition.     

Alternating Krylov subspace image restoration methods

Alternating methods for image deblurring and denoising have recently received considerable attention. The simplest of these methods are two-way methods that restore contaminated images by alternating between deblurring and denoising. This paper describes Krylov subspace-based two-way alternating iterative methods that allow the application of regularization operators different from the identity in both the deblurring and the denoising steps. Numerical examples show that this can improve the quality of the computed restorations. The methods are particularly attractive when matrix-vector products with a discrete blurring operator and its transpose can be evaluated rapidly, but the structure of these operators does not allow inexpensive diagonalization.     

Biconjugate direction methods in Krylov subspaces

The paper addresses the orthogonal and variational properties of a family of iterative algorithms in Krylov subspaces for solving the systems of linear algebraic equations (SLAE) with sparse nonsymmetric matrices. There are proposed and studied a biconjugate residual method, squared biconjugate residual method, and stabilized conjugate residual method. Some results of numerical experiments are given for a series of model problems as well.     

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.     

Relaxation strategies for nested Krylov methods

This paper studies computational aspects of Krylov methods for solving linear systems where the matrix–vector products dominate the cost of the solution process because they have to be computed via an expensive approximation procedure. In recent years, so-called relaxation strategies for tuning the precision of the matrix–vector multiplications in Krylov methods have proved to be effective for a range of problems. In this paper, we will argue that the gain obtained from such strategies is often limited. Another important strategy for reducing the work in the matrix–vector products is preconditioning the Krylov method by another iterative Krylov method. Flexible Krylov methods are Krylov methods designed for this situation. We combine these two approaches for reducing the work in the matrix–vector products. Specifically, we present strategies for choosing the precision of the matrix–vector products in several flexible Krylov methods as well as for choosing the accuracy of the variable preconditioner such that the overall method is as efficient as possible. We will illustrate this computational scheme with a Schur-complement system that arises in the modeling of global ocean circulation.     

Polynomial matrix-fraction representations for linear time-varying systems

This paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time-varying discrete-time systems. It is shown that the time-varying analog of the polynomial model always has a free state-space. Necessary and sufficient conditions for the existence of Bezout factorizations are obtained in terms system theoretic properties of state-space realizatios.     

Krylov projection methods for linear Hamiltonian systems

Numerical Algorithms - In this paper, we are concerned with the numerical treatment of a recent diffuse interface model for two-phase flow of electrolyte solutions (Campillo-Funollet et al., SIAM...     

Three-dimensional eddy-current computation using Krylov subspace methods

This paper considers the numerical solution of a transmissionboundary-value problem for the time-harmonic Maxwell equationswith the help of a special finite-volume discretization. Applyingthis technique to several three-dimensional test problems, weobtain large, sparse, complex linear systems, which are solvedby four types of algorithm, using biconjugate gradients, squaredconjugate gradients, stabilized conjugate gradients, and generalizedminimal residuals, respectively. Wecombine these methods withsuitably chosen preconditioning matrices and compare the speedof convergence.     

Rational Krylov sequence methods for eigenvalue computation

Algorithms to solve large sparse eigenvalue problems are considered. A new class of algorithms which is based on rational functions of the matrix is described. The Lanczos method, the Arnoldi method, the spectral transformation Lanczos method, and Rayleigh quotient iteration all are special cases, but there are also new algorithms which correspond to rational functions with several poles. In the simplest case a basis of a rational Krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of Hessenberg matrices.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.     

Recent advances in Krylov subspace spectral methods

This paper reviews the main properties, and most recent developments, of Krylov subspace spectral (KSS) methods for time-dependent variable-coefficient PDE. These methods use techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral domain in order to achieve high-order accuracy in time and stability characteristic of implicit time-stepping schemes, even though KSS methods themselves are explicit. In fact, for certain problems, 1-node KSS methods are unconditionally stable. Furthermore, these methods are equivalent to high-order operator splittings, thus offering another perspective for further analysis and enhancement. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal iteration methods utilizing the generalized Krylov basis

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 86–96.     

Fast generalized cross validation using Krylov subspace methods

The task of fitting smoothing spline surfaces to meteorological data such as temperature or rainfall observations is computationally intensive. The generalized cross validation (GCV) smoothing algorithm, if implemented using direct matrix techniques, is O(n ³) computationally, and memory requirements are O(n ²). Thus, for data sets larger than a few hundred observations, the algorithm is prohibitively slow. The core of the algorithm consists of solving series of shifted linear systems, and iterative techniques have been used to lower the computational complexity and facilitate implementation on a variety of supercomputer architectures. For large data sets though, the execution time is still quite high. In this paper we describe a Lanczos based approach that avoids explicitly solving the linear systems and dramatically reduces the amount of time required to fit surfaces to sets of data.      

Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.