Estimating the largest singular values of large sparse matrices via modified moments

We describe a procedure for determining a few of the largest singular values of a large sparse matrix. The method by Golub and Kent which uses the method of modified moments for estimating the eigenvalues of operators used in iterative methods for the solution of linear systems of equations is appropriately modified in order to generate a sequence of bidiagonal matrices whose singular values approximate those of the original sparse matrix. A simple Lanczos recursion is proposed for determining the corresponding left and right singular vectors. The potential asynchronous computation of the bidiagonal matrices using modified moments with the iterations of an adapted Chebyshev semi-iterative (CSI) method is an attractive feature for parallel computers. Comparisons in efficiency and accuracy with an appropriate Lanczos algorithm (with selective re-orthogonalization) are presented on large sparse (rectangular) matrices arising from applications such as information retrieval and seismic reflection tomography. This procedure is essentially motivated by the theory of moments and Gauss quadrature.This author's work was supported by the National Science Foundation under grants NSF CCR-8717492 and CCR-910000N (NCSA), the U.S. Department of Energy under grant DOE DE-FG02-85ER25001, and the Air Force Office of Scientific Research under grant AFOSR-90-0044 while at the University of Illinois at Urbana-Champaign Center for Supercomputing Research and Development.This author's work was supported by the U.S. Army Research Office under grant DAAL03-90-G-0105, and the National Science Foundation under grant NSF DCR-8412314.     

The existence of smooth densities for the prediction filtering and smoothing problems

The qualitative behavior of buckled states of two different models of elastic beams is studied. It is assumed that random imperfections affect the governing nonlinear equations. It is shown that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation whose coeffficients are Gaussian random variables. The corresponding asymptotic expansion for the displacement is to lowest order a Gaussian stochastic process.Work supported by NSF Grant No. DCR81-14726.Work supported by NSF Grant No. DMS87-01895.     

A Monte Carlo method for high dimensional integration

Summary A new method for the numerical integration of very high dimensional functions is introduced and implemented based on the Metropolis' Monte Carlo algorithm. The logarithm of the high dimensional integral is reduced to a 1-dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval. The improvement in accuracy is found to be substantial comparing to the conventional crude Monte Carlo integration. Several numerical demonstrations are made, and variability of the estimates are shown.     

Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1–3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.     

Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras

For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.     

A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

Noncommuting selfmaps and invariant approximations

Common fixed-point results are established for a new class of noncommuting selfmaps. We apply them to obtain several invariant approximation results which unify, extend, and complement well-known results.     

On the augmented system approach to sparse least-squares problems

Summary We study the augmented system approach for the solution of sparse linear least-squares problems. It is well known that this method has better numerical properties than the method based on the normal equations. We use recent work by Arioli et al. (1988) to introduce error bounds and estimates for the components of the solution of the augmented system. In particular, we find that, using iterative refinement, we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute. The final error and all our error estimates are much better than the classical or Skeel's error analysis (1979) indicates. Moreover, we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the Björck scaling (1967) on these estimates. We illustrate this with runs both on large-scale practical problems and contrived examples, comparing the numerical behaviour of the augmented systems approach with a code using the normal equations. These experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach, it is comparable or better when the least-squares matrix has a full row, and is, in any case, much more stable and robust.This author was visiting Harwell and was funded by a grant from the Italian National Council of Research (CNR), Istituto di Elaborazione dell'Informazione-CNR, via S. Maria 46, I-56100 Pisa, ItalyThis author was visiting Harwell from Faculty of Mathematics and Computer Science of the University of Amsterdam     

增算子的不动点和耦合不动点

Under some general continuous and compact conditions, the existence problems of fiked points andd coupled fixed points for increasing operators are studied. an application, we utilize the results obtained to study the existence of solutions for differential inclusions in Banach spaces.     

Limits of parallelism in explicit ODE methods

Numerical methods for ordinary initial value problems that do not depend on special properties of the system are usually found in the class of linear multistage multivalue methods, first formulated by J.C. Butcher. Among these the explicit methods are easiest to implement. For these reasons there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in parallel. Each such group of concurrent function evaluations can be regarded as a stage of the method. However, it turns out that parallelism affords only limited opportunity for reducing the computing time with such methods. This is most evident for the simple linear homogeneous constant-coefficient test problem, whose solution is essentially a matter of approximating the exponential by an algebraic function. For a given number of stages and a given number of saved values, parallelism offers a somewhat enlarged set of algebraic functions from which to choose. However, there is absolutely no benefit in having the degree of parallelism (number of processors) exceed the number of saved values of the method. Thus, in particular, parallel one-step methods offer no speedup over serial one-step methods for the standard linear test problem. Although the implication of this result for general nonlinear problems is unclear, there are indications that dramatic speedups are not possible in general. Also given are some results relevant to the construction of methods.Work supported in part by National Science Foundation grants DMS 89 11410 and DMS 90 15533 and by US Department of Energy grant DOE DEFG02-87ER25026. Work of the second author was completed while at the University of Illinois.