A generalization of parameterized inexact Uzawa method for generalized saddle point problems

Recently, a class of parameterized inexact Uzawa methods has been proposed for generalized saddle point problems by Bai and Wang [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932], and a generalization of the inexact parameterized Uzawa method has been studied for augmented linear systems by Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. (2008)]. This paper is concerned about a generalization of the parameterized inexact Uzawa method for solving the generalized saddle point problems with nonzero (2, 2) blocks. Some new iterative methods are presented and their convergence are studied in depth. By choosing different parameter matrices, we derive a series of existing and new iterative methods, including the preconditioned Uzawa method, the inexact Uzawa method, the SOR-like method, the GSOR method, the GIAOR method, the PIU method, the APIU method and so on. Numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact Uzawa methods.     

The corrected Uzawa method for solving saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by correcting the Uzawa algorithm, which has been proposed in [K. Arrow, L. Hurwicz, H. Uzawa, Studies in nonlinear programming, Stanford University Press, Stanford, CA, 1958]. We call this method as corrected Uzawa (CU) method. The convergence of the CU method is analyzed for solving nonsingular saddle point problem as well as the semi‐convergence for the singular case. First, the corrected model for the Uzawa algorithm is established, and the CU algorithm is presented. Then we study the geometric meaning of the CU model. Moreover, we introduce the overall reduction coefficient α to measure the effect of the CU process. It is shown that the CU method converges faster than the Uzawa method and several other methods if the overall reduction coefficient α satisfies certain conditions. Numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the CU method. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimum parameters for the generalized ADI method

Summary. In the context of the generalized ADI method, we are concerned with the problem of finding in the set of rational functions r with numerator degree m and denominator degree n an element that minimizes where E,F are disjoint real intervals. By extending a recent analysis by Levin and Saff, we present an explicit formula for choosing the pair (m,n) for given m +n. Furthermore, we provide a characterization of and a Remes type algorithm for its determination. Extensive numerical computations furnish some comparison of with asymptotically optimal solutions based on Fejér-Walsh and Leja-Bagby points. Received September 6, 1996 / revised version received June 30, 1997     

On a theorem of Stein-Rosenberg type in interval analysis

Summary In classical numerical analysis the asymptotic convergence factor (R ₁-factor) of an iterative processx ^m+1=Ax^m+b coincides with the spectral radius of then×n iteration matrixA. Thus the famous Theorem of Stein and Rosenberg can at least be partly reformulated in terms of asymptotic convergence factor. Forn×n interval matricesA with irreducible upper bound and nonnegative lower bound we compare the asymptotic convergence factor ( _T ) of the total step method in interval analysis with the factor _S of the corresponding single step method. We derive a result similar to that of the Theorem of Stein and Rosenberg. Furthermore we show that _S can be less than the spectral radius of the real single step matrix corresponding to the total step matrix |A| where |A| is the absolute value ofA. This answers an old question in interval analysis.     

On real and complex-valued bivariate Chebyshev polynomials

In the real uniform approximation of the function x^myⁿ by the space of bivariate polynomials of total degree m + n − 1 on the unit square, the product of monic univariate Chebyshev polynomials yields an optimal error. We exploit the fundamental Noether's theorem of algebraic curves theory to give necessary and sufficient conditions for unicity and to describe the set of optimal errors in case of nonuniqueness. Then, we extend these results to the complex approximation on biellipses. It turns out that the product of Chebyshev polynomials also provides an optimal error and that the same kind of uniqueness conditions prevail in the complex case. Yet, when nonuniqueness occurs, the characterization of the set of optimal errors presents peculiarities, compared to the real problem.     

On a class of order pick strategies in paternosters

We study the travel time needed to pick n items in a paternoster, operating under the m-step strategy. This means that the paternoster chooses the shortest route among the ones that change direction at most once, and after collecting at most m items. For random pick positions, we find the distribution and moments of the travel time, provided n>2m. It appears that, already for m=2, the m-step strategy is very close to optimal, and better than the nearest item heuristic.     

A proof of Parisi’s conjecture on the random assignment problem

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k ² conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n. AcknowledgementWe thank Mireille Bousquet-Mélou and Gilles Schaeffer for introducing the problem to us. We also thank an anonymous referee for suggesting some improvements of the exposition.     

Two algorithms for two-phase Stefan type problems

In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.     

Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat—Weber location problem is to find a point in ⁿ that minimizes the sum of the weighted Euclidean distances fromm given points in ⁿ . A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.     

Approximation in statistical sense to <Emphasis Type="Italic">n</Emphasis>-variate B-continuous functions by positive linear operators

Our primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.     

Uzawa iteration method for stokes type variational inequality of the second kind

In this paper,the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind.Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity.Moreover,the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ?.Subsequently,the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate.Finally,we give the numerical results to verify the feasibility of the Uzawa algorithm.     

Generalized Solution of Linear Systems and Image Restoration

Let n, m be positive integers; we consider m×n real linear systems. We define regularized solutions of a linear system as the minimizers of an optimization problem. The objective function of this optimization problem can be seen as the Tikhonov functional when the p-norm is considered instead of the Euclidean norm. The cases p=1 and p= are studied. This analysis is used to restore defocused synthetic images and real images with encouraging results.     

On the asymptotic convergence factor of the total step method in interval computation

For a special class of n×n interval matrices A we derive a necessary and sufficient condition for the asymptotic convergence factor α of the total step method x^(m+1)=Ax^(m)+b to be less than the spectral radius ϱ(|A|) of the absolute value |A| of A.     

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n ² L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Numerical differentiation inspired by a formula of R.P. Boas

First, we briefly discuss three classes of numerical differentiation formulae, namely finite difference methods, the method of contour integration, and sampling methods. Then we turn to an interpolation formula of R.P. Boas for the first derivative of an entire function of exponential type bounded on the real line. This formula may be classified as a sampling method. We improve it in two ways by incorporating a Gaussian multiplier for speeding up convergence and by extending it to higher derivatives. For derivatives of order s, we arrive at a differentiation formula with N^′ nodes that applies to all entire functions of exponential type without any additional restriction on their growth on the real line. It has an error bound that converges to zero like e^-αN/N^m as N→∞, where α>0 and N^′=2N, m=3/2 for odd s while N^′=2N+1, m=5/2 for even s. Comparable known formulae have stronger hypotheses and, for the same α, they have m=1/2 only. We also deduce a direct (error-free) generalization of Boas’ formula (Corollary 5). Furthermore, we give a modification of the main result for functions analytic in a domain and consider an extension to non-analytic functions as well. Finally, we illustrate the power of the method by examples.     

On the wiener type regularity of a boundary point for the polyharmonic operator

It is shown that the Wiener regularity of a boundary point with respect to the m-harmonic operator is a local property for the space dimensions n=2m,2m+1,2m+2 with m > 2 and n = 4,5,6,7 with m = 2. An estimate for the continuity modulus of the solution formulated in terms of the Wiener type m-capacitary integral is obtained for the same n and m.     

Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution

It is well known that if P is a nonnegative matrix, then its spectral radius is an eigenvalue of P (Perron-Frobenius theorem). In this paper it is shown that if P is an n × n nonnegative matrix and it commutes with a nonnegative symmetric involution when n=4m+3, then (1) P has at least two real eigenvalues if n=4m or 4m + 2, (2) P has at least one real eigenvalue if n=4m+1, and (3) P has at least three real eigenvalues if n=4m+3, where m is a nonnegative integer and n ? 1. Examples are given to show that these results are the best possible, and nonnegative symmetric involutions are classified.     

The number of limit cycles of cubic Hamiltonian system with perturbation

In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m) limit cycles, and has at most n limit cycles if m=1. If m=1, n=1 and m=1, n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles are given.     

Multigrid methods for Toeplitz linear systems with different size reduction

Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size reductions. We assume that the size n of the coefficient matrix is divisible by g≥2 such that at the lower level the system is reduced to one of size n/g, by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the two-grid method and of the multigrid method is proved, when the number θ∈ℕ of recursive calls is such that 1<θ<g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at “mirror points” and the standard two-grid method with g=2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas, both for circulant and Toeplitz matrices.