A saddle point variational formulation for projection–regularized parameter identification

Summary. This paper is concerned with the ill-posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection. As the theory has shown, the ansatz functions for the parameter have to be sufficiently smooth. In this paper we show that these – for a practical implementation unrealistic – smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation. We prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level goes to zero in the case of noisy data, as well as convergence rates under additional smoothness conditions. Received August 4, 2000 / Revised version received March 21, 2001 / Published online October 17, 2001     

On constrained Newton linearization and multigrid for variational inequalities

Summary. We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments. Received March 22, 1999 / Revised version received February 24, 2001 / Published online October 17, 2001     

Convergence acceleration by varying time‐step size using Bi‐CGSTAB method for turbulent flow computation

A design of varying step size approach both in time span and spatial coordinate systems to achieve fast convergence is demonstrated in this study. This method is based on the concept of minimization of residuals by the Bi‐CGSTAB algorithm, so that the convergence can be enforced by varying the time‐step size. The numerical results show that the time‐step size determined by the proposed method improves the convergence rate for turbulent computations using advanced turbulence models in low Reynolds‐number form, and the degree of improvement increases with the degree of the complexity of the turbulence models. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 454–474, 2001.     

A New Third-Order Iterative Processfor Solving Nonlinear Equations

 In this paper, we build up a modification of the Midpoint method, reducing its operational cost without losing its cubical convergence. Then we obtain a semilocal convergence result for this new iterative process and by means of several examples we compare it with other iterative processes. (Received 11 April 2000; in final form 27 March 2001)     

Asymptotic Estimates for Interpolation and Constrained Approximation in H^{2} by Diagonalization of Toeplitz Operators

Sharp convergence rates are provided for interpolation and approximation schemes in the Hardy space that use band-limited data. By means of new explicit formulae for the spectral decomposition of certain Toeplitz operators, sharp estimates for Carleman and Krein-Nudel'man approximation schemes are derived. In addition, pointwise convergence results are obtained. An illustrative example based on experimental data from a hyperfrequency filter is provided. Submitted: August 17, 2001? Revised: November 5, 2001.     

On the convergence of the Bermúdez-Moreno algorithm with constant parameters

Summary. Bermúdez-Moreno [5] presents a duality numerical algorithm for solving variational inequalities of the second kind. The performance of this algorithm strongly depends on the choice of two constant parameters. Assuming a further hypothesis of the inf-sup type, we present here a convergence theorem that improves on the one presented in [5]: we prove that the convergence is linear, and we give the expression of the asymptotic error constant and the explicit form of the optimal parameters, as a function of some constants related to the variational inequality. Finally, we present some numerical examples that confirm the theoretical results. Received June 28, 1999 / Revised version received February 19, 2001 / Published online October 17, 2001     

Cauchy completions of MV-algebras

Abstract. An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.?We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra. Received August 8, 2001; accepted in final form October 18, 2001.     

Weak set convergence and its application to projection methods

There are many examples in Numerical Analysis where convergence of approximate solutions to a solution of the original problem can not be shown in the sense of a norm topology but in the sense of weak convergence ([6], [9], [10]).

Moreover, (global) solutions are often not unique such that a concept of set convergence instead of convergence in the usual sense is more convenient and reasonable ([1], [2]). This particularly holds if weakly formulated problems are under consideration.

When dealing with problems where both situations coincide, a concept of weak set convergence seems to be adequate. Such a concept is developed and will be applied to certain projections methods.     

The Quality of Approximation of Brun’s Algorithm in Three Dimensions

 We show that for the three-dimensional multiplicative Brun’s algorithm, the exponent of convergence is , i.e. there is a such that for almost all ,

A new characterization of convergent multivariate subdivision schemes with nonnegative masks

We present in this paper new necessary and sufficient conditions for convergence of multivariate subdivision schemes with nonnegative finite masks, which simplify the assertion obtained by Wang in (J Approx Theory 113:207–220, 2001). Moreover, we construct an example, which shows that the convergence behaviors for univariate subdivision schemes and multivariate ones are essentially different.     

On the Power of Standard Information for Weighted Approximation

We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for standard information, as well as the equivalence of tractability and strong tractability for the two classes. April 11, 2001. Final version received: May 29, 2001.     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002     

Convergence of trajectories in fractal interpolation of stochastic processes

The notion of fractal interpolation functions (FIFs) can be applied to stochastic processes. Such construction is especially useful for the class of α-self-similar processes with stationary increments and for the class of α-fractional Brownian motions. For these classes, convergence of the Minkowski dimension of the graphs in fractal interpolation of the Hausdorff dimension of the graph of original process was studied in [Herburt I, Małysz R. On convergence of box dimensions of fractal interpolation stochastic processes. Demonstratio Math 2000;4:873–88. [11]], [Małysz R. A generalization of fractal interpolation stochastic processes to higher dimension. Fractals 2001;9:415–28. [15]], and [Herburt I. Box dimension of interpolations of self-similar processes with stationary increments. Probab Math Statist 2001;21:171–8. [10]].We prove that trajectories of fractal interpolation stochastic processes converge to the trajectory of the original process. We also show that convergence of the trajectories in fractal interpolation of stochastic processes is equivalent to the convergence of trajectories in linear interpolation.     

Discretization of backward semilinear stochastic evolution equations

The study on discretization and convergence of BSDEs rapidly developed in recent years. We especially mention the work of Ph. Briand, B. Delyon and J. Mémin [Donsker-type Theorem for BSDEs, Electron. Comm. Probab. 6 (2001) 1–14 (electronic)]. They got the convergence of the sequence Yⁿ$Y^n$ and pointed out that the weak convergence of filtrations was a powerful tool in this topic. In this paper, we first study the weak convergence of filtrations in Hilbert space. Using this tool, we get the convergence about discretization of backward semilinear stochastic evolution equations (BSSEEs for short).     

An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations

We revisit a one-step intermediate Newton method for the iterative computation of a zero of the sum of two nonlinear operators that was analyzed by Uko and Velásquez (Rev. Colomb. Mat. 35:21?C27, 2001). By utilizing weaker hypotheses of the Zabrejko-Nguen kind and a modified majorizing sequence we perform a semilocal convergence analysis which yields finer error bounds and more precise information on the location of the solution that the ones obtained in Rev. Colomb. Mat. 35:21?C27, 2001. This error analysis is obtained at the same computational cost as the analogous results of Uko and Velásquez (Rev. Colomb. Mat. 35:21?C27, 2001). We also give two generalizations of the well-known Kantorovich theorem on the solvability of nonlinear equations and the convergence of Newton??s method. Finally, we provide a numerical example to illustrate the predicted-by-theory performance of the Newton iterates involved in this paper.     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

A BFGS-IP algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach

This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of the iterates is progressively enforced thanks to shift variables and an exact penalty approach. Global and q-superlinear convergence is obtained for a fixed penalty parameter; global convergence to the analytic center of the optimal set is ensured when the barrier parameter tends to zero, provided strict complementarity holds. Received: December 21, 2000 / Accepted: July 13, 2001?Published online February 14, 2002     

A Mean Value Theorem for Dirichlet Series and a General Divisor Problem

 We consider the mean value formula for general Dirichlet series by applying the approximate functional equation of Ramachandra’s type, and derive the sum formula for its coefficients. The improvement of Landau’s classical results is established for the general divisor function. We also obtain the asymptotic behaviour of the mean value when the real part of s is near the abscissa of absolute convergence. Received 2 July 2001; in revised form 29 October 2001     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab