Stability and convergence properties of Bergman kernel methods for numerical conformal mapping

Summary In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conformal mapping of simply and doubly-connected domains. In particular, by using certain wellknown results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map.     

Constructing three-dimensional mappings onto the unit sphere with the hypercomplex Szegö kernel

In classical complex analysis the Szegö kernel method provides an explicit way to construct conformal maps from a given simply-connected domain G⊂C onto the unit disc. In this paper we revisit this method in the three-dimensional case. We investigate whether it is possible to construct three-dimensional mappings from some elementary domains into the three-dimensional unit ball by using the hypercomplex Szegö kernel. In the cases of rectangular domains, L-shaped domains, cylinders and the symmetric double-cone the proposed method leads surprisingly to qualitatively very good results. In the case of the cylinder we get even better results than those obtained by the hypercomplex Bergman method that was very recently proposed by several authors.We round off with also giving an explicit example of a domain, namely the T-piece, where the method does not lead to the desired result. This shows that one has to adapt the methods in accordance with different classes of domains.     

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind

Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential.     

A new trust region method for unconstrained optimization

In this paper, we propose a new trust region method for unconstrained optimization problems. The new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions. We also analyze the global linear convergence, local superlinear and quadratic convergence rate of the new method. Numerical results show that the new trust region method is available and efficient in practical computation.     

Analysis of finite element approximation and quadrature of Volterra integral equations

In this article we study Galerkin finite element approximations to integral equations of the Volterra type. Our prime concern is the noncoercive case, which is not covered by the standard finite element theory. The question of rates of convergence is studied for the case where an exact stiffness matrix is available, as well as the case where the latter is approximated via quadrature rules. The optimality of these rules is also considered from the point of view of the effect the choice of the quadrature has on the overall rate of convergence. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 663–672, 1997     

A numerical algorithm for the Stokes problem based on an integral equation for the pressure via conformal mappings

Summary In this paper we present an algorithm for solving numerically the Stokes problem in the plane. The known algorithms are all based on certain discretization schemes for the analytic equations. In contrast to this recent work our algorithm uses an explicit analytic solution of a certain approximating problem, which can easily be solved numerically up to machine accuracy. On the one hand this analytic formula is based on a complex representation of all solutions of the Stokes differential equations, and on the other hand it is based on the conformal mapping of the given domain on the unit disc. Therefore, a central prerequisite of our corresponding program is a program for computing this conformal mapping.     

Conformal mapping by the method of alternating projections

Summary The functional analytic principle of alternating projections is used to construct an iterative method for numerical conformal mapping of the unit disc onto regions with smooth boundaries. The result is a simple method which requires in each iterative step only two complex Fourier transforms. Local convergence can be proved using a theorem of Ostrowski. Convergence is linear. The asymptotic convergence factor is equal to the spectral radius of a certain operator. A version with overrelaxation as well as a discretized version are discussed along the same lines. For regions which are close to the unit disc convergence is fast. For some familiar regions the convergence factors can be calculated explicitly. Finally, the method is compared with Theodorsen's.Dedicated to the memory of Peter Henrici     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Rate of Convergence of Intermediate Order Statistics

Exact rates are derived for the uniform convergence of the density of intermediate order statistics towards the normal or lognormal density under certain smoothness conditions. Our methods also give the exact rate of convergence in the uniform metric and in the total variation metric.     

Persistence of attractors for one-step discretization of ordinary differential equations

We consider numerical one-step approximations of ordinary differentialequations and present two results on the persistence of attractorsappearing in the numerical system. First, we show that the upperlimit of a sequence of numerical attractors for a sequence ofvanishing time-steps is an attractor for the approximated systemif and only if for all these time-steps the numerical one-stepschemes admit attracting sets which approximate this upper limitset and attract with a uniform rate. Second, we show that ifthese numerical attractors themselves attract with a uniformrate, then they converge to some set if and only if this setis an attractor for the approximated system. In this case, wecan also give an estimate for the rate of convergence dependingon the rate of attraction and on the order of the numericalscheme.     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

The convergence of subspace trust region methods

The trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence. However, the subproblem in the trust region method is difficult or time-consuming to solve in practical computation, especially in large-scale problems. In this paper we consider a new class of trust region methods, specifically subspace trust region methods. The subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace. It is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially. We investigate the global convergence and convergence rate of these methods.     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.     

Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.     

The automorphism space Σ(G) of a domain without punctiform prime ends

Let G ⊆ ℂ be a simply connected domain and let Σ (G) be its group of conformal automorphisms with the topology of uniform chordal convergence on G. In 1984 Gaier raised the question whether the connectedness of the space Σ (G) implies that the domain G has only punctiform prime ends. As a contribution to answering this question in this paper the authors use suitable spike Junctions to construct a bounded domain without any punctiform prime end such that its automorphism space Σ (G) is not discrete, but totally disconnected.     

Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham. Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, 1090 Vienna, Austria Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany     

竞争风险场合PL型估计的强一致收敛性

本文构造了竞争风险场合分布函数的乘积极限（PL）型估计，运用经验过程的强逼近理论及Toylor展开方法，给出了PL型估计在全直线上的强一致收敛速度及其充分必要条件。     

SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally, we give an example of a contact problem where our proposed method can be applied.     

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence. Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000