Showing papers on "Rate of convergence published in 1984"

Analysis of mixed finite elements methods for the Stokes problem: a unified approach

Abstract: We develop a method for the analysis of mixed finite element methods for the Stokes problem in the velocity-pressure formulation. A technical "macroelement condition", which is sufficient for the classical Babuska-Brezzi inequality to be valid, is introduced. Using this condition,we are able to verify the stability, and optimal order of convergence, of several known mixed finite element methods.

Approximate solutions of the Bellman equation of deterministic control theory

Abstract: We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.

Persistence of excitation conditions and the convergence of adaptive schemes

Abstract: In the study of the behavior of adaptive filtering algorithms, persistence of excitation of the input process arises as a sufficient condition for convergence and, perhaps more importantly, for convergence rate of the parameter estimates. In this paper the underlying nature of the persistence requirement is presented and discussed, relating its normal specification in terms of moment conditions with covariance decays, etc., to sample path properties. Deterministic and stochastic persistence conditions and persistence measures are treated, as well as, persistence conditions for output-error, equation-error, and adaptive control schemes.

Local Convergence of Inexact Newton Methods

Abstract: A simple affine invariant condition which ensures local convergence of inexact Newton methods is provided. This condition is closely linked with the characterization of superlinear convergence given by Dennis and More. Radius of convergence and rate of convergence results are derived, and we show the theory is, in some sense, sharp. We apply the theory to various inexact Newton methods, including difference Newton-like methods and the general Newton-like method in which the iterates are perturbed by rounding errors in the computation.

On the rate of convergence to equilibrium in one-dimensional systems

Abstract: We determine the essential spectral radius of the Perron-Frobenius-operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one-dimensional systems to the greatest eigenvalues of generalized Perron-Frobenius-operators of the transformations (operators which yield singular invariant measures).

Differential dynamic programming and Newton's method for discrete optimal control problems

Abstract: The purpose of this paper is to draw a detailed comparison between Newton's method, as applied to discrete-time, unconstrained optimal control problems, and the second-order method known as differential dynamic programming (DDP). The main outcomes of the comparison are: (i) DDP does not coincide with Newton's method, but (ii) the methods are close enough that they have the same convergence rate, namely, quadratic. The comparison also reveals some other facts of theoretical and computational interest. For example, the methods differ only in that Newton's method operates on a linear approximation of the state at a certain point at which DDP operates on the exact value. This would suggest that DDP ought to be more accurate, an anticipation borne out in our computational example. Also, the positive definiteness of the Hessian of the objective function is easy to check within the framework of DDP. This enables one to propose a modification of DDP, so that a descent direction is produced at each iteration, regardless of the Hessian.

An elegant Lambert algorithm

Abstract: Of the many techniques extant for solving the two-body, two-point, time-constrained orbital boundary-value problem, commonly known today as Lambert's problem, none is more conceptually elegant than the classical method devised by Gauss. The simplicity of Gauss' method would certainly have been attractive to the modern astrodynamicist except for two major flaws — the method is singular for a transfer angle of 180 degrees and the convergence rate is extremely slow when that angle is not very small. In this paper a new algorithm is described which exactly parallels both the mechanics and the elegant simplicity of the classical one but is completely devoid of the two basic faults of the original. The equations of the new method are universal and not singular for the 180 degree transfer. (They are singular for a complete revolution through 360 degrees but this should not be cause for great alarm.) Furthermore, convergence is both remarkably rapid and almost uniform as well as being essentially independent of the initial guess. It should further be emphasized that all of the advantages of Gauss' method are inherent in the new method — notably, the preservation of numerical accuracy for small transfer angles (of the order of 2 or 3 degrees, for example).

A direct approach to second‐order MCSCF calculations using a norm extended optimization scheme

Abstract: Using configuration amplitudes and the unitary generators of orbital rotation the NEO algorithm has been derived. NEO (acronym for norm extended optimization) can be implemented as a direct second‐order restricted step MCSCF optimization procedure where the quadratic convergence is obtained through solving a Hessian‐type eigenvalue problem instead of a set of linear equations. Because configuration amplitudes are used as variables, the computations in each iteration can be made comparable to those of a direct CI calculation. The NEO is especially promising because convergence is assured to a state with the desired number of negative eigenvalues of the Hessian. With the NEO procedure one achieves: (1) Any set of configurations used in a direct CI can also be used for MCSCF; (2) excellent convergence characteristics including guaranteed convergence in ground state calculations; and (3) the converged state has the desired number of negative Hessian eigenvalues.

Superconvergence phenomenon in the finite element method arising from averaging gradients

Abstract: We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.

Rate of Convergence of the Method of Alternating Projections

Abstract: A proof is given of a rate of convergence theorem for the method of alternating projections The theorem had been announced earlier in [8] without proof

A Multilevel Algorithm for Mixed Problems

Abstract: We describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise, e.g. from mixed finite element approximations of the Stokes equation. The main difficulty, besides the indefiniteness, is the lack of regularity of the solution of the corresponding continuous problem. This is overcome by introducing a scale of mesh-dependent norms. The convergence rate of the described algorithm is bounded independently of the meshsize. For convenience we only discuss Jacobi relaxation as smoothing operator in detail. In the last section we comment on Lanczos-type smoothing procedures.

Rates of convergence of variational calculations and of expectation values

Abstract: We present a mathematical and numerical analysis of the rates of convergence of variational calculations and their impact on the issue of the convergence or divergence of expectation values obtained from variational wave functions. The rate of convergence of a variational calculation is critically dependent on the ability of finite linear combinations of basis functions to simulate the nonanalyticities (cusps) in the exact wave function being approximated. A slow rate of convergence of the variational energy can imply that the corresponding variational wave functions will yield divergent expectation values of physical operators not relatively bounded by the Hamiltonian. We illustrate the sorts of problems which can arise by examining Gauss‐type approximations to hydrogenic orbitals. Since all many‐electron wave functions have cusps similar to those in hydrogenic wave functions, this simple example is relevant to variational calculations performed on atoms and molecules. Finally, we offer suggestions on what types of variational wave functions are likely to yield rapid rates of convergence for the energy and reasonable rates of convergence for physical operators such as the dipole moment operator.

Correction to "The application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas"

Abstract: The method of conjugate gradients is applied to the analysis of radiation from thin-wire antennas. With this iterative technique, it is possible to solve electrically large arbitrarily oriented wire structures without storing any matrices as is conventionally done in the method of moments. The basic difference between the proposed method and Galerkin's method, for the same expansion functions, is that for the iterative technique we are solving a least squares problem. Hence, as the order of the approximation is increased, the proposed technique guarantees a monotonic decrease of the least squared error ( \parallel AI - Y\parallel^{2} ), whereas Galerkin's method does not. Even though the method converges for any initial guess, a good one may significantly reduce the time of computation. Also, explicit error formulas are given for the rate of convergence of this method. Hence, any problem can be solved to a prespecified degree of accuracy. It is shown that the method has the advantage of a direct solution as the final solution is obtained in a finite number of steps. The method is also suitable for solving singular operator equations in which case the method monotonically converges to the least squares solution with minimum norm. Numerical results are presented for the thin-wire antennas and are compared with the solution obtained by the method of moments.

Local Defect Correction Method and Domain Decomposition Techniques

Abstract: For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution.

The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation

Abstract: The numerical solution of the Poisson equation is treated by a multigrid method for a uniform grid. The convergence rate can be estimated even for the iteration with a V-cycle independently of the shape of the domain as long as it is convex and polygonal. The smoothing effect of the Gaus-Seidel relaxation is described by a discrete seminorm which is weaker than the energy norm.

On the asymptotic order of accuracy of Tikhonov regularization

Abstract: In this paper, the rate of convergence and the order of accuracy (with respect to the error level in the data) of Tikhonov's method for approximating the minimal-norm least-square solution of an ill-posed operator equation is investigated. It is shown that, in general, this rate of convergence is arbitrarily small. It is further shown how this rate depends on some smoothness properties of the solution. All results describe optimal orders.

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

Abstract: This paper considers' a finite-element approximation of a Poisson equation in a region with a curved boundary on which a Neumann condition is prescribed. Piecewise linear and bilinear elements are used on unfitted meshes with the region of integration being replaced by a polygonal approximation. It is shown, despite the variational crimes, that the rate of convergence is still order (h) in the H norm. Numerical examples show that the method is easy to implement and that the predicted rate of convergence is obtained.

Convergence proofs and error estimates for an iterative method for conformal mapping

Abstract: The iterative method as introduced in [8] and [9] for the determination of the conformal mapping ? of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve Γ ofG is parametrized by a function ? with Lipschitz continuous derivative $$\dot \eta e 0$$ then the method converges locally in the Sobolev spaceW of 2?-periodic absolutely continuous functions with square integrable derivative. If ? is in a Holder classC 2+μ, the order of convergence is at least 1+μ. If Γ is inC l+1+μ withl?1, 0<μ<1, then the iteration converges inC l+μ. For analytic boundary curves the convergence takes place in a space of analytic functions. For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N 1?l?μ) if Γ is inC l+1+μ, andO (exp (??N/2)) if Γ is an analytic curve. The number ? in the latter formula is bounded by ??logR, whereR is the radius of the largest circle into which ? can be extended analytically such that?'(z)?0 for |z|

Approximation of nonlinear filtering problems and order of convergence

Abstract: In this paper, we consider a filtering problem where the observation is a function of a diffusion corrupted by an independent white noise. We estimate the error caused by a discretization of the time interval ; we obtain some approximations of the optimal filter which can be computed with Monte-Carlo methods and we study the order of convergence.

Parameter choice by discrepancy principles for the approximate solution of Ill-posed problems

Abstract: A general a posteriori strategy for choosing the regularization parameter as a function of the error level is given which provides nearly the optimal rate of convergence.

On the convergence of difference schemes for the approximation of solutionsu ∈ W m 2 (m>0.5) of elliptic equations with mixed derivatives

Abstract: The paper deals with such estimates of the rate of convergence of difference methods, which are compatible with the smoothness of the exact solutionu ? W 2 m (Ω),m>0.5, of elliptic equations with mixed derivatives: The error in the norm of the discrete Sobolev spaceW 2 s (?), ? denoting the set of grid points, is shown to be of the orderO(|h| m?s), 0?s

Monte Carlo random walk calculations of unimolecular dissociation of methane

Abstract: Microcanonical rate coefficients and product energy distributions are computed for the CH3+H and CH2+H2 dissociation channels of CH4 by a random walk procedure. The formulation is based on Slater theory, but uses Metropolis Monte Carlo procedures to average over the reactant phase space. We find that the convergence rates of the calculations for the CH4 system are less rapid than that obtained in previous studies on more simple systems involving the dissociation of argon clusters. The convergence rate is found to decrease as the complexity of the process increases. Thus, convergence of the rate calculations for the simple two‐center elimination reaction to form CH3+H is found to be at least an order of magnitude faster than that for the three‐center channel leading to CH2+H2. When convergence is obtained, the computed rates and product translational energy distributions are in good accord with previously obtained quasiclassical trajectory results. The computer time required to obtain converged results for...

Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems

Abstract: A variety of iterative methods considered in [3] are applied to linear algebraic systems of the formAu=b, where the matrixA is consistently ordered [12] and the iteration matrix of the Jacobi method is skew-symmetric. The related theory of convergence is developed and the optimum values of the involved parameters for each considered scheme are determined. It reveals that under the aforementioned assumptions the Extrapolated Successive Underrelaxation method attains a rate of convergence which is clearly superior over the Successive Underrelaxation method [5] when the Jacobi iteration matrix is non-singular.

Analysis of a Multigrid Method as an Iterative Technique for Solving Linear Systems

Abstract: A general class of iterative methods is introduced for solving symmetric, positive definite linear systems. These methods use two different approximations to the inverse of the matrix of the problem, one of which involves the inverse of a smaller matrix. It is shown that the methods of this class reduce the error by a constant factor at each step and that under “ideal” circumstances this constant is equal to ${{(\kappa ' - 1)} / {(\kappa ' + 1)}}$, where $\kappa '$ is the ratio of the largest eigenvalue to the $(J + 1)$ eigenvalue of the matrix, J being the dimension of the smaller matrix involved. A multigrid method is presented as an example of a method of this class, and it is shown that while the multigrid method does not quite achieve this optimal rate of convergence, it does reduce the error at each step by a constant factor independent of the mesh spacing h. The size of this constant factor and properties of the differential equation and the discretization that affect it are also discussed.

Asymptotic convergence rate of arcangeli's method for III-posed problems

Abstract: We provide asymptotic convergence rates, in terms of the error level in the data, for Arcangeli's method of regularizing an ill-posed operator equation of the first kind

An explicit analysis of Stummel's patch test examples

Abstract: A constructive approach for the treatment of Stummel's examples to the patch test is presented, together with generalizations of these examples. The strange convergence phenomenon of nonconforming approximations is discussed in detail, and a kind of modified variational equation for remedying these defects is introduced. Our variational equation uses a combination of hybrid and penalty methods that define a convergent procedure with optimal order of convergence to the exact solution with respect to a mesh-dependent norm.

A parallelized point rowwise successive over-relaxation method on a multiprocessor

Abstract: A parallelized point rowwise Successive Over-Relaxation (SOR) iterative algorithm is developed for the Heterogeneous Element Processor (HEP) multiple instruction stream computer. The classical point SOR method is not easily vectorizable with rowwise ordering of the grid points, but it can be effectively parallelized on a multiple instruction stream machine without suffering in computational and convergence rate. The details of the implementation including restructuring of a serial FORTRAN program and techniques needed to exploit the parallel processing architectural concept of the HEP are presented. The parallelized algorithm is analyzed in detail. The lessons learned in this study are documented and may provide some guidelines for similar future coding since new approaches and restructuring techniques are required for programming a multiple instruction stream machine, which are totally different than those required for programming an algorithm on a vector processor. To assess the capabilitiesof the parallelized algorithm it was used to solve the Laplace's equation on a rectangular field with Dirichlet boundary conditions. Computer run times are presented which indicate significant speed gain over a scalar version of the code. For a moderate to large size problem seventeen or more processes are required to make efficient use of the parallel processing hardware. Also, to demonstrate the capability of the algorithm for a realistic problem, it was used to obtain the numerical solution of a viscous incompressible fluid in a square cavity. Since point iterative relaxation schemes are at the core of many systems of elliptic as well as non-elliptic partial differential equations occuring in engineering and scientific applications, the present study suggests the possibility of both reducing the real time processing and increasing the scope of computational modeling.

Interpolation of the Coefficients in Nonlinear Elliptic Galerkin Procedures

Abstract: A continuity argument is employed to prove that the interpolation of the coefficients in nonlinear Galerkin procedures does not result in a reduction of the order of convergence.

A collocation method for boundary value problems of differential equations with functional arguments

Abstract: A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC 1 and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.