Showing papers on "Rate of convergence published in 2002"
TL;DR: In this article, a point interpolation meshless method is proposed based on combining radial and polynomial basis functions, which makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least-squares approximation.
Abstract: A point interpolation meshless method is proposed based on combining radial and polynomial basis functions. Involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis. This non-singularity is useful in constructing well-performed shape functions. Furthermore, the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property. This makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least-squares approximation. In addition, the partial derivatives of shape functions are easily obtained, thus improving computational efficiency. Examples on curve/surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high. Copyright © 2002 John Wiley & Sons, Ltd.
909 citations
TL;DR: The method allows for discontinuities, internal to the elements, in the approximation across the interface, and it is shown that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh.
832 citations
TL;DR: In this paper, it was shown that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschnitz condition.
Abstract: Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.
570 citations
TL;DR: A coupled multiphase propagation that imposes the idea of mutually exclusive propagating curves and increases the robustness as well as the convergence rate is proposed and has been validated using three important applications in computer vision.
406 citations
TL;DR: Under fairly general conditions, this paper proves the almost sure convergence of the complete algorithm due to Longstaff and Schwartz and determines the rate of convergence of approximation two and proves that its normalized error is asymptotically Gaussian.
Abstract: Recently, various authors proposed Monte-Carlo methods for the computation of American option prices, based on least squares regression. The purpose of this paper is to analyze an algorithm due to Longstaff and Schwartz. This algorithm involves two types of approximation. Approximation one: replace the conditional expectations in the dynamic programming principle by projections on a finite set of functions. Approximation two: use Monte-Carlo simulations and least squares regression to compute the value function of approximation one. Under fairly general conditions, we prove the almost sure convergence of the complete algorithm. We also determine the rate of convergence of approximation two and prove that its normalized error is asymptotically Gaussian.
359 citations
TL;DR: It is proved that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varePSilon\to 0$.
Abstract: We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$.
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example.
347 citations
TL;DR: In this article, the authors established the R-linear convergence of the Barzilai and Borwein (BB) method for any-dimensional strongly convex quadratics and showed that the BB method is also locally Rlinear convergent for general objective functions.
Abstract: Combined with non-monotone line search, the Barzilai and Borwein (BB) gradient method has been successfully extended for solving unconstrained optimization problems and is competitive with conjugate gradient methods. In this paper, we establish the R-linear convergence of the BB method for any-dimensional strongly convex quadratics. One corollary of this result is that the BB method is also locally R-linear convergent for general objective functions, and hence the stepsize in the BB method will always be accepted by the non-monotone line search when the iterate is close to the solution.
296 citations
TL;DR: It is shown that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space, thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges.
Abstract: We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.
223 citations
TL;DR: General results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi- Bellman equations with variable coecients are obtained using systematically a tricky idea of N.V. Krylov.
Abstract: Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.
197 citations
TL;DR: An algorithm, and a 2D implementation for a fully automatic hp-adaptive strategy for elliptic problems, which confirms optimal, exponential convergence rates predicted by the theory of hp methods.
Abstract: We present an algorithm, and a 2D implementation for a fully automatic hp-adaptive strategy for elliptic problems. Given a mesh, the next, optimally refined mesh, is determined by maximizing the rate of decrease of the hp-interpolation error for a reference solution. Numerical results confirm optimal, exponential convergence rates predicted by the theory of hp methods.
192 citations
TL;DR: It is found that in general, the possibility of coordination is more likely to arise when the overall level of noise is low and when the public information is relatively informative, and higher-order uncertainty vanishes, as the noise in the signals disappears.
TL;DR: Two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners and the obtained estimates are partially sharper than the known estimates in literature.
Abstract: In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.
TL;DR: In this paper, the authors studied the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and proved that any finite energy solution converges to the corresponding Schrodinger equation in the energy space, after the infinite oscillation in time is removed.
Abstract: We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schrodinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in \(L^2\).
TL;DR: In this article, the authors deal with the computational analysis of strain localization problems using nonlocal continuum damage models of the integral type and present a general framework for a consistent derivation of the tangent stiffness matrix.
TL;DR: A space-time domain decomposition iteration is analyzed, for a model advection diffusion equation in one and two dimensions, the block red-black variant of the waveform relaxation method, and the analysis provides new convergence results for this scheme.
Abstract: We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.
TL;DR: In this article, Lagrangian reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations were considered and it was shown that, for a logarithmic-(quasi-)uniform distribution of sample points, the reducedbasis approximation converges exponentially to the exact solution uniformly in parameter space.
07 Aug 2002
TL;DR: An extension of the SAGE (space-alternating generalized expectation-maximization) algorithm that allows for joint estimation of the complex weight, the relative delay, the direction of departure and of incidence, as well as the Doppler frequency of waves propagating from the transmitter to the receiver in mobile radio environments is presented.
Abstract: This contribution presents an extension of the SAGE (space-alternating generalized expectation-maximization) algorithm originally published in Fleury et al. (1999) that allows for joint estimation of the complex weight, the relative delay, the direction (i.e. azimuths and co-elevations) of departure and of incidence, as well as the Doppler frequency of waves propagating from the transmitter to the receiver in mobile radio environments. The scheme is particularly well suited for MIMO (multiple-input multiple-output) channel investigations. Its performance, in terms of convergence rate and asymptotic behaviour of the root-mean-square estimation errors, is assessed by means of Monte-Carlo simulations in synthetic time-invariant channels. The results demonstrate rapid convergence (six SAGE iteration cycles) of the root-mean-square estimation errors towards values close to the root of the corresponding Cramer-Rao lower bounds for the "one-wave" scenario, even when the waves only slightly differ either in delay, in direction of departure, or in direction of incidence. The SAGE algorithm is also applied to measurement data to assess the propagation constellation in a non-line-of-sight and an obstructed line-of-sight situation. Most of the estimated waves can be easily related to the propagation environments. Finally, the computational expense of the scheme is shortly discussed.
TL;DR: The order of convergence of the Decomposition method is contemplated, and the results are applied to some problems.
TL;DR: The study of the local field map provides an understanding of why methods that do not use the staircase approximation converge faster than methods that use it, and a theoretical analysis is proposed in the limit when the number of slices tends to infinity, which shows that even in that case the stairs approximation is not well suited to describe the real profile.
Abstract: An electromagnetic study of the staircase approximation of arbitrary shaped gratings is conducted with three different grating theories. Numerical results on a deep aluminum sinusoidal grating show that the staircase approximation introduces sharp maxima in the local field map close to the edges of the profile. These maxima are especially pronounced in TM polarization and do not exist with the original sinusoidal profile. Their existence is not an algorithmic artifact, since they are found with different grating theories and numerical implementations. Since the number of the maxima increases with the number of the slices, a greater number of Fourier components is required to correctly represent the electromagnetic field, and thus a worsening of the convergence rate is observed. The study of the local field map provides an understanding of why methods that do not use the staircase approximation (e.g., the differential theory) converge faster than methods that use it. As a consequence, a 1% accuracy in the efficiencies of a deep sinusoidal metallic grating is obtained 30 times faster when the differential theory is used in comparison with the use of the rigorous coupled-wave theory. A theoretical analysis is proposed in the limit when the number of slices tends to infinity, which shows that even in that case the staircase approximation is not well suited to describe the real profile.
TL;DR: In this paper, the principle of the decomposition method is described and its advantages as well as drawbacks are discussed, and an aftertreatment technique (AT) is proposed, which yields the analytic approximate solution with fast convergence rate and high accuracy through the application of Pade approximation to the series solution derived from ADM.
Abstract: Adomian's decomposition method (ADM) is a nonnumerical method which can be adapted for solving nonlinear ordinary differential equations. In this paper, the principle of the decomposition method is described, and its advantages as well as drawbacks are discussed. Then an aftertreatment technique (AT) is proposed, which yields the analytic approximate solution with fast convergence rate and high accuracy through the application of Pade approximation to the series solution derived from ADM. Some concrete examples are also studied to show with numerical results how the AT works efficiently.
TL;DR: Weak convergence of the Euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of Lebesgue measure zero as mentioned in this paper, and the rate of convergence is presented for coefficients are Holder continuous.
Abstract: Weak convergence of the Euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of Lebesgue measure zero. The rate of convergence is presented when coefficients are Holder continuous. Monte Carlo simulations are also discussed.
01 May 2002
TL;DR: This work uses theoretical analysis and numerical experiments to investigate the convergence rate of the iterative split-operator approach for solving nonlinear reactive transport problems.
Abstract: Numerical solutions to nonlinear reactive solute transport problems are often computed using split-operator (SO) approaches, which separate the transport and reaction processes. This uncoupling introduces an additional source of numerical error, known as the splitting error. The iterative split-operator (ISO) algorithm removes the splitting error through iteration. Although the ISO algorithm is often used, there has been very little analysis of its convergence behavior. This work uses theoretical analysis and numerical experiments to investigate the convergence rate of the iterative split-operator approach for solving nonlinear reactive transport problems.
TL;DR: The modified PSO algorithm was empirically studied with a suite of four well‐known benchmark functions, and was further examined with a practical application case, a neural‐network‐based modeling of aerodynamic data, demonstrating that the modified algorithm statistically outperforms the original one.
Abstract: In this paper, a modification strategy is proposed for the particle swarm optimization (PSO) algorithm. The strategy adds an adaptive scaling term into the algorithm, which aims to increase its convergence rate and thereby to obtain an acceptable solution with a lower number of objective function evaluations. Such an improvement can be useful in many practical engineering optimizations where the evaluation of a candidate solution is a computationally expensive operation and consequently finding the global optimum or a good sub‐optimal solution with the algorithm is too time consuming, or even impossible within the time available. The modified PSO algorithm was empirically studied with a suite of four well‐known benchmark functions, and was further examined with a practical application case, a neural‐network‐based modeling of aerodynamic data. The numerical simulation demonstrates that the modified algorithm statistically outperforms the original one.
TL;DR: Derived from the idea of stochastic approximation, recursive algorithms to identify a Hammerstein system are presented and recover the characteristic of the nonlinear memoryless subsystem, while the third one estimates the impulse response of the linear dynamic part.
Abstract: Derived from the idea of stochastic approximation, recursive algorithms to identify a Hammerstein system are presented. Two of them recover the characteristic of the nonlinear memoryless subsystem, while the third one estimates the impulse response of the linear dynamic part. The a priori information about both subsystems is nonparametric. Consistency in quadratic mean is shown, and the convergence rate is examined. Results of numerical simulation are also presented.
TL;DR: This paper shows that the inexact Levenberg-Marquardt method (ILMM), which does not require computing exact search directions, has a superlinear rate of convergence under the same local error bound assumption and proposes the ILMM with Armijo's stepsize rule that has global convergence under mild conditions.
Abstract: In this paper, we consider convergence properties of the Levenberg-Marquardt method for solving nonlinear equations. It is well-known that the nonsingularity of Jacobian at a solution guarantees that the Levenberg-Marquardt method has a quadratic rate of convergence. Recently, Yamashita and Fukushima showed that the Levenberg-Marquardt method has a quadratic rate of convergence under the local error bound assumption, which is milder than the nonsingularity of Jacobian. In this paper, we show that the inexact Levenberg-Marquardt method (ILMM), which does not require computing exact search directions, has a superlinear rate of convergence under the same local error bound assumption. Moreover, we propose the ILMM with Armijo's stepsize rule that has global convergence under mild conditions.
01 Jan 2002
TL;DR: In this paper, it was shown that applying the baker's transformation to lattice rules gives O(N - 2+∈) convergence for nonperiodic integrands with sufficient smoothness.
Abstract: Good lattice quadrature rules are known to have O(N - 2+∈) convergence for periodic integrands with sufficient smoothness. Here it is shown that applying the baker's transformation to lattice rules gives O(N - 2+∈) convergence for nonperiodic integrands with sufficient smoothness. This approach is philosophically different than making a periodizing transformation of the integrand as it results in a different error analysis.
TL;DR: In this paper, the authors present a simple quantitative bound on the total variation distance after k iterations between two Markov chains with different initial distributions but identical transition probabilities, which is a simplified and improved version of the result in Rosenthal (1995), which also takes into account the $epsilon$-improvement of Roberts and Tweedie (1999), and which follows as a special case of the more complicated time-inhomogeneous results of Douc et al. (2002).
Abstract: We state and prove a simple quantitative bound on the total variation distance after k iterations between two Markov chains with different initial distributions but identical transition probabilities. The result is a simplified and improved version of the result in Rosenthal (1995), which also takes into account the $epsilon$-improvement of Roberts and Tweedie (1999), and which follows as a special case of the more complicated time-inhomogeneous results of Douc et al. (2002). However, the proof we present is very short and simple; and we feel that it is worthwhile to boil the proof down to its essence. This paper is purely expository; no new results are presented.
TL;DR: An answer for an m -adic topology when the ideal m can be chosen generic enough is proposed: compared to a smooth case, this work proves quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.
Abstract: Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.
TL;DR: In this article, a new simple method for choosing regularization parameters is proposed based on the conditional stability estimate for this ill-posed problem, and it has an almost optimal convergence rate when the exact solution is in H2.
Abstract: In this paper, we discuss a classical ill-posed problem—numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H2. The advantages of our method are (1) we can get similar computational results with much less computation, in comparison with other methods, and (2) we can find the discontinuous points numerically.
TL;DR: An adaptive wavelet scheme for saddle point problems is developed and analyzed and under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements.
Abstract: In this paper an adaptive wavelet scheme for saddle point problems is developed and analyzed. Under the assumption that the underlying continuous problem satisfies the inf-sup condition, it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the error of the best wavelet N-term approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the Ladyzhenskaya--Babuska--Brezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.