Showing papers on "Rate of convergence published in 1997"
TL;DR: A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided and Mathematical results on conditions for uniqueness of sparse solutions are also given.
Abstract: We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging.
1,864 citations
Book•
01 Jan 1997
TL;DR: Applications and issues application to learning, state dependent noise and queueing applications to signal processing and adaptive control mathematical background convergence with probability one, introduction weak convergence methods for general algorithms applications, proofs of convergence rate of convergence averaging of the iterates distributed/decentralized and asynchronous algorithms.
Abstract: Applications and issues application to learning, state dependent noise and queueing applications to signal processing and adaptive control mathematical background convergence with probability one - Martingale difference noise convergence with probability one - correlated noise weak convergence - introduction weak convergence methods for general algorithms applications - proofs of convergence rate of convergence averaging of the iterates distributed/decentralized and asynchronous algorithms.
1,172 citations
538 citations
TL;DR: Exact computable rates of convergence for Gaussian target distributions are obtained and different random and non‐random updating strategies and blocking combinations are compared using the rates.
Abstract: In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image analysis problem several updating strategies are described and compared. For problems in Bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice.
448 citations
TL;DR: This paper considers the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems, and shows that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled.
Abstract: In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.
396 citations
TL;DR: This work embeds both LMS and steepest descent, as well as other intermediate methods, within a one-parameter class of algorithms, and proposes a hybrid class of methods that combine the faster early convergence rate of LMS with the faster ultimate linear convergence rates of steepmost descent.
Abstract: The least mean squares (LMS) method for linear least squares problems differs from the steepest descent method in that it processes data blocks one-by-one, with intermediate adjustment of the parameter vector under optimization. This mode of operation often leads to faster convergence when far from the eventual limit and to slower (sublinear) convergence when close to the optimal solution. We embed both LMS and steepest descent, as well as other intermediate methods, within a one-parameter class of algorithms, and we propose a hybrid class of methods that combine the faster early convergence rate of LMS with the faster ultimate linear convergence rate of steepest descent. These methods are well suited for neural network training problems with large data sets. Furthermore, these methods allow the effective use of scaling based, for example, on diagonal or other approximations of the Hessian matrix.
339 citations
TL;DR: Several new implicit schemes for the solution of the compressible Navier?Stokes equations are presented, with attention on the development of a new implicit scheme using a positivity-preserving version of Toroet al.'s HLLC scheme, which is the simplest average-state solver capable of exactly preserving isolated shock, contact, and shear waves.
320 citations
TL;DR: In this paper, it was shown that the iteratively regularized Gauss-Newton method is a locally convergent method for solving nonlinear ill-posed problems, provided the nonlinear operator satisfies a certain smoothness condition.
Abstract: In this paper we prove that the iteratively regularized Gauss-Newton method is a locally convergent method for solving nonlinear ill-posed problems, provided the nonlinear operator satisfies a certain smoothness condition. For perturbed data we propose a priori and a posteriori stopping rules that guarantee convergence of the iterates, if the noise level goes to zero. Under appropriate closeness and smoothness conditions on the exact solution we obtain the same convergence rates as for linear ill-posed problems.
249 citations
TL;DR: Another tree reconstruction method, the Witness-Antiwitness Method, or WAM, is presented, which is significantly faster than DCM, especially on random trees, and converges at the same rate as DCM.
Abstract: Inferring evolutionary trees is an interesting and important problem in biology that is very difficult from a computational point of view as most associated optimization problems are NP-hard. Although it is known that many methods are provably statistically consistent (i.e. the probability of recovering the correct tree converges on 1 as the sequence length increases), the actual rate of convergence for different methods has not been well understood. In a recent paper we introduced a new method for reconstructing evolutionary trees called the Dyadic Closure Method (DCM), and we showed that DCM has a very fast convergence rate. DCM runs in O(n^5 log n) time, where n is the number of sequences, so although it is polynomial it has computational requirements that are potentially too large to be of use in practice. In this paper we present another tree reconstruction method, the Witness-Antiwitness Method, or WAM. WAM is significantly faster than DCM, especially on random trees, and converges at the same rate as DCM. We also compare WAM to other methods used to reconstruct trees, including Neighbor Joining (possibly the most popular method among molecular biologists), and new methods introduced in the computer science literature.
222 citations
TL;DR: In this paper, the convergence of Pade approximants is studied under two types of assumptions: in the first case the function f to be approximated has to have all its singularities in a compact set E ⊆ C of capacity zero (the function may be multi-valued in C \ E ), and in the second case f has to be analytic in a domain possessing a certain symmetry property (this notion is defined and discussed below).
212 citations
TL;DR: A stochastic particle method for the McKean-Vlasov and the Burgers equation is introduced and numerical experiments are presented which confirm the theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.
Abstract: In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in [0,T] x R with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure μ t , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with N particles provides a discrete measure which approximates μkΔt for each time kΔt (where Δt is a discretization step of the time interval [0, T]). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of μkΔt. We show that the convergence rate is O (1/√N+ for the approximation in L i (Ω x R) of the cumulative distribution function at time T, and of order O (e 2 +1/e) (1/√N + √Δt for the approximation in L l (Ω x R) of the density at time T (Ω is the underlying probability space, e is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate O (1/√N+√Δt). This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.
TL;DR: In this paper, the authors show that the convergence rate is polynomial when the feedback is a function of time, and they also show that exponential convergence is obtained by considering time-varying feedbacks which are only continuous.
Abstract: Rigid body models with two controls cannot be locally asymptotically stabilized by continuous feedbacks which are functions of the state only. This impossibility no longer holds when the feedback is also a function of time, and time-varying asymptotically stabilizing feedbacks have already been proposed. However, due to the smoothness of the feedbacks, the convergence rate is only polynomial. In this paper, exponential convergence is obtained by considering time-varying feedbacks which are only continuous.
TL;DR: An efficient smoother for the solution of the Stokes problem by multigrid methods is presented, obtained from a variant of the pressure correction steps in SIMPLE type algorithms, understood as a Jacobi type iteration.
TL;DR: In this paper, the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator is given, and the nonlinear martingale problem is solved.
Abstract: We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniques and a representation of the nonlinear process on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then consider a nonlinear equation with both Vlasov and Boltzmann terms and give a weak pathwise propagation of chaos result using a compactness-uniqueness method which necessitates some regularity. These results imply functional laws of large numbers and extend to multitype models. We give algorithms simulating or approximating the particle systems.
TL;DR: In this article, the authors consider numerical solutions of second-order elliptic partial differential equations, such as Laplace's equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG).
TL;DR: In this paper, a meshless Petrov-Galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of Shepard functions, and conditions on test functions and trial functions for nonintegrable pseudo-derivatives for Petrov Galerkin method which pass the patch test.
Proceedings Article•
01 Dec 1997
TL;DR: It is shown that for discounted MDPs with discount factor γ > 1/2 the asymptotic rate of convergence of Q-learning is O(1/tR(1-γ)) if R(1 - γ) 0, where pmin and pmax now become the minimum and maximum state-action occupation frequencies corresponding to the stationary distribution.
Abstract: In this paper we show that for discounted MDPs with discount factor γ > 1/2 the asymptotic rate of convergence of Q-learning is O(1/tR(1-γ)) if R(1 - γ) 0, where pmin and pmax now become the minimum and maximum state-action occupation frequencies corresponding to the stationary distribution.
TL;DR: A solution method is proposed that alternates between a proximal step and a projection-type step for the monotone variational inequality part of this mixed problem and its convergence and rate of convergence are analyzed.
Abstract: We consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem. We propose a solution method for this problem that alternates between a proximal step (for the maximal monotone operator part) and a projection-type step (for the monotone variational inequality part) and analyze its convergence and rate of convergence. This method extends a decomposition method of Chen and Teboulle [Math. Programming, 64 (1994), pp. 81--101] for convex programming and yields, as a by-product, new decomposition methods.
TL;DR: This paper explains the methodology used to develop a high-resolution, multi-dimensional Euler solver that is capable of handling non-ideal equation of state and stiff chemical source terms and provides details on the verification of the integrated set of algorithms that resulted in an application code.
Abstract: This paper explains the methodology used to develop a high-resolution, multi-dimensional Euler solver that is capable of handling non-ideal equation of state and stiff chemical source terms. We have developed a pointwise implementation that has computational advantages for our intended applications, as opposed to a finite volume implementation. Our solver allows for the placement of internal reflective boundaries and the standard inflow and outflow and reflective boundaries at the edge of the domain. We discuss the spatial discretization and the temporal integration schemes, upwinding and flux splitting and the combined use of the Lax - Friedrichs and Roe schemes to solve for the required fluxes. A complete description of the pointwise internal boundary method is given. An overall summary of a representative code structure is given. We provide details on the verification of our integrated set of algorithms that resulted in an application code. We demonstrate the order of convergence for test problems. Two...
CNET1
TL;DR: A novel approach to the blind linear equalization of possibly nonminimum phase and time-varying communication channels through the concept of mutually referenced equalizers (MREs), in which several filters are considered, the outputs of which act as training signals for each other.
Abstract: This paper presents a novel approach to the blind linear equalization of possibly nonminimum phase and time-varying communication channels. In the context of channel diversity, we introduce the concept of mutually referenced equalizers (MREs) in which several filters are considered, the outputs of which act as training signals for each other. A corresponding (constrained) multidimensional mean-square error (MSE) cost function is derived, the minimization of which is shown to be a necessary and sufficient condition for equalization. The links with a standard linear prediction problem are demonstrated. The proposed technique exhibits properties of important practical concern: 1) the proposed algorithm is globally convergent. 2) Simple closed-form solutions exist for the MREs, but the MREs also lend themselves readily to adaptive implementation. In particular, the recursive least-squares (RLS) algorithm can be used to offer optimal convergence rate. 3) The MRE method provides a solution for all equalization delays, which results in robustness properties with respect to SNR and ill-defined channel lengths.
TL;DR: In this paper, a collocation method for approximating integrals of rapidly oscillatory functions is analyzed, which is efficient for integrals involving Bessel functions J,(rx) with a large oscillation frequency parameter r, as well as for many other one-and multi-dimensional integrals with rapid irregular oscillations.
01 Jan 1997
TL;DR: In this article, the authors present a decentralized approach to power load management by modeling direct load management as a computational market and demonstrate that their approach is very efficient with a superlinear rate of convergence to equilibrium and an excellent scalability.
Abstract: Power load management enables energy utilities to reduce peak loads and thereby save money. Due to the large number of different loads, power load management is a complicated optimization problem. We present a new decentralized approach to this problem by modeling direct load management as a computational market. Our simulation results demonstrate that our approach is very efficient with a superlinear rate of convergence to equilibrium and an excellent scalability, requiring few iterations even when the number of agents is in the order of one thousand. Aframework for analysis of this and similar problems is given which shows how nonlinear optimization and numerical mathematics can be exploited to characterize, compare, and tailor problem-solving strategies in market-oriented programming.
TL;DR: A Newton-type method for solving a semismooth reformulation of monotone complementarity problems, which has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions and some numerical results are presented.
Abstract: In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented.
TL;DR: In this paper, finite element methods of least-squares type for the stationary, incompressible Navier-Stokes equations in two and three dimensions were studied and optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions were obtained.
Abstract: In this paper we study finite element methods of least-squares type for the stationary, incompressible Navier--Stokes equations in two and three dimensions. We consider methods based on velocity-vorticity-pressure form of the Navier--Stokes equations augmented with several nonstandard boundary conditions. Least-squares minimization principles for these boundary value problems are developed with the aid of the Agmon--Douglis--Nirenberg (ADN) elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions. Results of several computational experiments with least-squares methods which illustrate, among other things, the optimal convergence rates are also reported.
TL;DR: Easy verifiable sufficient conditions of robust asymptotic stability of linear time-delay systems subject to parametric unstructured or highly-structured perturbations are given.
TL;DR: A Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold is considered and an orthonormal wavelet basis is described to compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain the same convergence rates.
Abstract: We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.
TL;DR: This paper uses the preconditioning matrix I + S(α) to show that if a coefficient matrix A is an irreducibly diagonally dominant Z-matrix, then [I + S (α)]A is also a strictly diagonal dominant Z -matrix and is shown that the proposed method is also superior to other iterative methods.
TL;DR: Viewing the error as a combination of two terms, the approximation error measuring the adequacy of the model, and the estimation error resulting from the finiteness of the sample size, upper bounds are derived to the expected total error, thus obtaining bounds for the rate of convergence.
TL;DR: In this article, a useful error bound for alternating projections was developed for the method of alternating projections which is relatively easy to compute and remember, and a counterexample to a conjecture of Kayalar and Weinert was presented.
TL;DR: Weighted averages of Kiefer--Wolfowitz-type procedures, which are driven by larger step lengths than usual, can achieve the optimal rate of convergence because a priori knowledge of a lower bound on the smallest eigenvalue of the Hessian matrix is avoided.
Abstract: Weighted averages of Kiefer--Wolfowitz-type procedures, which are driven by larger step lengths than usual, can achieve the optimal rate of convergence. A priori knowledge of a lower bound on the smallest eigenvalue of the Hessian matrix is avoided. The asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a Newton-type adaptive algorithm. Several different gradient estimates are considered; one of them leads to a vanishing asymptotic bias. This gradient estimate applied with the weighted averaging algorithm usually yields a better asymptotic mean squared error than applied with the standard algorithm.