General Convergence Theory Research Articles

Convergence Rates for Identification of Robin Coefficient from Terminal Observations

This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou (2000) and obtain a convergence rate of O(δ12)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\delta ^\\frac{1}{2})$$\\end{document}, where δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that the above mentioned convergence rate is obtained without the need for characterizing the range space of modelling operator and its Fréchet derivative, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators.

On the nature of Bregman functions

Let C⊂Rn be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.

Nonparametric inference for reversed mean models with panel count data

Panel count data typically refer to data arising from studies with recurrent events, in which subjects are observed only at discrete time points rather than under continuous observations. We investigate a general situation where a recurrent event process is eventually truncated by an informative terminal event and we are particularly interested in behaviors of the recurrent event process near the terminal event. We propose a reversed mean model for estimating the mean function of the recurrent event process. We develop a two-stage sieve likelihood-based method to estimate the mean function, which overcomes the computational difficulties arising from a nuisance functional parameter involved in the likelihood. The consistency and the convergence rate of the two-stage estimator are established. Allowing for the convergence rate slower than the standard rate, we develop the general weak convergence theory of M-estimators with a nuisance functional parameter, and then apply it to the proposed estimator for deriving the asymptotic normality. Furthermore, a class of two-sample tests is developed. The proposed methods are evaluated with extensive simulation studies and illustrated with panel count data from the Chinese Longitudinal Healthy Longevity Study.

On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field

Abstract We study a general convergence theory for the analysis of numerical solutions to a magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension $d$$(=2,3)$. First, we introduce the concept of dissipative weak (DW) solutions and prove the weak–strong uniqueness property for DW solutions, meaning a DW solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the DW solution, as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax equivalence theorem. Finally, as an application of this theory, we show the convergence analysis of two numerical methods.

Accelerated derivative-free nonlinear least-squares applied to the estimation of Manning coefficients

A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. One of them is based on iterative subspace minimization and the other one is based on spline interpolation with variable nodes. Each iteration based on those procedures is followed by an acceleration step inspired in the Sequential Secant Method. The practical motivation for this work is the estimation of parameters in Hydraulic models applied to dam breaking problems. Numerical examples of the application of the new method to those problems are given.

Net convergence structures with applications to vector lattices

Convergence is a fundamental topic in analysis that is most commonly modeled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.

-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

Local asymptotics for nonlocal convective Cahn-Hilliard equations with W1,1 kernel and singular potential

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W1,1 convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.

The Convergence of the Generalized Lanczos Trust-Region Method for the Trust-Region Subproblem

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. The generalized Lanczos trust-region (GLTR) method is a well-known Lanczos type approach for solving a large-scale TRS. The method projects the original large-scale TRS onto a sequence of lower dimensional Krylov subspaces, whose orthonormal bases are generated by the symmetric Lanczos process, and computes approximate solutions from the underlying subspaces. There have been some a priori bounds available for the errors of the approximate solutions and approximate objective values obtained by the GLTR method, but no a priori bound exists on the errors of the approximate Lagrangian multipliers and the residual norms of approximate solutions obtained by the GLTR method. In this paper, a general convergence theory of the GLTR method is established for the TRS in the easy case, showing that the a priori bounds for these four quantities are closely interrelated and the one for the computable residual norm is of crucial importance in both theory and practice as it can predict the sizes of other three uncomputable errors reliably. Numerical experiments demonstrate that our bounds are realistic and predict the convergence rates of the four quantities accurately.

Convergence analysis of penalty based numerical methods for constrained inequality problems

This paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems, including variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. The constraint is relaxed by a penalty formulation and is re-stored as the penalty parameter tends to zero. The main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh-size and the penalty parameter approach zero simultaneously but independently. The convergence of the penalty based numerical methods is first established for a general elliptic variational-hemivariational inequality with constraints, and then for hemivariational inequalities and variational inequalities as special cases. Applications to problems in contact mechanics are described.

Unbounded topologies and uo-convergence in locally solid vector lattices

Suppose X is a vector lattice and there is a notion of convergence x α → σ x in X . Then we can speak of an “unbounded” version of this convergence by saying that x α → u σ x if | x α − x | ∧ u → σ 0 for every u ∈ X + . In the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo -convergence and those convergences deriving from locally solid topologies. We will see that, not only do the majority of recent results on unbounded norm convergence generalize, but they do so effortlessly. Not only that, but the structure of unbounded topologies is clearer without a norm. We demonstrate this by removing metrizability, completeness, and local convexity from nearly all arguments, while at the same time making the proofs simpler and more general. We also give characterizations of minimal topologies in terms of unbounded topologies and uo -convergence.

Asymptotic Convergence Algorithms

The subject of general theory of convergence is to study, if possible, the behavior of a class of algorithms, particularly that the algorithms differ in most cases by details of implementation.This leads naturally to a general pattern in which algorithms or classes of algorithms are represented by multifunctions, i. e. the maps ▪.An algorithm is defined by a map ▪, which associates to each point xk already know a new point xk+1 = A(xk), ▪.The asymptotic convergence describes how fast the sequence {xk } could arrive to an optimal solution if exists.

Going off grid: computationally efficient inference for log-Gaussian Cox processes

This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, whereas an approximation based on a counting process on a partition of the domain achieves only first-order convergence. The results improve upon the general theory of convergence for stochastic partial differential equation models introduced by Lindgren et al. (2011). The new method is demonstrated on a standard point pattern dataset, and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of Chakraborty et al. (2011). The second extension constructs a log-Gaussian Cox process on the world's oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

Estimation of arbitrary order central statistical moments by the multilevel Monte Carlo method

We extend the general framework of the Multilevel Monte Carlo method to multilevel estimation of arbitrary order central statistical moments. In particular, we prove that under certain assumptions, the total cost of an MLMC central moment estimator is asymptotically the same as the cost of the multilevel sample mean estimator and thereby is asymptotically the same as the cost of a single deterministic forward solve. The general convergence theory is applied to a class of obstacle problems with rough random obstacle profiles. Numerical experiments confirm theoretical findings.

Stratified LMN-convergence tower spaces

We develop a general theory of convergence for lattice-valued spaces based on the concept of s-stratified LM-filters. For different choices of the frames L and M, different kinds of filters arise, and suitable choices of the lattice N allow to view many existing fuzzy and probabilistic convergence spaces as special examples of our stratified LMN-convergence tower spaces. The stratification requires a certain relation between the frames L and M and we use a so-called stratification mapping to this end. The stratification condition is used to show that the resulting category is fiber-small and hence Cartesian closedness is equivalent to the existence of natural function spaces. We give several examples for our spaces for different choices of the lattices L, M and N with a special focus on enriched LM-fuzzy topological spaces. Finally, we study diagonal axioms and regularity.

Convergence of Inner-Iteration GMRES Methods for Rank-Deficient Least Squares Problems

We develop a general convergence theory for the generalized minimal residual method preconditioned by inner iterations for solving least squares problems. The inner iterations are performed by stationary iterative methods. We also present theoretical justifications for using specific inner iterations such as the Jacobi and SOR-type methods. The theory improves previous work [K. Morikuni and K. Hayami, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1--22], particularly in the rank-deficient case. We also characterize the spectrum of the preconditioned coefficient matrix by the spectral radius of the iteration matrix for the inner iterations and give a convergence bound for the proposed methods. Finally, numerical experiments show that the proposed methods are more robust and efficient compared to previous methods for some rank-deficient problems.

Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

We develop a novel convergence theory for the multilevel sample variance estimators in the framework of the multilevel Monte Carlo methods. We prove that, dependent on the regularity of the quantity of interest, the multilevel sample variance estimator may achieve the same asymptotic cost/error relation as the multilevel sample mean, which is superior to the standard Monte Carlo method. Weaker regularity assumptions result in reduced convergence rates, quantified in our analysis. The general convergence theory is applied to a class of scalar elliptic obstacle problems with rough random obstacle profiles, which is a simple model of contact between a deformable body with a rough uncertain substrate. Numerical experiments confirm theoretical convergence proofs.

A Monte Carlo method for variance estimation for estimators based on induced smoothing.

An important issue in statistical inference for semiparametric models is how to provide reliable and consistent variance estimation. Brown and Wang (2005. Standard errors and covariance matrices for smoothed rank estimators. Biometrika 92: , 732-746) proposed a variance estimation procedure based on an induced smoothing for non-smooth estimating functions. Herein a Monte Carlo version is developed that does not require any explicit form for the estimating function itself, as long as numerical evaluation can be carried out. A general convergence theory is established, showing that any one-step iteration leads to a consistent variance estimator and continuation of the iterations converges at an exponential rate. The method is demonstrated through the Buckley-James estimator and the weighted log-rank estimators for censored linear regression, and rank estimation for multiple event times data.

Weaker convergence for Newton's method under Hölder differentiability

We use more precise majorizing sequences than in earlier studies such as [J. Appell, E. De Pascale, J.V. Lysenko, and P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997), pp. 1–17; I.K. Argyros, Concerning the ‘terra incognita’ between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; F. Cianciaruso, A further journey in the ‘terra incognita’ of the Newton–Kantorovich method, Nonlinear Funct. Anal. Appl. 15 (2010), pp. 173–183; F. Cianciaruso and E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso, E. De Pascale, and P.P. Zabrejko, Some remarks on the Newton–Kantorovich approximations, Atti Sem. Mat. Fis. Univ. Modena 48 (2000), pp. 207–215; E. De Pascale and P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280; J.A. Ezquerro and M.A. Hernández, On the R-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math. 197 (2006), pp. 53–61; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations (in Russian), Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24; P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems, J. Complexity 26 (2010), pp. 3–42; J. Rokne, Newton's method under mild differentiability conditions with error analysis, Numer. Math. 18 (1971/72), pp. 401–412; B.A. Vertgeim, On conditions for the applicability of Newton's method, (in Russian), Dokl. Akad. N., SSSR 110 (1956), pp. 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); English transl.: Amer. Math. Soc. Transl. 16 (1960), pp. 378–382; P.P. Zabrejko and D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim. 9 (1987), pp. 671–684; A.I. Zinc˘enko, Some approximate methods of solving equations with non-differentiable operators (Ukrainian), Dopovidi Akad. Nauk Ukraïn. RSR (1963), pp. 156–161] to provide a semilocal convergence analysis for Newton's method under Hölder differentiability conditions. Our sufficient convergence conditions are also weaker even in the Lipschitz differentiability case. Moreover, the results are obtained under the same or less computational cost. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.

Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method

Using a new analysis approach, we establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a given target $\sigma$ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, that is, the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with {\em low} or {\em modest} accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi--Davidson (JD) method with the fixed target $\sigma$ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.