Convergence Rate Estimates Research Articles

Predicting Convergence Rate of Namaklan Twin Tunnels Using Machine Learning Methods

Convergence prediction of tunnels has always been one of the important issues of geotechnical projects. Developing prediction models is a good approach to predict convergence, when there is no knowledge of the possibility of convergence occurrence in the future. In this study, convergence rates of two tunnels from the Namaklan twin tunnel were predicted by different types of machine learning methods. Artificial neural networks (ANNs), multivariate linear regression (MLR), multivariate nonlinear regression (MNR), support vector regression (SVR), Gaussian process regression (GPR), regression trees and ensembles of trees (ET) were applied to predict the convergence rate (CR). The six parameters of cohesion (c), internal friction angle (Φ), uniaxial compressive strength of rock mass (σc), rock mass rating, overburden height (H) and the number of installed rock bolts (NB) were selected as predictor parameters. The dataset was collected via field investigations and laboratory experiments. The results showed that the MLP–ANN model can successfully predict the CR with the determination coefficient (R2) of 0.93. The RBF–ANN model is also successful in predicting the CR with R2 of 0.81. The SVR, MNR and MLR models were also constructed to obtain an empirical formula for predicting the CR. Comparing among the three models showed that the SVR model is more successful (R2 = 0.66) than the MNR and MLR models with R2 of 0.65 and 0.61, respectively. However, the SVR model is placed in the next rank of the ANN models. Among the rest models, except the ET model (R2 = 0.66), the RT and GPR models have no good capability for the prediction of the CR. In total, assessing the statistical indices indicated that the ANNs are superior to the other models in predicting the CR. However, the SVR model could be considered to be a reliable predictive model for convergence rate estimation.

The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general [formula omitted]-Hessian equations

In this paper, we focus on the convergence analysis of the unique solution for a Dirichlet problem of the general k-Hessian equation in a ball. By introducing some suitable growth conditions and developing a new iterative technique, the unique solution of the k-Hessian equation is obtained. Then we carry out the convergence analysis for the iterative sequences and further obtain the convergence rate and error estimate for the unique solution. The numerical result indicates that the convergence rate is very fast.

Three-scale singular limits of the rotating stratified Boussinesq equations

ABSTRACT Singular limits of the rotating stratified Boussinesq equations are considered as the Froude number and Rossby number tend to zero at different rates. The limiting dynamics are validated for the rotation-dominant and the stratification-dominant cases, respectively. Furthermore, the convergence rate estimates are obtained.

Why random reshuffling beats stochastic gradient descent

We analyze the convergence rate of the random reshuffling (RR) method, which is a randomized first-order incremental algorithm for minimizing a finite sum of convex component functions. RR proceeds in cycles, picking a uniformly random order (permutation) and processing the component functions one at a time according to this order, i.e., at each cycle, each component function is sampled without replacement from the collection. Though RR has been numerically observed to outperform its with-replacement counterpart stochastic gradient descent (SGD), characterization of its convergence rate has been a long standing open question. In this paper, we answer this question by showing that when the component functions are quadratics or smooth and the sum function is strongly convex, RR with iterate averaging and a diminishing stepsize $\alpha_k=\Theta(1/k^s)$ for $s\in (1/2,1)$ converges at rate $\Theta(1/k^{2s})$ with probability one in the suboptimality of the objective value, thus improving upon the $\Omega(1/k)$ rate of SGD. Our analysis draws on the theory of Polyak-Ruppert averaging and relies on decoupling the dependent cycle gradient error into an independent term over cycles and another term dominated by $\alpha_k^2$. This allows us to apply law of large numbers to an appropriately weighted version of the cycle gradient errors, where the weights depend on the stepsize. We also provide high probability convergence rate estimates that shows decay rate of different terms and allows us to propose a modification of RR with convergence rate ${\cal O}(\frac{1}{k^2})$.

A Bilevel Programming Approach to the Convergence Analysis of Control-Lyapunov Functions

This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min-max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush-Kuhn-Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption.

Convergence rates of symmetric scan Gibbs sampler

We consider the symmetric scan Gibbs sampler, and give some explicit estimates of convergence rates on the Wasserstein distance for this Markov chain Monte Carlo under the Dobrushin uniqueness condition.

L^p-estimates and regularity for SPDEs with monotone semilinearity

Semilinear stochastic partial differential equations on bounded domains {mathscr {D}} are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are L^p-estimates for such equations. The L^p-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space H^2({mathscr {D}}') and ell ^2-integrable with values in H^3({mathscr {D}}'), for any compact {mathscr {D}}' subset {mathscr {D}}. Using results from L^p-theory of SPDEs obtained by Kim (Stoch Proc Appl 112:261–283, 2004) we get analogous results in weighted Sobolev spaces on the whole {mathscr {D}}. Finally it is shown that the solution is Hölder continuous in time of order frac{1}{2} - frac{2}{q} as a process with values in a weighted L^q-space, where q arises from the integrability assumptions imposed on the initial condition and forcing terms.

Economic and environmental convergence of transformation economy: the case of China

Rapid economic reforms and proper GDP growth in China has affected the regional development of Chinese provinces. This study aims to estimate the degree of economic and environmental disparities within Chinese provinces for developing policy recommendations of regional transformation. The reduced log-linear specification of endogenous growth model is used for the estimation of convergence rates within Chinese provinces. The empirical results prove that an increase of 1% in GDP per capita basic year reduces the economic growth rate by 0.1% in the reference year. Thus, the ratio of the average per capita income in the wealthiest group to poorest provinces accounted for the factor 9.6 in 1995 and factor 4.1 in the year 2015, which means a reduction of disproportionate development. Environmental convergence trends were also found and less polluted provinces eventually increase emissions at higher rates than the initially polluted ones. With the pass of time, all provinces do move to the same steady state in environmental parameters. The speed of the economic and environmental convergence in China provinces is rather slow, and the economic growth was achieved by great sacrifices of an environment, since all provinces are striving to the same steady state in terms of pollution increase. The industrialized regions due to the presence of significant financial resources should pay more attention to the protection of the environment using all the available economic potential. At the same time, both initially poor provinces and rich have to develop more profoundly agriculture, tourism, recreation, and other environmentally friendly industries to improve economic performance.

Necessary and sufficient condition for the existence of positive solutions of a coupled system for reaction–diffusion equations

In this article, by using fixed point theorem and monotone iterative technique, the author presents‘ necessary and sufficient conditions of the existence of positive solutions for a class of nonlinear singular reaction–diffusion system of differential equations under reasonable conditions. Moreover, some uniqueness results of positive solution and estimate of convergent rate of iterative sequence of solution are obtained.

Convergence rates of the blocked Gibbs sampler with random scan in the Wasserstein metric

ABSTRACTThis paper establishes explicit estimates of convergence rates for the blocked Gibbs sampler with random scan under the Dobrushin conditions. The estimates of convergence in the Wasserstein metric are obtained by taking purely analytic approaches.

Reachability of optimal convergence rate estimates for high-order numerical convex optimization methods

The Monteiro-Svaiter accelerated hybrid proximal extragradient method (2013) with one step of Newton’s method used at every iteration for the approximate solution of an auxiliary problem is considered. The Monteiro-Svaiter method is optimal (with respect to the number of gradient and Hessian evaluations for the optimized function) for sufficiently smooth convex optimization problems in the class of methods using only the gradient and Hessian of the optimized function. An optimal tensor method involving higher derivatives is proposed by replacing Newton’s step with a step of Yu.E. Nesterov’s recently proposed tensor method (2018) and by using a special generalization of the step size selection condition in the outer accelerated proximal extragradient method. This tensor method with derivatives up to the third order inclusive is found fairly practical, since the complexity of its iteration is comparable with that of Newton’s one. Thus, a constructive solution is obtained for Nesterov’s problem (2018) of closing the gap between tight lower and overstated upper bounds for the convergence rate of existing tensor methods of order p ≥ 3.

Continuous second order minimization method with variable metric projection operator

The paper examines a new continuous projection second order method of minimization of continuously Frechet differentiable convex functions on the convex closed simple set in separable, normed Hilbert space with variable metric. This method accelerates common continuous projection minimization method by means of quasi-Newton matrices. In the method, apart from variable metric operator, vector of search direction for motion to minimum, constructed in auxiliary extrapolated point, is used. By other word, complex continuous extragradient variable metric method is investigated. Short review of allied methods is presented and their connections with given method are indicated. Also some auxiliary inequalities are presented which are used for theoretical reasoning of the method. With their help, under given supplemental conditions, including requirements on operator of metric and on method parameters, convergence of the method for convex smooth functions is proved. Under conditions completely identical to those in convergence theorem, without additional requirements to the function, estimates of the method's convergence rate are obtained for convex smooth functions. It is pointed out, that one must execute computational implementation of the method by means of numerical methods for ODEs solution and by taking into account the conditions of proved theorems.

An Approximate Method for Determining the Harmonic Barycentric Coordinates for Arbitrary Polygons

A relation for finding the harmonic barycentric coordinates for an arbitrary polygon is obtained. The solution is approximate analytical. In the proposed statement, the harmonic barycentric coordinates are determined in terms of the logarithmic potential of a double layer by solving the Dirichlet problem by the Fredholm method. The approximate nature of the solution is determined by the expansion of the kernel of the integral Fredholm equation of the second kind for the unknown density of potential on the boundary of the domain in the orthogonal Legendre polynomials and the expansion of Green’s function; these expansions are used for the calculation of the potential. An estimate of convergence rate and the error of the solution is obtained. The approximate solutions obtained by the proposed method are compared with the known exact solutions of some benchmark problems.

Приближение кусочно гладких функций тригонометрическими суммами Фурье

We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.

Convergence Rate Estimates for Aleksandrov's Solution to the Monge--Ampère Equation

In this paper, we establish convergence rate estimates for convex solutions to the Dirichlet problem of the Monge--Ampere equation $\det D^2u=f$ in $\Omega$, where $f$ is a positive and continuous ...

An unexpected connection between Bayes A-optimal designs and the group lasso

We show that the A-optimal design optimization problem over m design points in $${\mathbb {R}}^n$$ is equivalent to minimizing a quadratic function plus a group lasso sparsity inducing term over $$n\times m$$ real matrices. This observation allows to describe several new algorithms for A-optimal design based on splitting and block coordinate decomposition. These techniques are well known and proved powerful to treat large scale problems in machine learning and signal processing communities. The proposed algorithms come with rigorous convergence guarantees and convergence rate estimate stemming from the optimization literature. Performances are illustrated on synthetic benchmarks and compared to existing methods for solving the optimal design problem.

An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

Let C be a nonempty closed convex subset of a real Hilbert space mathcal{H} with inner product langle cdot , cdot rangle , and let f: mathcal{H}rightarrow mathcal{H} be a nonlinear operator. Consider the inverse variational inequality (in short, operatorname{IVI}(C,f)) problem of finding a point xi ^{*}in mathcal{H} such that \t\t\tf(ξ∗)∈C,〈ξ∗,v−f(ξ∗)〉≥0,∀v∈C.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ f\\bigl(\\xi ^{*}\\bigr)\\in C, \\quad \\bigl\\langle \\xi ^{*}, v-f \\bigl(\\xi ^{*}\\bigr)\\bigr\\rangle \\geq 0, \\quad \\forall v\\in C. $$\\end{document} In this paper, we prove that operatorname{IVI}(C,f) has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261–1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator P_{C} is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.

Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations

Optimal control of systems governed by elliptic partial differential equations (PDEs) without constraint on the set of controls can be equivalently reformulated as a coupled system of second order elliptic PDEs, which has been considered to solve by a non-overlapping Schwarz domain decomposition method with the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions by Benamou (SIAM J Numer Anal 33:2401–2416, 1996) where a convergence analysis had been performed. Towards fast convergence of the overlapping and non-overlapping Schwarz subdomain iterations, in this paper we firstly perform, for fixed Tikhonov parameter $$\mu $$ , rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions, as well as those involved in a Ventcell-like and a two-sided Robin-like transmission condition that we propose to accelerate the Schwarz subdomain iterations, and meanwhile we obtain also the corresponding asymptotic convergence rate estimates. The results show that the Tikhonov parameter $$\mu $$ occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition methods significantly: when $$\mu $$ is less than a certain threshold value, with the decreasing of the Tikhonov parameter $$\mu $$ , the subdomain iteration converges more and more fast, though the regularity of the system deteriorates in this process. We lastly investigate the case where the Tikhonov parameter $$\mu =h^4$$ that is suggested by Benamou (where h is the mesh size). We obtain as well the optimized transmission parameters involved in the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions, and find that they lead to optimized Schwarz methods that are very robust in the mesh size. The analysis also sheds light on optimizing the Schwarz domain decomposition methods for biharmonic equations, since they can also be reformulated as a system of second order elliptic PDEs. We use various numerical experiments to illustrate the theoretical findings.

Solving the backward problem in Riesz–Feller fractional diffusion by a new nonlocal regularization method

In this paper, we consider the backward problem introduced in [48] for Riesz–Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the Lp estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introducedJ(φ)=12‖u(φ(x);x,T)−fδ(x)‖2+β2‖[Pα1⁎φ](x)‖2, where β∈(0,1) is a regularization parameter, “⁎” denotes the convolution operation and Pα1(x) is called a convolution kernel with parameter α1, which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information.

Non-linear tyre model–based non-singular terminal sliding mode observer for vehicle velocity and side-slip angle estimation

Vehicle velocity and side-slip angle are important vehicle states for the electronic stability programme and traction control system in vehicle safety control system and for the control allocation method of electric vehicles with in-wheel motors. This paper proposes an innovative side-slip angle estimator based on the non-linear Dugoff tyre model and non-singular terminal sliding mode observer. The proposed estimation method based on the non-linear tyre model can accurately present the tyre’s non-linear characteristics and can show advantages over estimation methods based on the linear tyre model. The utilised Dugoff tyre model has a relatively simple structure with few parameters, and the proposed non-linear observer can be applied in various vehicle tyres and various road conditions. Precise determination of the Dugoff tyre model parameters is not required and the proposed observer can still perform good estimation results even though tyre parameters and the tyre–road friction coefficient are not accurate. The proposed non-singular terminal sliding mode observer can achieve fast convergence rate and better estimation performance than the traditional sliding mode observer. At the end of this paper, simulations in various conditions are presented to validate the proposed non-linear estimator.