Quality Of Approximation Research Articles

An adaptive finite element model for wellbore breakout simulation

SummaryA finite element formulation is proposed and implemented for analysing the stability of excavated wells using the DiMaggio‐Sandler constitutive elastoplastic model with a typical carbonate reservoir configuration. The quality of the finite element approximation is ensured by applying smooth curved elements adapted to the wellbore geometry, and h − p adaptive finite element meshes in the plastic zone. General purpose procedures are defined to transfer the elastoplastic deformation history to newly created integration points. A breakout damage criterion is proposed based on the second invariant of the deviatoric plastic deformation tensor. This damage criterion is used to apply a mesh movement algorithm to represent material collapse. The automatic successive application of the breakout damage criterion results in elliptical realistically looking geometries obtained in experiments reported in the literature.

Efficient algorithms for solving nonlinear fractional programming problems

In this paper, an efficient algorithm based on the Pascoletti-Serafini scalarization (PS) approach is proposed to obtain almost uniform approximations of the entire Pareto front of bi-objective optimization problems. Five test problems with convex, non-convex, connected, and disconnected Pareto fronts are applied to evaluate the quality of approximations obtained by the proposed algorithm. Results are compared with results of some algorithms including the normal constraint (NC), weighted constraint (WC), Benson type, differential evolution (DE) with binomial crossover, non-dominated sorting genetic algorithm-II (NSGA-II), and S metric selection evolutionary multiobjective algorithm (SMS-EMOA). The results confirm the effectiveness of the presented bi-objective algorithm in terms of the quality of approximations of the Pareto front and CPU time. In addition, two algorithms are presented for approximately solving fractional programming (FP) problems. The first algorithm is based on an objective space cut and bound method for solving convex FP problems and the second algorithm is based on the proposed bi-objective algorithm for solving nonlinear FP problems. In addition, several examples are provided to demonstrate the performance of these suggested fractional algorithms.

Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum.

Stochastic Model-Based Minimization of Weakly Convex Functions

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and re...

Interaction of the simple carbene c-C3H2 with H2: potential energy surface and low-energy scattering.

Cyclopropenylidene, c-C3H2, is a simple hydrocarbon, ubiquitous in astrophysical gases, and possessing a permanent electric dipole moment. Its readily observed rotational transitions make it an excellent probe for the physics and history of interstellar matter. The collisional properties of c-C3H2 with the main background gas, H2, are computed here. We present a full 5-D Potential Energy Surface in the rigid molecule approach, and fit it to relevant functionals for subsequent scattering. We perform low-energy quantum scattering at energies less than 50 cm-1. We use both ortho and para H2 as projectiles. We determine the quality of the various approximations to exact coupled channel scattering and examine paths to go towards higher energy scattering relevant for astrophysics. We compare the results obtained here with earlier ones for scattering with helium.

VSP data inversion for vertical velocity gradient and elliptical anisotropy model

Inversion of velocity parameters for the walkaway VSP data in a multilayered medium can be impeded by velocity gradients and anisotropy in some layers. A problem occurs if we compare velocities obtained from borehole seismic profiling which are equal to their vertical components with the velocities calculated with paths coming from far offsets where the horizontal component plays an important role, especially when the vertical gradient exists and the ray paths are curve-shaped. In this contribution we present the results of velocity model inversion for VSP data considering velocity gradient and elliptical anisotropy. The algorithm consists of two steps, optimization of velocity parameters and optimization of ray paths for the given model. Both procedures use the Nelder-Mead simplex method which finds local minima. Due to the character of optimization we performed also multistart analysis which can provide information about possible equivalences between parameters. Analysis was conducted for different parameterizations, in some cases allowing introduction of additional parameters: vertical gradient and elliptical anisotropy coefficient. The optimal model for a specific set of data is chosen with the help of Bayesian Information Criterion to balance complexity of model with quality of approximation of traveltimes.

Parametrization of Spreading Processes Within Complex Networks with the Use of Knowledge Acquired from Network Samples

Information spreading processes are main drivers of viral campaigns. They are usually conducted within large scale social networks. Parametrisation of online campaigns is usually related to allocation of budgets, number of seeds and strategies of their selection. It is hard to predict campaign effects. Proposed in this paper approach uses network samples to select appropriate strategy for use within complete network. Results showed how scaling of samples is affecting approximation quality. Multi-criteria analysis of performance allows for strategy selection according to preferences and goals.

Analysis of the (μ/μI,λ)-σ-Self-Adaptation Evolution Strategy with Repair by Projection Applied to a Conically Constrained Problem

A theoretical performance analysis of the $(\mu /\mu _{I},\lambda) - \sigma $ -self-adaptation evolution strategy ( $\sigma $ SA-ES) is presented considering a conically constrained problem. Infeasible offspring are repaired using projection onto the boundary of the feasibility region. Closed-form approximations are used for the one-generation progress of the evolution strategy. Approximate deterministic evolution equations are formulated for analyzing the strategy’s dynamics. By iterating the evolution equations with the approximate one-generation expressions, the evolution strategy’s dynamics can be predicted. The derived theoretical results are compared to experiments for assessing the approximation quality. It is shown that in the steady state the $(\mu /\mu _{I},\lambda) - \sigma $ SA-ES exhibits a performance as if the ES were optimizing a sphere model. Unlike the nonrecombinative $(1,\lambda)$ -ES, the parental steady state behavior does not evolve on the cone boundary but stays away from the boundary to a certain extent.

Method of smoothing photoluminescence spectra

The smoothing method for experimental photoluminescence spectra, which uses Tikhonov regularization method, is proposed. Smoothing problem statement is performed as a statement of the minimization problem based on the least squares criterion. The functional to be minimized is a compound of two terms. The first one is the basic term, which determines the quality of data approximation provided by smooth function. The second one is the regularization term, which sets a constraint on the energy by smooth function derivative of a given order. A solution of the smoothing problem is obtained in the closed form analytically. The solution can be implemented by a simple array of matrix operations; one of those is an operation of matrix inversion. The necessity of matrix inversion imposes constraints on the order of derivative used in the proposed method and on noise level. It is related to the increase of either the order of derivative at the given noise level in data or the noise level at the given order of derivative, that deteriorates the conditionality of a matrix to be inverted and limits the scope of the proposed method, accordingly. Efficiency comparison of the proposed smoothing method, the Savitzky-Golay method, and the moving average method was performed by the numerical simulation of two models of photoluminescence spectra. Their advantages, disadvantages, and also the scopes of applicability are noted. The best values of operating parameters of these methods for the different noise levels in data were obtained. The conditions, under which the potential capabilities of the proposed smoothing method excel the potential possibilities of Savitzky-Golay method, were indicated. The results of smoothing the experimental photoluminescence spectra obtained for ZnO:Mn nanocrystals and ZnS:Mn crystals are provided.

Structure‐preserving local optimal control of mechanical systems

SummaryWhile system dynamics are usually derived in continuous time, respective model‐based optimal control problems can only be solved numerically, ie, as discrete‐time approximations. Thus, the performance of control methods depends on the choice of numerical integration scheme. In this paper, we present a first‐order discretization of linear quadratic optimal control problems for mechanical systems that is structure preserving and hence preferable to standard methods. Our approach is based on symplectic integration schemes and thereby inherits structure from the original continuous‐time problem. Starting from a symplectic discretization of the system dynamics, modified discrete‐time Riccati equations are derived, which preserve the Hamiltonian structure of optimal control problems in addition to the mechanical structure of the control system. The method is extended to optimal tracking problems for nonlinear mechanical systems and evaluated in several numerical examples. Compared to standard discretization, it improves the approximation quality by orders of magnitude. This enables low‐bandwidth control and sensing in real‐time autonomous control applications.

An adaptive sampling method for global sensitivity analysis based on least-squares support vector regression

In the field of engineering, surrogate models are commonly used for approximating the behavior of a physical phenomenon in order to reduce the computational costs. Generally, a surrogate model is created based on a set of training data, where a typical method for the statistical design is the Latin hypercube sampling (LHS). Even though a space-filling distribution of the training data is reached, the sampling process takes no information on the underlying behavior of the physical phenomenon into account and new data cannot be sampled in the same distribution if the approximation quality is not sufficient. Therefore, in this study we present a novel adaptive sampling method based on a specific surrogate model, the least-squares support vector regression. The adaptive sampling method generates training data based on the uncertainty in local prognosis capabilities of the surrogate model - areas of higher uncertainty require more sample data. The approach offers a cost efficient calculation due to the properties of the least-squares support vector regression. The opportunities of the adaptive sampling method are proven in comparison with the LHS on different analytical examples. Furthermore, the adaptive sampling method is applied to the calculation of global sensitivity values according to Sobol, where it shows faster convergence than the LHS method. With the applications in this paper it is shown that the presented adaptive sampling method improves the estimation of global sensitivity values, hence reducing the overall computational costs visibly.

Analysis of the (1,λ)-σ-Self-Adaptation Evolution Strategy with repair by projection applied to a conically constrained problem

A thorough theoretical analysis of evolution strategies with constraint handling is important for the understanding of the inner workings of evolution strategies applied to constrained problems. Simple problems are of interest for the first analyses. To this end, the behavior of the (1,λ)-σ-Self-Adaptation Evolution Strategy applied to a conically constrained problem is analyzed. For handling infeasible offspring, a repair approach that projects infeasible offspring onto the boundary of the feasibility region is considered. Closed-form approximations are derived for the expected changes of an individual's parameter vector and mutation strength from one generation to the next. For analyzing the strategy's behavior over multiple generations, deterministic evolution equations are derived. It is shown that those evolution equations together with the approximate one-generation expressions allow to approximately predict the evolution dynamics using closed-form approximations. Those derived approximations are compared to simulations in order to visualize the approximation quality.

Approximations of Algebraic Irrationalities with Matrices

We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic irrationalities giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we briefly perform a numerical comparison with well-known iterative methods (such as Newton and Halley ones), showing that the approximations provided by regular representations appear more accurate for the same size of the denominator.

Non-smooth regularization in radial artificial neural networks

We propose a new approach to the approximation of an arbitrary function by the sum of radial basis functions using non-smooth regularization method combined with a new algorithm in determining the initial approximation which ensures an acceptable quality of approximation uniformly over the data area.

The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients

In this work, a meshfree technique based on radial basis functions (RBFs) is proposed for the study of time-fractional higher order partial differential equations (PDEs) of constant and variable coefficients. The RBFs used containing shape parameter, the selection of which is not an easy task and affects stability and accuracy of the results. The required shape parameter has been computed with the help of recently developed algorithm in literature. Computer simulations are performed for six different time-fractional PDEs which features excellent agreement with the exact solutions and earlier works. Approximation quality of the computed solutions are assessed via $$E_{2}$$ , $$E_{\infty }$$ and $$E_{\text {rms}}$$ error norms.

Top-k Critical Vertices Query on Shortest Path

Shortest path query is one of the most fundamental and classic problems in graph analytics, which returns the complete shortest path between any two vertices. However, in many real-life scenarios, only critical vertices on the shortest path are desirable and it is unnecessary to search for the complete path. This paper investigates the shortest path sketch by defining a top- $k$ critical vertices ( $k$ CV) query on the shortest path. Given a source vertex $s$ and target vertex $t$ in a graph, $k$ CV query can return the top- $k$ significant vertices on the shortest path $SP(s,t)$ . The significance of the vertices can be predefined. The key strategy for seeking the sketch is to apply off-line preprocessed distance oracle to accelerate on-line real-time queries. This allows us to omit unnecessary vertices and obtain the most representative sketch of the shortest path directly. We further explore a series of methods and optimizations to answer $k$ CV query on both centralized and distributed platforms, using exact and approximate approaches, respectively. We evaluate our methods in terms of time, space complexity and approximation quality. Experiments on large-scale real-world networks validate that our algorithms are of high efficiency and accuracy.

LATE: A Level-Set Method Based on Local Approximation of Taylor Expansion for Segmenting Intensity Inhomogeneous Images.

Intensity inhomogeneity is common in real-world images and inevitably leads to many difficulties for accurate image segmentation. Numerous level-set methods have been proposed to segment images with intensity inhomogeneity. However, most of these methods are based on linear approximation, such as locally weighted mean, which may cause problems when handling images with severe intensity inhomogeneities. In this paper, we view segmentation of such images as a nonconvex optimization problem, since the intensity variation in such an image follows a nonlinear distribution. Then, we propose a novel level-set method named local approximation of Taylor expansion (LATE), which is a nonlinear approximation method to solve the nonconvex optimization problem. In LATE, we use the statistical information of the local region as a fidelity term and the differentials of intensity inhomogeneity as an adjusting term to model the approximation function. In particular, since the first-order differential is represented by the variation degree of intensity inhomogeneity, LATE can improve the approximation quality and enhance the local intensity contrast of images with severe intensity inhomogeneity. Moreover, LATE solves the optimization of function fitting by relaxing the constraint condition. In addition, LATE can be viewed as a constraint relaxation of classical methods, such as the region-scalable fitting model and the local intensity clustering model. Finally, the level-set energy functional is constructed based on the Taylor expansion approximation. To validate the effectiveness of our method, we conduct thorough experiments on synthetic and real images. Experimental results show that the proposed method clearly outperforms other solutions in comparison.

Inspection tables for single acceptance sampling with crisp and fuzzy formulation of quality limits

PurposeThe purpose of this paper is to construct innovative exact and approximative sampling plans for acceptance sampling in statistical quality control. These sampling plans are determined for crisp and fuzzy formulation of quality limits, various lot sizes and common α- and β-levels.Design/methodology/approachThe authors use generalized fuzzy hypothesis testing to determine sampling plans with fuzzified quality limits. This test method allows a consideration of the indifference zone related to expert opinion or user priorities. In addition to the exact sampling plans calculated with the hypergeometric operating characteristic function, the authors consider approximative sampling plans using a little known, but excellent operating characteristic function. Further, a comprehensive sensitivity analysis of calculated sampling plans is performed, in order to examine how the inspection effort depends on crisp and fuzzy formulation of quality limits, the lot size and specifications of the producer’s and consumer’s risks.FindingsThe results related the parametric sensitivity analysis of the calculated sampling plans and the conclusions regarding the approximation quality provide the user a comprehensive basis for a direct implementation of the sampling plans in practice.Originality/valueThe constructed sampling plans ensure the simultaneous control of producer’s and consumer’s risks with the smallest possible inspection effort on the one hand and a consideration of expert opinion or user priorities on the other hand.

Production Functions: Entrepreneurial Sector of Russia's Regions

In the context of transformation of the Russian economy there is an urgent need for accelerated development of small and medium enterprises. Solving this problem is very urgent in each of Russia’s regions. The aim of the study was to assess the two-factor production functions that describe the dependence of the volume of production (turnover) of small and medium enterprises on the wages of their employees and investments in fixed assets. The study was based on empirical spatial data characterizing the activities of medium enterprises, small enterprises and microenterprises. Official statistical information on 82 regions of Russia for 2016 was used. The study allowed to determine the factors influencing turnover of SMEs located in all regions, to prove the high quality of approximation of the initial data by the two-factor production functions, to prove that the economy of the Russian regions has not reached saturation with small and medium enterprises. The results of the study, namely new knowledge and tools for assessing production activities of small and medium enterprises in the regions, are of scientific and practical importance. They can be used in monitoring of business climate in regions, determining resource needs, as well as the implementation of the Russian strategy for its development up to 2030.

Understanding the drivers of sensitive behavior using Poisson regression from quantitative randomized response technique data.

Understanding sensitive behaviors—those that are socially unacceptable or non-compliant with rules or regulations—is essential for creating effective interventions. Sensitive behaviors are challenging to study, because participants are unlikely to disclose sensitive behaviors for fear of retribution or due to social undesirability. Methods for studying sensitive behavior include randomized response techniques, which provide anonymity to interviewees who answer sensitive questions. A variation on this approach, the quantitative randomized response technique (QRRT), allows researchers to estimate the frequency or quantity of sensitive behaviors. However, to date no studies have used QRRT to identify potential drivers of non-compliant behavior because regression methodology has not been developed for the nonnegative count data produced by QRRT. We develop a Poisson regression methodology for QRRT data, based on maximum likelihood estimation computed via the expectation-maximization (EM) algorithm. The methodology can be implemented with relatively minor modification of existing software for generalized linear models. We derive the Fisher information matrix in this setting and use it to obtain the asymptotic variance-covariance matrix of the regression parameter estimates. Simulation results demonstrate the quality of the asymptotic approximations. The method is illustrated with a case study examining potential drivers of non-compliance with hunting regulations in Sierra Leone. The new methodology allows assessment of the importance of potential drivers of different quantities of non-compliant behavior, using a likelihood-based, information-theoretic approach. Free, open-source software is provided to support QRRT regression.