Accurate Approximation Research Articles

Efficient acceleration of the convergence of the minimum free energy path via a path-planning generated initial guess.

We demonstrate that combining a shifted clustering algorithm with a fast-marching-based algorithm can generate accurate approximations of the minimum energy path (MEP) given a free energy landscape (FEL). Using this approximation as the initial guess for the MEP, followed by further refinement with the string method (referred to as the fast marching tree (FMT)-string combined approach), significantly reduces the number of iterations required for MEP convergence. This approach saves substantial time compared to using linear interpolation (LI) for the initial guess. Our method offers a viable solution for obtaining an effective initial guess of the MEP when an approximate or converged FEL is available. This work highlights the potential of applying FMT-based approaches to extract the MEP in chemical reactions.

On the Univariate Vector-Valued Rational Interpolation and Recovery Problems

In this paper, we consider a novel vector-valued rational interpolation algorithm and its application. Compared to the classic vector-valued rational interpolation algorithm, the proposed algorithm relaxes the constraint that the denominators of components of the interpolation function must be identical. Furthermore, this algorithm can be applied to construct the vector-valued interpolation function component-wise, with the help of the common divisors among the denominators of components. Through experimental comparisons with the classic vector-valued rational interpolation algorithm, it is found that the proposed algorithm exhibits low construction cost, low degree of the interpolation function, and high approximation accuracy.

Evaluation of geomorphological classification uncertainty using rough set theory: A case study of Shaanxi Province, China

AbstractGeomorphological classification is affected by classification principles, indicators, methods, and data resolution, which can lead to uncertainty in the results. Such uncertainty directly affects the quality and subsequent applications of geomorphological classification. To quantify and control the uncertainty, it is important to select an appropriate and effective method for evaluating the uncertainty of geomorphological classification. This study evaluated the uncertainty of geomorphological classification of Shaanxi Province at the ground‐feature class and image scales, which derived from rough set theory: rough entropy, approximate classification quality, and approximate classification accuracy. The three indicators helped effectively assess the uncertainty of geomorphological classification at multi‐scale and measured the degree to which different factors affected the uncertainty of geomorphological classification. The relative impacts of three factors on the uncertainty of classification decreased in the order of classification methods, data resolution, and classification indicators. This finding is helpful to objectively evaluate and control the uncertainty generated in the process and results of geomorphological classification, and can provide targeted reference and guidance for future geomorphological classification work, which is more conducive to decision‐making and application. At the same time, this study is also a beneficial supplement to the geomorphological research based on digital terrain analysis.

A new contribution to the Newton-Leibnitz differentiation of polynomial approximations in thermal processes

This paper presents the determining of the critical cooling time and preheating temperature for welding. The primary equation which represents the cooling temperature is transcendent and, as such, it is not suitable for explicit solving. Hence, the author has investigated the appropriate interpolation polynomial which should align the polynomial level with the possibility of providing good accuracy of the approximation. Since the application and design of the welding included calculation of the derivatives of the polynomial, it is concluded that the accuracy of the derivative is not precise, nor does it follow the accuracy of the polynomial approximation. This research showed that the current theory for differentiation of the polynomial approximation does not provide precise results, although all the requirements of differentiability are provided. There are certain shortcomings in these actions, which imply that the currently known method of differentiation cannot be used. The new method of differentiating includes relatively small differentiating error regarding the exact value of the derivative, and the error for the polynomial approximation relating to the exact formula is not surpassed. The attempt to solve the design problem in thermal processes also makes a contribution in differentiation.

Generalized Multiscale Finite Element Method for discrete network (graph) models

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

Implicit-Explicit schemes for decoupling multicontinuum problems in porous media

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct a coarse-scale approximation. The NLMC method is a multiscale method for developing an accurate and physically meaningful coarse-scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.

Multiobjective enterprise development algorithm for optimizing structural design by weight and displacement

This study presents a novel metaheuristic optimization algorithm for complex multiobjective engineering problems. By integrating advanced population and nondominated sorting techniques into the existing single-objective enterprise development algorithm, this new multiobjective approach effectively identifies Pareto-optimal solutions. The algorithm leverages these techniques to explore engineering solutions within multiobjective search spaces. We evaluated its performance using 29 multiobjective mathematical benchmark problems and conducted a comparative analysis against ten established metaheuristic optimization algorithms. The results demonstrate that the algorithm produces highly accurate approximations of Pareto-optimal fronts while maintaining a uniform distribution of solutions. Additionally, the algorithm was applied to real-world engineering challenges, including the optimization of structures such as the 942-bar tower, the 1536-bar double-layer barrel vault, and the 1410-bar double-layer dome truss, with the primary objectives of minimizing both structural weight and maximum nodal deflection. The findings highlight the algorithm's effectiveness in solving practical engineering problems and consistently achieving optimal Pareto fronts.

Saddle Point Techniques to Create Resilient Estimators for Cross-Variograms and Accurate Approximations of Their Distributions

Saddle Point Techniques to Create Resilient Estimators for Cross-Variograms and Accurate Approximations of Their Distributions

Towards optimal spatio‐temporal decomposition of control‐related sum‐of‐squares programs

AbstractThis paper presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad‐hoc selection of split locations. In this work, we eliminate the need for this ad‐hoc selection by introducing an optimization‐based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.

Taylor theory in quantum calculus: a general approach

Let the function β be strictly increasing and continuous on an interval I ⊂ ℝ. The β-difference operator is defined by Dβ f (t) = (f(β(t)) − f(t)/(β(t) – t), where t ≠ β(t), and Dβ f (t) = f′ t(t) when t = s 0 is a fixed point of the function β. This quantum operator is a generalization of q-Jackson, Hahn, power and other quantum operators. As a convenience of the β-function: β(t) turns into the probability distribution function with the probability measure 1, and the sample space ℝ, in the case of its conditions are relaxed to be increasing and continuous from the right, that is, lim t →∞ β(t) = 1 and lim t →∞ β(t) = 0, and also by using the Lebesgue-Stieltjes measure of the interval [a, b] to be β(b) − β(a). In this paper, we investigate a β-Taylor’s formula associated with the operator Dβ when the function β has a unique fixed point s 0 ∈ I, which may allow for more flexible and accurate approximations of functions. An estimation of its remainder is given. Additionally, the β-power series is defined. Furthermore, as application, the β-expansion form of some fundamental functions is introduced. Finally, we find the unique solution of the β-shifting problem.

Discretisation of the Hough parameter space for fitting and recognising geometric primitives in 3D point clouds

Research in recognising and fitting simple geometric shapes has been ongoing since the 1970s, with various approaches proposed, including stochastic methods, parameter methods, primitive-based registration techniques, and more recently, deep learning. The Hough transform is a method of interest due to its demonstrated robustness to noise and outliers, ability to handle missing data, and support for multiple model instances. Unfortunately, one of the main limitations of the Hough transform is how to properly discretise its parameter space, as increasing their number or decreasing the sampling frequency can make it computationally expensive.The relationship between the approximation accuracy and the parameter space’s discretisation is investigated to address this. We present two distinct discretisations to illustrate how the fitting and recognition quality can be improved by selecting an appropriate parameter discretisation. Our parameter-driven space discretisation is shown to significantly improve the parameter recognition quality over the classical method and reduce computational time and space by decreasing the discretisation’s dimension, as demonstrated by an extensive validation on a benchmark of geometric primitives. Preliminary experiments are also presented on segmenting datasets from urban buildings and CAD objects.

Local Second Order Mo̷ller-Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: A Distributed Memory Implementation.

In order to alleviate the computational burden associated with superlinear compute scalings with molecular size in electron correlation methods, researchers have developed local correlation methods that wisely treat relatively small contributions as zeros but still yield accurate energy approximation. Such local correlation techniques can also be combined with parallel computing resources to obtain further efficiency and scalability. This work focuses on the distributed memory parallel implementation of a local correlation method for second order Mo̷ller-Plesset (MP2) theory. This method also only has a single threshold to control the dropping of terms and accuracy of different computing kernels in the algorithm. The process partitioning strategy and distributed parallel implementation with the message passing interface (MPI) are discussed. In particular, the algorithm relies on a fixed sparsity pattern matrix multiplication and a corresponding distributed conjugate gradient solver, which exhibits almost linear scaling in both strong and weak scaling analyses. Numerical experiments on a range of molecules, including linear chains and molecules with 2 and 3-dimensional characters, are reported. For example, with only 32 MPI ranks, this MP2 implementation can calculate the correlation energy of vancomycin in def2-TZVP basis within 0.003% accuracy (10-6.5 threshold) in half an hour, where the same problem is unfeasible to solve with sequential or pure shared memory implementations.

Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

Domain Decomposition for Data-Driven Reduced Modeling of Large-Scale Systems

This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training datasets. In these settings, standard, single-domain approaches may be too inaccurate or may overfit and hence generalize poorly. Moreover, processing large-scale datasets typically requires significant memory and computing resources, which can render single-domain approaches computationally prohibitive. To address these challenges, we introduce a domain-decomposition formulation into the construction of a data-driven reduced model. In doing so, the basis functions used in the reduced model approximation become localized in space, which can increase the accuracy of the domain-decomposed approximation of the complex dynamics. The decomposition furthermore reduces the memory and computing requirements to process the underlying large-scale training dataset. We demonstrate the effectiveness and scalability of our approach in a large-scale three-dimensional unsteady rotating-detonation rocket engine simulation scenario with more than 75 million degrees of freedom and a sparse training dataset. Our results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure by up to 13% and up to 5% for other key quantities, such as temperature, and fuel, and oxidizer mass fractions. Lastly, our approach decreases the memory requirements for processing by almost a factor of four, which in turn reduces the computing requirements as well.

Basics of the reaction kinetics on metallic alloys.

Kinetics of heterogeneous catalytic reactions are often complicated by various factors, and from the perspective of statistical physics the development of the corresponding models is frequently challenging especially in the case of practically important alloy catalysts. To extend the basics in this area, I use a generic kinetic model of N_{2} formation inthe NO reaction with such species as CO or H_{2}. The focus is onthe reaction on the surface of a bimetallic alloy with low integral fraction of one of the metals so that the alloy is formed only at the surface. The main goal is to illustrate the specifics of the reaction kinetics and to clarify the accuracy of various approximations which are often inevitable for the analysis of such systems. The key results are as follows. (i) In the baseline one-metal case with the Langmuir-type equations, the model predicts the existence of the optimal adsorbate binding energy corresponding to the maximal reaction rate. (ii) With suitable substitution of the symbols, the equations employed for a uniform surface [item (i)] can be used for the mean-field description of reaction on the surface of a random alloy. In particular, the maximum in the dependence of the reaction rate on the binding energy is converted to the maximum with respect to the alloy composition. (iii) The reaction kinetics on the random-alloy surface have been described exactly. The corresponding maximum in the reaction rate is lower and somewhat smeared compared to that predicted in the mean-field approximation for an alloy or for the optimal monometallic catalyst. (iv) The reaction kinetics have been scrutinized also in the case of two-dimensional (2D) segregation of metal atoms at the surface in the limits of slow and rapid adsorbate diffusion between the spots. The kinetics are shown to be qualitatively different in these limits and also compared to the random-alloy case. (v) The thermodynamic criteria for 2D segregation of metal atoms at the surface have been derived as well.

New slow-roll approximations for inflation in Einstein-Gauss-Bonnet gravity

We propose new slow-roll approximations for inflationary models with the Gauss-Bonnet term. We find more accurate expressions of the standard slow-roll parameters as functions of the scalar field. To check the accuracy of approximations considered we construct inflationary models with quadratic and quartic monomial potentials and the Gauss-Bonnet term. Numerical analysis of these models indicates that the proposed inflationary scenarios do not contradict to the observation data. New slow-roll approximations show that the constructed inflationary models are in agreement with the observation data, whereas one does not get allowed observational parameters at the same values of parameters of the constructed models in the standard slow-roll approximation.

Relaxed TS Fuzzy Model Transformation to Improve the Approximation Accuracy/Complexity Tradeoff and Relax the Computation Complexity

Relaxed TS Fuzzy Model Transformation to Improve the Approximation Accuracy/Complexity Tradeoff and Relax the Computation Complexity

Метод обработки сигналов токов статора асинхронного двигателя для диагностики обрыва стержня ротора

The study is relevant due to high requirements to the reliability of the operation of auxiliary mechanisms of thermal power plants (TPP). The main source of mechanical power of auxiliary mechanisms of TPPs is high-voltage induction motor with squirrel cage rotor (IMSC). One of the most difficult causes of emergency shutdown to detect early is a broken rotor bar. The breakage of IMSC rotor bar is of latent character. It can be for a long time without considerable influence on the machine operation mode. Nevertheless, the fact of the breakage can be considered as an emergency condition of IMSC, and the consequences of leaving the groove of the magnetic core are irreversible. Scientifically, early diagnostics of IMSC rotor bars breakage of TPP auxiliary needs is both a difficult and extremely urgent task. The existing methods of diagnostics of such damage are based on measurement and analysis of vibration and acoustic physical quantities. The most common electrical parameter, used as a source of information, as a rule, is current consumption. And most of methods are based on Fourier analysis. The purpose of the study is to develop a mathematical algorithm to process digitized curves of stator phase currents to diagnose high-voltage IMSC. The theory of electric machines, the method of analytical approximation of experimental data, the method of regression analysis have been used to solve the problem. Experimental data is obtained, and testing of the algorithm is conducted on an experimental installation with a 0,4 kV induction motor with a modified design of the active part of the rotor developed for physical simulation of IMSC broken rotor bar. This paper proposes a new method of digital processing of stator current signals based on the regression analysis method for diagnostics of IMSC rotor winding. The authors have developed an algorithm for mathematical processing of digitized stator current curves based on regression analysis in the cosine-sine basis to identify the diagnostic sign of failure of one rotor bar of IMSC rotor. The authors have determined the number of harmonic components that best describe the initial signal from the point of view of the balance between saving computational resources and accuracy of approximation. A mathematical algorithm to detect the rotor bar breakage of an induction motor based on regression analysis of stator currents has been obtained and tested. It allows to detect rotor circuit damage with 97 % accuracy. At the same time the diagnostic sign changes its level when one bar breaks by 5 %. It is recommended to adapt the algorithm to the programming languages of microprocessor relay protection units.

On the use of the cumulant generating function for inference on time series

Innovative inference procedures for analyzing time series data are introduced. The methodology covers density approximation and composite hypothesis testing based on Whittle's estimator, which is a widely applied M-estimator in the frequency domain. Its core feature involves the cumulant generating function of Whittle's score obtained using an approximated distribution of the periodogram ordinates. A testing algorithm not only significantly expands the applicability of the state-of-the-art saddlepoint test, but also maintains the numerical accuracy of the saddlepoint approximation. Connections are made with three other prevalent frequency domain techniques: the bootstrap, empirical likelihood, and exponential tilting. Numerical examples using both simulated and real data illustrate the advantages and accuracy of the saddlepoint methods.