Linear Approximation Research Articles

Effective R-Matrix Parameterizations for Nuclear Data

We derive an effective Reich-Moore approximation (RMA) of the Wigner-EisenbudR-matrix formalism parameterized by complex-valued resonance energies and widths; this RMA exactly reproduces the total eliminated cross section. We show that resonance parameters evaluated for a conventional boundary conditions (BCs),Bc=Sc(E),are approximately equal to theR-matrix parameters in Park’s formalism by employing a linear approximation of the shift function therein [T.-S. Park, Phys. Rev. C106(2021) 064612]. We outline a method for converting Park’s observed reduced width amplitudes (RWAs) and their covariance matrix into Brune’s alternativeR-matrix RWAs and their covariance matrix [C. Brune, Phys. Rev. C66(2002) 044611]. We extend the Park’sR-matrix formalism into the complex plane by introducing a complex-valued basis set of eigenfunctions of a complex-symmetric (non-Hermitian) Hamiltonian in theR-matrix interior. We observe that itsR-matrix resonance energies and widths are directly related to the poles and residues, respectively, of Hwang’s sum-over-poles representation of cross sections [R.N. Hwang, Nucl. Sci. Eng.96(1987) 192].

One-step sparse ridge estimation with folded concave penalty

The local linear approximation algorithm is an effective algorithm for computing a global solution of the folded concave penalization problem. However, the effectiveness of this method is highly dependent on a reasonably good initial estimator. It will lose efficacy when the correlation among predictors is high. In this paper, we propose a new local linear approximation ridge algorithm designed to deal with highly correlated predictors. The ridge estimator is chosen as an initial estimator, the local linear approximation ridge algorithm is stable and effective. Simulation studies and a real data analysis show that the proposed algorithm has better performance than the local linear approximation algorithm in the presence of highly correlated predictors.

A log-linearization of the p-median fortification problem for healthcare system

This paper focuses on developing an investment strategy to enhance the resilience of healthcare systems. This problem, called p-median fortification problem, has been introduced by Starita and Scaparra (2022). It aggregates two sub-problems which are to minimize the travel cost and to minimize the number of patients not treated by the system. The problem is formulated as a non-linear model to determine an effective distribution of protective resources among facilities in service or supply system to improve the system reliability. In this paper, we present a new linear approximation of the p-median fortification problem based on log-linearization techniques. The proposed approach is tested based on a benchmark inspired form the literature and the obtained results illustrate the effectiveness the new approximation approach.

Perishables Liner Bunkering Strategy and Route Optimization Considering Carbon Tax and Sulfur Emissions

For the issue of perishable goods liner shipping route selection and bunkering strategy, this paper gives full consideration to the characteristics of perishable goods, in the context of the carbon tax and the setting of emission control zones, constructs a mixed-integer nonlinear planning model with the goal of minimizing the total weekly cost of liner service, designs a segmented linear approximation algorithm for model solution, and conducts an arithmetic analysis based on the actual liner voyage as an example, which provides a reference for the actual operation of shipping.

Data-Driven Control of Highly Interactive Systems using 3DoF Model-On-Demand MPC: Application to a MIMO CSTR

This paper presents a Three-Degree-of-Freedom Model Predictive Control (3DoF MPC) framework based on Multi-Input-Multi-Output (MIMO) “Model-on-Demand” (MoD) estimation. MoD is a data-centric weighted regression algorithm that generates local models over adaptively varying neighborhoods of changing operating conditions. The 3DoF formulation enables individualized tuning of parameters relating to setpoint tracking and measured and unmeasured disturbance rejection. Online estimation of system dynamics using MIMO MoD and augmentation with the 3DoF MPC structure allows the generation of control laws based on efficient locally linear approximations of system nonlinearities. This paper evaluates the framework through a case study involving a nonlinear MIMO Continuous Stirred Tank Reactor (CSTR) model. The MIMO CSTR system is highly interactive, making data-driven estimation and control notably more challenging than its SISO counterpart. The generation of an informative database using modified “zippered” multisines is presented. The paper concludes with a case study demonstrating the effectiveness of 3DoF MoD MPC in achieving constrained MIMO control of reactor concentration and temperature in the presence of disturbances through a flexible and intuitive approach.

Using infeasible path cuts to solve Electric Vehicle Routing Problems with realistic charging functions exactly within a branch-and-cut framework

The paper investigates the Electric Vehicle Routing Problem with a non-linear concave and strictly monotonic increasing charging function. In the literature, the non-linear charging function is typically approximated by a piecewise linear charging function which does not overestimate the real charging function in any point. As the piecewise linear charging function underestimates the real state-of-charge in some points, such an approximation excludes feasible solutions from the solution space. To overcome this drawback we introduce a new method to determine a piecewise linear charging function overestimating the real charging function in a way that the area between both functions is minimized as well as an adaptation of a known linearization to include the piecewise linear charging function in a branch-and-cut approach. Thereby, we include infeasible solutions in the solution space. To declare them infeasible again we check every integer solution obtained in the branch-and-cut procedure and add an infeasible path cut if the solution is infeasible for the real charging function such that the procedure terminates with an optimal solution for the real charging function. Our approach is evaluated in a computational study in which instances with up to 100 customers were solved to optimality. Moreover, we evaluate the trade-off between a more complex model formulation due to more binary variables if the number of supporting points for the piecewise linear approximation is increased and the higher approximation error if fewer supporting points are used.

Mathematical approximator based on basic spline approximation

The article considers the problem of constructing a mathematical piecewise linear approximator based on approximation by basis splines. An algorithm has been developed designed to implement a class of special functions and create parallel microprogramming for the algorithm for calculating spline functions and, on their basis, parallel operation of a microprogram automaton with associative memory.

Estimates for Piecewise Linear Approximation of Derivative Functions of Sobolev Classes

Estimates for Piecewise Linear Approximation of Derivative Functions of Sobolev Classes

Low-Power High Precision Floating-Point Divider With Bidimensional Linear Approximation

Low-Power High Precision Floating-Point Divider With Bidimensional Linear Approximation

Nonlinear generalized master equations: quantum case

A system of N≫1 interacting spinless quantum particles, described by a statistical operator F(t), is considered. A time-dependent projection operator formalism for a family of projectors, which select a statistical operator FS(t) for a group of S < N relevant particles by integration of the variables of the irrelevant N − S ‘environment’ particles, is presented. This formalism results in a nonlinear version of the inhomogeneous Nakajima–Zwanzig generalized master equation (GME) for the relevant part of the statistical operator F(t), which contains an undesirable initial condition term related to the irrelevant part of the statistical operator depending on N variables. By introducing an additional operator identity, the obtained nonlinear inhomogeneous Nakajima–Zwanzig equation is exactly converted into a homogeneous nonlinear equation, accounting for initial correlations in the kernel governing its evolution. Both of these equations are equivalent to the corresponding evolution equations for FS(t) and take into account the dynamics of the environment particles. The equations for FS(t) are specialized in the linear approximation for the particle density n and analyzed for a one-particle statistical operator F1(t) . It turns out that, in addition to the correlations caused by interparticle interaction, the irrelevant initial condition term in the inhomogeneous equation for F1(t) , defined by the two-particle correlations, also contains quantum correlations conditioned by the quantum particle statistics. The latter do not vanish with time and cannot be disregarded in the conventional way by applying e.g. Bogoliubov’s principle of weakening of initial correlations. In order to take initial quantum and other correlations (commonly discarded in an unconvincing way) into consideration, the obtained nonlinear homogeneous GME has been applied. Exact in the first approximation in n, we obtain a new homogeneous equation for F1(t) , which accounts for correlations and is valid for any time moment t. It is shown how this reversible-in-time equation changes at different timescales and converts into the irreversible quantum Boltzmann equation on the macroscopic timescale without the use of the conventional approximations.

Substantiating the directions of health tourism development in Czechia

&lt;abstract&gt; &lt;p&gt;The purpose of the article is to substantiate the directions of health tourism development in Czechia. The purpose of the study is the subjects of health tourism: well-known rehabilitation centers in Czechia, medical institutions, health resorts. The methodological basis was formed by the following methods: abstract-logical; statistical; expert evaluation; cluster analysis; piecewise linear approximation. Among the most popular regions of Czechia for health tourism, the main places are Prague, Moravian-Silesia, Olomouc, Plzeň, South Moravian and South Bohemian. The increase in the number of patients among residents of the country and foreigners who use health tourism services is studied. The five leaders among the countries that used the services are identified. The main problems and advantages of health tourism in Czechia are identified. The Czech tourism development strategy for 2021–2030 is analyzed and directions for its implementation are proposed. &lt;bold&gt;Originality / value&lt;/bold&gt;: the originality of the study lies in forecasting the directions of health tourism development in Czechia using the method of piecewise linear approximation in the following areas: adults with health insurance; adults at their own expense, children and teenagers, which will increase the volume of income for health tourism institutions in Czechia. According to the forecasting results, the demand for health services will increase by 3% annually. The results of this study can be useful for substantiating the directions of further health tourism development in the country, which will ensure the competitiveness and investment attractiveness of the country.&lt;/p&gt; &lt;/abstract&gt;

Image Matching by Bare Homography.

This paper presents Slime, a novel non-deep image matching framework which models the scene as rough local overlapping planes. This intermediate representation sits in-between the local affine approximation of the keypoint patches and the global matching based on both spatial and similarity constraints, providing a progressive pruning of the correspondences, as planes are easier to handle with respect to general scenes. Slime decomposes the images into overlapping regions at different scales and computes loose planar homographies. Planes are mutually extended by compatible matches and the images are split into fixed tiles, with only the best homographies retained for each pair of tiles. Stable matches are identified according to the consensus of the admissible stereo configurations provided by pairwise homographies. Within tiles, the rough planes are then merged according to their overlap in terms of matches and further consistent correspondences are extracted. The whole process only involves homography constraints. As a result, both the coverage and the stability of correct matches over the scene are amplified, together with the ability to spot matches in challenging scenes, allowing traditional hybrid matching pipelines to make up lost ground against recent end-to-end deep matching methods. In addition, the paper gives a thorough comparative analysis of recent state-of-the-art in image matching represented by end-to-end deep networks and hybrid pipelines. The evaluation considers both planar and non-planar scenes, taking into account critical and challenging scenarios including abrupt temporal image changes and strong variations in relative image rotations. According to this analysis, although the impressive progress done in this field, there is still a wide room for improvements to be investigated in future research.

Error Bounds for Compositions of Piecewise affine Approximations

Nonlinear expressions in dynamic models are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. Rather than directly approximating complicated multivariate functions in a high-dimensional space, these can instead be decomposed into a collection of simpler functions, which are then approximated individually and recomposed. This paper provides efficient methods to generate PWA approximations of nonlinear functions via functional decomposition. The key contributions focus on placing breakpoints for PWA approximations to satisfy a desired error tolerance and on bounding the error of PWA compositions as a function of the error tolerance for each component. The proposed methods are used to systematically construct a PWA approximation for a complicated function, either to within a desired error tolerance or to a given level of complexity in the approximation.

PREDICTION OF CLIMATOLOGY USING NEURO-FUZZY TECHNIQUES

For social and economic activists, weather forecasting is the most talked about topic right now. Due to its applicability in a number of public and private industries, such as forestry, air traffic, agriculture, and maritime, it is also garnering widespread interest. Recent events have caused alterations in the climate. Occurat a rapid pace, rendering traditional weather forecasting techniques less accurate, more time-consuming, and erratic. To solve these, more advanced and effective weather forecast techniques are required. Challenges. Through actual evidence, we show that artificial neural networks generate significantly less deviations than GDAS assessment. Thus, almost exact weather forecast results are predicted. This article investigates the forecasting performance of neural networks in comparison to linear and polynomial approximations for a time series produced by the chaotic Mackey-Glass differential delay equation. There is one step ahead in the predicting horizon. A basic neural network with two neurons is used in a series of regressions with polynomial approximators, and the numerous correlation coefficients are compared. A nearly perfect forecast is produced by the neural network, a very basic neural network that outperforms polynomial expansions. Lastly, over a broad range of realizations, the neural network outperforms the other techniques in terms of precision.

Non-average Stability Analysis for a Boost Converter

The purpose of this paper is to present a method for analyzing the stability of DC-DC boost converters providing a universal transfer function. This method relies on the application of the Laplace transform without any ad-hoc linear approximation or system linearization. A common methodology for evaluating the stability of DC-DC converters is the average method, which is essentially a linearization. In this paper, the Laplace transform is applied directly, resulting in a universal Z-transform model being used to design controllers and stability analyses as a discrete unifying transfer function. The examples cover both power and low voltage converters. Matlab simulations have been conducted to verify the theoretical findings. In particular, the second example considers a small 5V to 12V boost converter, which was previously examined in a Texas Instruments application note. The transfer functions and the Bode plot are provided.

Analytical Prescribed Performance Guidance with Field-of-View and Impact-Angle Constraints

Aiming at the impact-angle-control guidance with field-of-view constraint in the three-dimensional space, a novel analytical prescribed performance guidance law is proposed. First, a new prescribed performance function is designed in the relative range domain, and the common drawbacks of most existing prescribed performance functions, such as large overshoot, time dependency, and infinite-time convergence, are avoided. Second, the three-dimensional biased pure proportional navigation guidance frame is selected. The biased term is derived based on the prescribed performance control theory and the optimal error dynamic method, for realizing the terminal impact-angle constraint. After that, the analytical solution of missile lead angle is derived under some linearization approximations, and two methods are respectively developed for obtaining the range of the guidance gain and updating its value to guarantee the satisfaction of the field-of-view constraint during the guidance process. Moreover, for the proposed guidance law and methods, the tracking error could be ensured to remain in the predefined performance boundaries. Finally, numerical simulations are conducted to verify the effectiveness of the new theoretical findings.

Formation Model for Acoustic Characteristics of Solid Media with Ordered Fracturing

Introduction. The development of new structural materials and improvement of existing technologies for the production of new products on their basis lead to the emergence of new types of medium discontinuities. Therefore, the development of new models of discontinuities that take the previously ignored parameters into account seems to be relevant for the purposes of nondestructive testing and structural measurements. This concerns, e.g., the roughness of adjacent surfaces of microcrack ordered sets.Aim. Theoretical substantiation for the processes of elastic waves propagation through an elastic medium containing an ordered lattice of microcracks with boundary conditions in the linear slip approximation, modified by taking into account the parameters of micro-convexities of microcrack rough boundaries. Database formation for experimental studies aimed at determining the physical and mechanical characteristics of structural materials.Materials and methods. The acoustic characteristics of materials were determined based on the derivation and solutions of dispersion equations describing the formation and propagation of effective longitudinal, transverse, and Rayleigh surface elastic waves in elastic media with ordered cracking. Their values were also used to determine the effective speed of Rayleigh surface waves.Results. The conducted simulation of elastic wave formation processes showed that an increase in the concentration of microcracks leads to a decrease in the phase velocities of effective longitudinal, transverse, and surface waves, as well as to an increase in the attenuation coefficients at given ultrasound frequencies and material parameters.Conclusion. The radius of the microsphere that replaces the surface micro-convexity and the roughness parameter Rz have a significant impact on the formation of physical and mechanical characteristics of materials, which are determined by the results of ultrasonic measurements. The developed model can be recommended as a basis for interpreting the results of ultrasonic measurements.

НЕЛИНЕЙНЫЙ АНАЛИЗ МАССИВНЫХ ЖЕЛЕЗОБЕТОННЫХ КОНСТРУКЦИЙ С УЧЕТОМ ТРЕЩИНООБРАЗОВАНИЯ

A finite element method, as well as the algorithm and the program for solid reinforced concrete structures analysis have been developed, taking into account plastic deformations of concrete. A modified Willam &amp; Warnke failure criterion was used, supplemented by a flow criterion. Two models of volumetric deformation of concrete have been developed: an elastic model under brittle fracture and an ideal elastic-plastic model. An eight-node solid finite element with linear approximation of displacement functions, which implements the deformation models above mentioned, is constructed. This finite element is adapted to the PRINS computational software, and as part of this program it can be used for physically nonlinear analysis of building structures containing three-dimensional reinforced concrete elements. Modern building codes prescribe to carry out calculations of concrete and reinforced concrete structures in a nonlinear formulation, taking into account the real properties of concrete and reinforcement. To verify the developed finite element, a series of test calculations of a beam in the condition of pure bending was carried out. Comparison of the calculation results with experimental data confirmed the high accuracy and reliability of the results obtained.

On the power of iid information for linear approximation

This survey is concerned with the power of random information for approximation in the (deterministic) worst-case setting, with special emphasis on information consisting of functionals selected independently and identically distributed (iid) at random on a class of admissible information functionals. We present a general result based on a weighted least squares method and derive consequences for special cases. Improvements are available if the information is "Gaussian" or if we consider iid function values for Sobolev spaces. We include open questions to guide future research on the power of random information in the context of information-based complexity.

Efficient computational strategies for a mathematical programming model for multi-echelon inventory optimization based on the guaranteed-service approach

This paper presents a Multi-Echelon Inventory Optimization (MEIO) framework, based on the Guaranteed-Service Model (GSM), to allocate safety stocks across a supply chain with several locations and products, minimizing costs while meeting service level objectives. Extending previous work by Achkar et al. (2023), this paper enhances the Mixed-Integer Quadratically Constrained Program (MIQCP) with a highly efficient solution approach. The model introduces a piecewise linear approximation, significantly improving computational efficiency and the accuracy of the approximation for the fill rate function. It also introduces a different and more efficient approach to account for stochastic lead times using a discrete function. Moreover, an extension of the approach to account for non-normally distributed demands is proposed. The model is applied to several instances of a real-world case study from a pharmaceutical company, with more than 7300 product-location combinations, showing that optimal solutions can be obtained within few seconds of computational time.