Generalized Approximation Method Research Articles

Asymptotic Methods for Transaction Costs

We propose a general approximation method for the determination of optimal trading strategies in markets with proportional transaction costs, with a polynomial approximation of the residual value function. The method is exemplified by several problems, from optimally tracking benchmarks and hedging the log contract to maximizing utility from terminal wealth. Strategies are also approximated by practically executable, discrete trades. We identify the necessary trade-off between the trading frequency and trade size to ensure satisfactory agreement with the theoretically optimal, continuous strategies of infinite activity.

Optimal Structured Matrix Approximation for Robustness to Incomplete Biosequence Data.

We propose a general method for optimally approximating an arbitrary matrix M by a structured matrix T (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database [1]. The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.

A general approximation method for accurate evaluation of the intermolecular potential parameters by using density functional theory

In this work, a new approach is developed and presented for determining the intermolecular potential parameters from intermolecular interaction energy data obtained using the Density Functional Theory (DFT) method over a different basis set. The proposed method in this work is especially useful for the evaluation of all intermolecular potential parameters of arbitrary atoms and molecules. The method developed in this study can be used for evaluating some thermodynamic properties of molecules. In general, this approach has great advantages for the determination of all types of potential parameters of atoms and molecules. As an application, the Lennard-Jones (12-6) and Morse potential parameters of the Ga(CH3)3 molecule have been determined first by using the basis set with density functional method (DFT/B3LYP) method over the 6-31G basis set. The speed of sound, heat capacities, and second virial coefficient over the Lennard-Jones (L-J) (12-6) potential of Ga(CH3)3 have been calculated. The results obtained for the speed of sound and heat capacities of Ga(CH3)3 are compared with theoretical and experimental data. The results indicate that our data are in good agreement with data from the literature.

Ab initio study of structural, mechanical and electronic properties of 3d transitional metal carbide in cubic rocksalt (rs), zincblende (zb), and cesium chloride (cc) structures by using LDA and GGA Approximation.

This study rigorously investigates three 3d transition metal carbide (TMC) structures via LDA and GGA approximations. It examines cohesive energy (Ecoh), Vickers hardness (Hv), mechanical stability, and electronic properties. Notably, most 3d TMCs exhibit higher cohesive energy than nitrides, and rs-TiC demonstrates a Vickers hardness of 25.66 GPa, outperforming its nitride counterpart. The study employs theoretical calculations to expedite research, revealing mechanical stability in CrC and MnC (GGA) and CrC (LDA in cc structure), while all 3d TMCs in rs and seven in zb structures show stability. Charge transfer and bonding analysis reveal enhanced covalency along the series, influenced by the interplay between p orbitals of carbon and d orbitals of the metal. Most 3d TMCs exhibit metallic properties, excluding zb-TiC and zb-FeC in all phases. An inverse correlation between elastic constant C44 and electronic states near the Fermi level (EF) emerges, guiding applications and design. This study efficiently uncovers 3d TMC properties, offering insights for applications and design. We employed the Vienna ab initio Simulation software (VASP) to perform computations based on density functional theory (DFT). Our approach incorporated both the projector augmented wave (PAW) and PW91 general gradient approximation (GGA) methods within the local density approximation (LDA).

A Unifying Approximate Method of Multipliers for Distributed Composite Optimization

This article investigates solving convex composite optimization on an undirected network, where each node, privately endowed with a smooth component function and a nonsmooth one, is required to minimize the sum of all the component functions throughout the network. To address such a problem, a general approximate method of multipliers (AMM) is developed, which attempts to approximate the method of multipliers by virtue of a surrogate function with numerous options. We then design the possibly nonseparable, time-varying surrogate function in various ways, leading to different distributed realizations of AMM. We demonstrate that AMM generalizes more than ten state-of-the-art distributed optimization algorithms, and certain specific designs of its surrogate function result in a variety of new algorithms to the literature. Furthermore, we show that AMM is able to achieve an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1/k)$</tex-math></inline-formula> rate of convergence to optimality, and the convergence rate becomes linear when the problem is locally restricted strongly convex and smooth. Such convergence rates provide new or stronger convergence results to many prior methods that can be viewed as specializations of AMM.

A general approximation method for optimal stopping and random delay

AbstractThis study examines the continuous‐time optimal stopping problem with an infinite horizon under Markov processes. Existing research focuses on finding explicit solutions under certain assumptions of the reward function or underlying process; however, these assumptions may either not be fulfilled or be difficult to validate in practice. We developed a continuous‐time Markov chain (CTMC) approximation method to find the optimal solution, which applies to general reward functions and underlying Markov processes. We demonstrated that our method can be used to solve the optimal stopping problem with a random delay, in which the delay could be either an independent random variable or a function of the underlying process. We established a theoretical upper bound for the approximation error to facilitate error control. Furthermore, we designed a two‐stage scheme to implement our method efficiently. The numerical results show that the proposed method is accurate and rapid under various model specifications.

The Analysis of English Sentence Components Based on Decision Tree Classification Algorithm

Decision tree is an important classification method in data excavation technology. It is a predictive analysis model expressed in the form of a tree structure (including binary trees and poly trees). The decision tree method is a more general classification function approximation method. It is an algorithm commonly used in predictive models to find some potentially valuable information by purposefully classifying a large amount of data. In this article, the author tries to analyze the English sentence components based on the decision tree classification algorithm. The author starts with the decision tree, extracts the decision tree rules, and generates a classifier by effectively sorting the decision tree rules, and applies it to classification prediction.

Mesh stiffness and dynamic response analysis of modified gear system with thin web and weight reduction holes

Though the special gear foundation structure and tooth modification are effectively applied to weight optimization and vibration control, the lack of an effective mathematical model limits the investigation of their effects. The present work develops a general polygonal approximation method for weight reduction holes, which is introduced into an analytical-finite element (AFE) method along with the decomposition method to calculate the time-varying mesh stiffness (TVMS) of modified spur gear pairs with complex foundation (thin webs and weight reduction holes). The comparisons with the traditional finite element (FE) method show the superiority of the model in calculation accuracy and efficiency. Furthermore, the actual TVMS is introduced into the dynamics model of a high-speed gear-rotor-bearing system to analyze the dynamic responses, and the effectiveness of the model is verified by experiments. The results indicate that the thin web has a greater influence on the reduction of average amplitude in stiffness than holes. The thin web significantly controls the system vibration, while the tooth modification reduces the vibration fluctuation caused by holes at medium and high speeds. The analysis and experimental results provide an important reference for gear weight optimization and vibration reduction.

Dynamics of diffusion on monoplex and multiplex networks: a message-passing approach

New ideas and technologies adopted by a small number of individuals occasionally spread globally through a complex web of social ties. Here, we present a simple and general approximation method, namely, a message-passing approach, that allows us to describe the diffusion processes on (sparse) random networks in an almost exact manner. We consider two classes of binary-action games where the best pure strategies for individual players are characterized as variants of the threshold rule. We verify that the dynamics of diffusion observed on synthetic networks are accurately replicated by the message-passing equation, whose fixed point corresponds to a Nash equilibrium, while the conventional mean-field method tends to overestimate the size and frequency of diffusion. Generalized cascade conditions under which a global diffusion can occur are also provided. We extend the framework to analyze multiplex networks in which social interactions take place in multiple layers.

Polynomial approximation of dynamic signals of single platform railway scales

Purpose. Obtaining an approximating function (or a system of approximating equations), which, with a minimum error, will make approximations to the available data on a train of railway objects through 1 platform scales. Methodology. To solve this problem, numerical methods are used, namely, the approximation by polynomial functions of the nth order. The experimental data on the basis of which the experiments were carried out were obtained from the weighing and identification system of wagon in motion on a single platform scale. The approximation process is automated using a program written in the Python programming language in which the polyPit and polyid functions of the numPy library are used to obtain the polynomial coefficients. Findings. Due to the use of polynomial approximation in data processing from tensometric railroad weighing systems, it was possible to obtain a system of linear equations that, with minimal error, restored the experimental data that were obtained from the existing system of the Severny GOK: Metinvest enterprise. When normalizing the readings of the sensors from conventional units, obtained from the summing box to the range of values [0; 1] it became possible, in percentage terms, to describe a railway object. This makes it possible to avoid the dependence of the final results on the travel speed of the carriage or locomotive, which leads to an increase in the accuracy of the identification of cars in the rolling stock due to the use of the percentage of the axles staying on the weighing platform (approach / exit). It became possible to determine the type of carriage with the same number of axles, but different characteristics of the center space and the base of the rolling stock. Originality. The novelty is to obtain a general method of approximation of experimental data of the passage of wagons through a single-platform scales, which can be used to train intelligent systems and generate close to real data of the passage of a car (due to the imposition of noise, etc.). Practical value. Improving the accuracy and speed of the carriage identification as a whole, which reduces the plant downtime, contributes to an increase in the number of weighed and identified moving objects, as well as the ability to identify the type of carriage with the same number of axles in the train. The methods presented in the work can be used both for identification and for tasks, the end result of which is the classification of input data (neural networks, etc.).

High order weak approximation for irregular functionals of time-inhomogeneous SDEs

Abstract This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.

Dynamics of Diffusion on Monoplex and Multiplex Networks: A Message-Passing Approach

New ideas and technologies adopted by a small number of individuals occasionally spread globally through a complex web of social ties. Here, we present a simple and general approximation method, namely, a message-passing approach, that allows us to describe the diffusion processes on complex networks in an almost exact manner. We consider two classes of binary-action games in each of which the best pure strategy for individual players is characterized as a variant of the threshold rule. We show that the dynamics of diffusion observed on synthetic networks are accurately replicated by the message-passing equation, whose fixed point corresponds to a Nash equilibrium. In contrast, the mean-field method tends to overestimate the size and frequency of diffusion. We extend the framework to analyze multiplex networks in which social interactions take place in multiple layers.

General rational approximation of Gaussian wavelet series and continuous‐timegm‐C filter implementation

SummaryA general method of rational approximation for Gaussian wavelet series and Gaussian wavelet filter circuit design with simplegm‐C integrators is presented in this work. Firstly, the multiorder derivatives of Gaussian function are analyzed and proved as wavelet base functions. Then a high‐accuracy general approximation model of Gaussian wavelet series is constructed, and the transfer function of first‐order derivative of Gaussian wavelet filter is obtained using quantum differential evolution (QDE) algorithm. Thirdly, as an example, a fifth‐order continuous‐time analog first‐order derivative of Gaussian wavelet filter circuit is designed based on multiple loop feedback structure with a simplegm‐C integrator as the basic blocks. Finally, simulation results demonstrate that the proposed method is an excellent way for the wavelet transform implementation. The designed first‐order derivative of Gaussian wavelet filter circuit operates from a 0.53‐V supply voltage and a bias current 2.5 nA. The power dissipation of the wavelet filter circuit at the basic scale is 41.1 nW. Moreover, the high‐accuracy QRS detection based on the designed wavelet filter has been validated in application analysis.

All–Anisotropic Spheroidal Photonic Antennas: Theory and Modeling

We introduce the concept of all-anisotropic spheroidal photonic antennas, where the nanoantenna is formed by a uniaxial anisotropic crystal, and study their modeling and optimization. For this purpose, we develop, for the first time, a rigorous theory for the electromagnetic (EM) scattering by anisotropic spheroids, using a full-wave method which features spheroidal eigenvectors with a set of discrete wavenumbers for the expansion of the fields in the region of anisotropy. We fully validate our theory by comparisons with the commercial HFSS software, as well as with a general purpose discrete dipole approximation method. Our implementation turns out to be fast, and serves as an efficient tool for modeling the resonant properties of spheroidal photonic antennas. In this context, we show that geometrical tailoring, by employing high-aspect ratio prolate and oblate spheroids, allows for optimizing the nanoantenna and achieving maximal forward-to-back scattering or maximal forward-scattering with reduced back-scattering. In addition, we demonstrate how the use of anisotropic materials enables us to set this optimum state at smaller wavelengths, even close to ultraviolet (UV), as compared to isotropic dielectrics, and we reveal the key role of the incident field's polarization in achieving optimized operation for both prolate and oblate configurations.

Efficient spread-size approximation of opinion spreading in general social networks.

In contemporary society, understanding how information, such as trends and viruses, spreads in various social networks is an important topic in many areas. However, it is difficult to mathematically measure how widespread the information is, especially for a general network structure. There have been studies on opinion spreading, but many studies are limited to specific spreading models such as the susceptible-infected-recovered model and the independent cascade model, and it is difficult to apply these studies to various situations. In this paper, we first suggest a general opinion spreading model (GOSM) that generalizes a large class of popular spreading models. In this model, each node has one of several states, and the state changes through interaction with neighboring nodes at discrete time intervals. Next, we show that many GOSMs have a stable property that is a GOSM version of a uniform equicontinuity. Then, we provide an approximation method to approximate the expected spread size for stable GOSMs. For the approximation method, we propose a concentration theorem that guarantees that a generalized mean-field theorem calculates the expected spreading size within small error bounds for finite time steps for a slightly dense network structure. Furthermore, we prove that a “single simulation” of running the Monte Carlo simulation is sufficient to approximate the expected spreading size. We conduct experiments on both synthetic and real-world networks and show that our generalized approximation method well predicts the state density of the various models, especially in graphs with a large number of nodes. Experimental results show that the generalized mean-field approximation and a single Monte Carlo simulation converge as shown in the concentration theorem.

Engineering Subwavelength Nanoantennas in the Visible by Employing Resonant Anisotropic Nanospheroids

We analyze the engineering of subwavelength nanoantennas composed of anisotropic nanospheroids, for the development of photonic devices. Instead of using conventional isotropic dielectrics, we introduce resonant anisotropic nanoparticles, allowing for shifting Kerker condition points further inside the visible. To address this study, we construct a perturbation-based discrete eigenfunction method, for the electromagnetic scattering of a plane wave by a prolate or oblate uniaxial anisotropic spheroid. The method is fast and yields the solution for the bistatic radar and total scattering cross sections, which is valid for small eccentricities of the spheroid. The validity of this technique is verified by the alternative general purpose discrete dipole approximation method. We investigate the engineering of subwavelength nanoantennas due to material and geometry shaping, like the change of anisotropy type, anisotropy ratio, and deviation of the nanoantenna from sphericity.

General viscosity approximation methods for quasi-nonexpansive mappings with applications

The purpose of this paper is to introduce and study the general viscosity approximation methods for quasi-nonexpansive mappings in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converge strongly to a fixed point of quasi-nonexpansive mappings in Hilbert spaces, which is also the unique solution of some variational inequality. Then this result is used to study the split equality fixed point problems, the split equality common fixed point problems, the split equality null point problems, etc. Our results improve and generalize many results in the literature and they should have many applications in nonlinear science.

Optimal parameter selections for a general Halpern iteration

Let C be a closed affine subset of a real Hilbert space H and $T:C \rightarrow C$ be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process: $$\left\{\begin{array}{ll} x_{0} \in C,\\ x_{n + 1}=t_{n}u+(1-t_{n})Tx_{n},n\geq 0, \end{array}\right. $$ is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, $\mathbb {R}$. The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.

A general approximation method for bicriteria minimization problems

We present a general technique for approximating bicriteria minimization problems with positive-valued, polynomially computable objective functions. Given 0<ϵ≤1 and a polynomial-time α-approximation algorithm for the corresponding weighted sum problem, we show how to obtain a bicriteria (α⋅(1+2ϵ),α⋅(1+2ϵ))-approximation algorithm for the budget-constrained problem whose running time is polynomial in the encoding length of the input and linear in 1ϵ.Moreover, we show that our method can be extended to compute an (α⋅(1+2ϵ),α⋅(1+2ϵ))-approximate Pareto curve under the same assumptions. Our technique applies to many minimization problems to which most previous algorithms for computing approximate Pareto curves cannot be applied because the corresponding gap problem is NP-hard to solve. For maximization problems, however, we show that approximation results similar to the ones presented here for minimization problems are impossible to obtain in polynomial time unless P=NP.

INFLUENCE OF SPATIAL INTERACTIONS OF INCLUSIONS ON THE EFFECTIVE ELASTIC TENSOR OF CRACKED POROUS MEDIUM

The determination of effective stiffness tensor of microinhomogeneous and, in general, macroscopically homogeneous composite medium is related to so-called problem of many-body interaction. Solution to the problem can be found only as an approximation. In this paper we consider a solution to such a problem for porous-cracked medium that is a terrigenous rock having anisotropic elastic properties. The elastic anisotropy is a result of many factors including anisotropic properties of clay minerals and preferential orientation of non-isometric heterogeneities. Different Effective Medium Theories for calculating effective stiffness tensor of cracked porous medium use so called Effective Field Hypothesis (H1, H2 and H3). For example, T-matrix method, Mori-Tanaka method, General Singular Approximation method, and Effective Field Methods use the Effective Field Hypothesis. Thus, different methods produce similar results. When constructing models of rock’s effective properties the rock is treated a composite “made by nature”. In this case of importance is a proper approximation of the real medium by a parametric model medium that reflects specific features of rock’s microstructure. The microstructure is a result of rock evolution. Therefore, the model of the medium and the model parameters play very important roles in the modelling. To prove this statement, two models of a cracked- porous medium’s properties were created using two different methods: the T-matrix method and General Singular Approximation Method. The methods were applied for two different parametric models of one and the same rock. The models were build based on visual analysis of rock’s thin sections. Each of the constructed models has different number of parameters. The parameters are also different. However, a common feature of the two models is that for rocks of this type it is necessary to take into account a rigidity of contact between mineral grains and organic material. Besides, a connectivity of different heterogeneities should be also parametrized. For each model a set of parameters was found and a porosity interval where the models produce similar results in terms of elastic wave velocities is determined.