1,637 results
Search Results
2. Patterns Relating to Complete Symbols.
- Author
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Hilton, Peter, Pedersen, Jean, Quintanar-Zilinskas, Victor, Velarde, Linda, and Walden, Byron
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SIGNS & symbols , *PAPER arts , *ALGORITHMS , *APPROXIMATION theory , *POLYGONS - Abstract
In this paper, the authors look for patterns among the symbols, including complete symbols. These symbols arise from a paper-folding algorithm which produces approsimations to regular polygons, to any desired level of accuracy. In order to quickly generate these symbols, they use a Maple program, which is included in Section 4 of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2008
3. Unsupervised Attribute Reduction Algorithm for Mixed Data Based on Fuzzy Optimal Approximation Set.
- Author
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Wen, Haotong, Zhao, Shixin, and Liang, Meishe
- Subjects
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ROUGH sets , *FUZZY sets , *MACHINE learning , *ALGORITHMS , *APPROXIMATION theory , *APPROXIMATION algorithms - Abstract
Fuzzy rough set theory has been successfully applied to many attribute reduction methods, in which the lower approximation set plays a pivotal role. However, the definition of lower approximation used has ignored the information conveyed by the upper approximation and the boundary region. This oversight has resulted in an unreasonable relation representation of the target set. Despite the fact that scholars have proposed numerous enhancements to rough set models, such as the variable precision model, none have successfully resolved the issues inherent in the classical models. To address this limitation, this paper proposes an unsupervised attribute reduction algorithm for mixed data based on an improved optimal approximation set. Firstly, the theory of an improved optimal approximation set and its associated algorithm are proposed. Subsequently, we extend the classical theory of optimal approximation sets to fuzzy rough set theory, leading to the development of a fuzzy improved approximation set method. Finally, building on the proposed theory, we introduce a novel, fuzzy optimal approximation-set-based unsupervised attribute reduction algorithm (FOUAR). Comparative experiments conducted with all the proposed algorithms indicate the efficacy of FOUAR in selecting fewer attributes while maintaining and improving the performance of the machine learning algorithm. Furthermore, they highlight the advantage of the improved optimal approximation set algorithm, which offers higher similarity to the target set and provides a more concise expression. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies.
- Author
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Yang Yang, Degao Peng, Jianfeng Lu, and Weitao Yang
- Subjects
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DENSITY functional theory , *ALGORITHMS , *CHARGE transfer , *GROUND state (Quantum mechanics) , *APPROXIMATION theory , *PARTICLES , *EXCITATION energy (In situ microanalysis) - Abstract
The particle-particle random phase approximation (pp-RPA) has been used to investigate excitation problems in our recent paper [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. It has been shown to be capable of describing double, Rydberg, and charge transfer excitations, which are challenging for conventional time-dependent density functional theory (TDDFT). However, its performance on larger molecules is unknown as a result of its expensive O(N6) scaling. In this article, we derive and implement a Davidson iterative algorithm for the pp-RPA to calculate the lowest few excitations for large systems. The formal scaling is reduced to O(N4), which is comparable with the commonly used configuration interaction singles (CIS) and TDDFT methods. With this iterative algorithm, we carried out benchmark tests on molecules that are significantly larger than the molecules in our previous paper with a reasonably large basis set. Despite some self-consistent field convergence problems with ground state calculations of (N - 2)-electron systems, we are able to accurately capture lowest few excitations for systems with converged calculations. Compared to CIS and TDDFT, there is no systematic bias for the pp-RPA with the mean signed error close to zero. The mean absolute error of pp-RPA with B3LYP or PBE references is similar to that of TDDFT, which suggests that the pp-RPA is a comparable method to TDDFT for large molecules. Moreover, excitations with relatively large non-HOMO excitation contributions are also well described in terms of excitation energies, as long as there is also a relatively large HOMO excitation contribution. These findings, in conjunction with the capability of pp-RPA for describing challenging excitations shown earlier, further demonstrate the potential of pp-RPA as a reliable and general method to describe excitations, and to be a good alternative to TDDFT methods. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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5. Approximability results for the p-centdian and the converse centdian problems.
- Author
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Yen Hung Chen
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UNDIRECTED graphs , *APPROXIMATION theory , *MATHEMATICAL functions , *INTEGERS , *ALGORITHMS - Abstract
Given an undirected graph G = (V,E) with a nonnegative edge length function and an integer p, 0 < p < |V |, the p-centdian problem is to find p vertices (called the centdian set) of V such that the eccentricity plus median-distance is minimized, in which the eccentricity is the maximum (length) distance of all vertices to their nearest centdian set and the median-distance is the total (length) distance of all vertices to their nearest centdian set. The eccentricity plus median-distance is called the centdian-distance. The purpose of the p-centdian problem is to find p open facilities (servers) which satisfy the quality-of-service of the minimum total distance (median-distance) and the maximum distance (eccentricity) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the centdian-distance and the objective function is to minimize the cardinality of the centdian set, this problem is called the converse centdian problem. In this paper, we prove the p-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the p-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems. [ABSTRACT FROM AUTHOR]
- Published
- 2022
6. A temporal segmentation algorithm for restoring the controllability of networked control systems.
- Author
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Cui, Yulong, Wu, Mincheng, Shao, Cunqi, and He, Shibo
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POLYNOMIALS , *APPROXIMATION theory , *FEASIBILITY studies , *ALGORITHMS - Abstract
Restoring the controllability of networked control systems is a fundamental issue that needs to be settled, especially when malicious attacks or malfunctions occur. Previous studies tried to address such an issue by adding extra driver nodes or rewiring edges, which will inevitably change the original network structure. In this paper, an algorithm to restore the controllability of networked systems while preserving the integrity of the original network structure is proposed. Specifically, a static uncontrollable network will be transformed into a controllable temporal network by means of the cactus‐based segmentation method. The original problem will be equivalent to the classical set cover problem, which is known to be NP‐Hard, if the least number of segmentations is considered. An approximation algorithm with polynomial time complexity is proposed and it is proved that the solution to the problem is two‐optimal. Finally, simulations are carried out to verify the effectiveness and feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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7. Variational path integral molecular dynamics and hybrid Monte Carlo algorithms using a fourth order propagator with applications to molecular systems.
- Author
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Yuki Kamibayashi and Shinichi Miura
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MOLECULAR dynamics , *PATH integrals , *HARMONIC oscillators , *HESSIAN matrices , *APPROXIMATION theory , *MONTE Carlo method , *ALGORITHMS - Abstract
In the present study, variational path integral molecular dynamics and associated hybrid Monte Carlo (HMC) methods have been developed on the basis of a fourth order approximation of a density operator. To reveal various parameter dependence of physical quantities, we analytically solve one dimensional harmonic oscillators by the variational path integral; as a byproduct, we obtain the analytical expression of the discretized density matrix using the fourth order approximation for the oscillators. Then, we apply our methods to realistic systems like a water molecule and a para-hydrogen cluster. In the HMC, we adopt two level description to avoid the time consuming Hessian evaluation. For the systems examined in this paper, the HMC method is found to be about three times more efficient than the molecular dynamics method if appropriate HMC parameters are adopted; the advantage of the HMC method is suggested to be more evident for systems described by many body interaction. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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8. EXPOSITORY RESEARCH PAPERS.
- Author
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Ipsen, Ilse
- Subjects
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ALGORITHMS , *APPROXIMATION theory , *MATHEMATICS problems & exercises , *GAUSSIAN quadrature formulas , *MATHEMATICAL analysis - Abstract
The article highlights two papers published within the issue titled "Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm," by Steven Brams, Michael Jones and Christian Klamler, and "Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples," by Rodrigo Platte, Lloyd Trefethen and Arno Kuijlaars. The first paper discussed the mathematical problem of proportional division. The second one talked about potential theory, matrix iterations in the form of Krylov space methods and quadrature formulae.
- Published
- 2011
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9. Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations.
- Author
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Fuke Wu, Tianhai Tian, Rawlings, James B., and Yin, George
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CHEMICAL kinetics , *APPROXIMATION theory , *LANGEVIN equations , *CHEMICAL species , *STOCHASTIC processes , *ALGORITHMS - Abstract
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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10. Multivalue-multistage Method for Second-order ODEs.
- Author
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Ismail, Ainathon and Rabiei, Faranak
- Subjects
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DIFFERENTIAL equations , *ALGORITHMS , *NUMERICAL analysis , *VECTOR analysis , *APPROXIMATION theory - Abstract
The aim of this paper is to generate the needed framework to develop order conditions for second-order Ordinary Differential Equations (ODEs) by class of multivalue-multistage Nystrom method. A general approach to study the order conditions of the methods for solving second-order initial value problems is investigated. Our investigation will be carried out by adapting the theory of Nystrom-series and using the sets of second order rooted trees for solving second-order ODEs which leads to a general set of order conditions. In this paper the method of order three using constant step-size algorithm is derived. The stability region of method is also proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
11. The Conjugate Gradient Viscosity Approximation Algorithm for Split Generalized Equilibrium and Variational Inequality Problems.
- Author
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Li, Meixia, Che, Haitao, and Tan, Jingjing
- Subjects
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VISCOSITY , *APPROXIMATION theory , *ALGORITHMS , *GENERALIZATION , *FEASIBILITY studies - Abstract
In this paper, we study a kind of conjugate gradient viscosity approximation algorithm for finding a common solution of split generalized equilibrium problem and variational inequality problem. Under mild conditions, we prove that the sequence generated by the proposed iterative algorithm converges strongly to the common solution. The conclusion presented in this paper is the generalization, extension, and supplement of the previously known results in the corresponding references. Some numerical results are illustrated to show the feasibility and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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12. Successive-approximation algorithm for estimating capacity of Li-ion batteries.
- Author
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Goh, Taedong, Park, Minjun, Seo, Minhwan, Kim, Jun Gu, and Kim, Sang Woo
- Subjects
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APPROXIMATION theory , *LITHIUM-ion batteries , *OPEN-circuit voltage , *KALMAN filtering , *ALGORITHMS - Abstract
This paper proposes a capacity estimation algorithm for Li-ion batteries (LIBs) using the successive approximation method. Model-based capacity estimation method can be applied to a variety of current profiles because the capacity is calculated from state of charge (SOC) estimated accurately using the functional relationship between open circuit voltage (OCV) and the SOC. However, with aging, the OCV-SOC table changes, which worsens the estimation accuracy. Therefore, additional experiments are necessary to compensate the errors whenever the capacity should be identified. To overcome this restriction, this paper proposes an algorithm for estimating both the capacity and the corresponding OCV-SOC table based on the preliminary OCV-SOC tables obtained from other batteries. The capacity and the table are updated successively based on the prior capacity estimate. This work proposes two algorithms for voltage characteristics: OCV measurement and SOC estimation cases. The former uses the measured OCV to calculate the SOCs directly, while the latter estimates the SOCs using a dual extended Kalman filter (DEKF). Aging data from five LIB packs are analyzed, and the capacity estimation errors are less than 2.2% for the OCV measurement case and 3.06% until 20% loss of capacity estimate for SOC estimation case. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
13. SVD-BASED ALGORITHMS FOR THE BEST RANK-1 APPROXIMATION OF A SYMMETRIC TENSOR.
- Author
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YU GUAN, CHU, MOODY T., and DELIN CHU
- Subjects
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ALGORITHMS , *APPROXIMATION theory , *MATHEMATICAL symmetry , *TENSOR algebra , *ITERATIVE methods (Mathematics) - Abstract
This paper revisits the problem of finding the best rank-1 approximation to a symmetric tensor and makes three contributions. First, in contrast to the many long and lingering arguments in the literature, it offers a straightforward justification that generically the best rank-1 approximation to a symmetric tensor is symmetric. Second, in contrast to the typical workhorse in the practice for the low-rank tensor approximation, namely, the alternating least squares (ALS) technique which improves one factor a time, this paper proposes three alternative algorithms based on the singular value decomposition (SVD) that modifies two factors a time. One step of SVD-based iteration is superior to two steps of ALS iterations. Third, it is not only proved that the generalized Rayleigh quotients generated from the three SVD-based algorithms enjoy monotone convergence, but also that the iterates themselves converge. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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14. Determining the Optimal Order Quantity with Compound Erlang Demand under (T,Q) Policy.
- Author
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Jiang, Aiping, Tam, Kwok Leung, Bao, Yingzi, and Lu, Jialing
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ELECTRIC power , *ALGORITHMS , *APPROXIMATION theory , *POWER resources , *ALGEBRA - Abstract
Management of electric equipment has a direct impact on companies’ performance and profitability. Considering the critical role that electric power materials play in supporting maintenance operations and preventing equipment failure, it is essential to maintain an inventory to a reasonable level. However, a majority of these electric power materials exhibit an intermittent demand pattern characterized by random arrivals interspersed with time periods with no demand at all. These characteristics cause additional difficulty for companies in managing these electric power material inventories. In response to the above problem, this paper, based on the electric power material demand data of Shanghai Electric Power Company, develops a new method to determine the optimal order quantity Q⁎ in a cost-oriented periodic review (T,Q) system, whereby unsatisfied demands are backordered and demand follows a compound Erlang distribution. Q⁎corresponds to the value of Q that gives the minimum expected total inventory holding and backordering cost. Subsequently, an empirical investigation is conducted to compare this method with the Newsvendor model. Results verify its superiority in cost savings. Ultimately, considering the complicated calculation and low efficiency of that algorithm, this paper proposes an approximation and a heuristic algorithm which have a higher level of utility in a real industrial context. The approximation algorithm simplifies the calculation process by reducing iterative times while the heuristic algorithm achieves it by generalizing the relationship between the optimal order quantity Q⁎ and mean demand interarrival rate λ. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
15. APPROXIMATE SOLUTIONS OF INTERVAL-VALUED OPTIMIZATION PROBLEMS.
- Author
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Van Tuyen, Nguyen
- Subjects
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APPROXIMATE solutions (Logic) , *EXISTENCE theorems , *MATHEMATICAL optimization , *MATHEMATICAL programming , *MULTIPLE criteria decision making , *APPROXIMATION theory , *ALGORITHMS - Abstract
This paper deals with approximate solutions of an optimization problem with interval-valued objective function. Four types of approximate solution concepts of the problem are proposed by considering the partial ordering LU on the set of all closed and bounded intervals. We show that these solutions exist under very weak conditions. Under suitable constraint qualifications, we derive Karush-Kuhn-Tucker necessary and sufficient optimality conditions for convex interval-valued optimization problems. [ABSTRACT FROM AUTHOR]
- Published
- 2021
16. Tensor hypercontracted ppRPA: Reducing the cost of the particle-particle random phase approximation from O(r6) to O(r4).
- Author
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Shenvi, Neil, van Aggelen, Helen, Yang Yang, and Weitao Yang
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FUNCTIONAL analysis , *ALGORITHMS , *ELECTRONIC structure , *APPROXIMATION theory , *TENSOR algebra - Abstract
In recent years, interest in the random-phase approximation (RPA) has grown rapidly. At the same time, tensor hypercontraction has emerged as an intriguing method to reduce the computational cost of electronic structure algorithms. In this paper, we combine the particle-particle random phase approximation with tensor hypercontraction to produce the tensor-hypercontracted particle-particle RPA (THC-ppRPA) algorithm. Unlike previous implementations of ppRPA which scale as O(r6), the THC-ppRPA algorithm scales asymptotically as only O(r4), albeit with a much larger prefactor than the traditional algorithm.We apply THC-ppRPA to several model systems and show that it yields the same results as traditional ppRPA to within mH accuracy. Our method opens the door to the development of post-Kohn Sham functionals based on ppRPA without the excessive asymptotic cost of traditional ppRPA implementations. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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- View/download PDF
17. Output regulation problem of a class of pure-feedback nonlinear systems via adaptive neural control.
- Author
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Jia, Fujin, Lu, Junwei, and Li, Yongmin
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ADAPTIVE control systems , *NONLINEAR systems , *ADAPTIVE fuzzy control , *APPROXIMATION theory , *ALGORITHMS , *CLOSED loop systems - Abstract
In the paper, a control algorithm for output regulation problem of nonlinear pure-feedback systems with unknown functions is proposed. The main contributions of the proposed method are not only to avoid Assumptions of unknown functions, but also adopt a non-backstepping control scheme. First, a high-gain state observer with disturbance signals is designed based on the new system that has been converted. Second, an internal model with the observer state is established. Finally, based on Lyapunov analysis and the neural network approximation theory, the control algorithm is proposed to ensure that all the signals of the closed-loop system are the semi-globally uniformly ultimately bounded, and the tracking error converges to a small neighborhood of the origin. Three simulation studies are worked out to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
18. Multi-Objective Optimization Design for Electromagnetic Devices With Permanent Magnet Based on Approximation Model and Distributed Cooperative Particle Swarm Optimization Algorithm.
- Author
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Xuerong, Ye, Hao, Chen, Huimin, Liang, Xinjun, Chen, and Jiaxin, You
- Subjects
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ELECTROMAGNETIC devices , *MULTIDISCIPLINARY design optimization , *PERMANENT magnet motors , *APPROXIMATION theory , *PARTICLE swarm optimization , *ALGORITHMS , *ENERGY consumption ,MOTOR design & construction - Abstract
Electromagnetic devices containing permanent magnet (PM) are common and typical electromagnetic devices that have advantages of low-power consumption, small size, high sensitivity, and its optimization problem has aroused widespread concern. Calculation accuracy of electromagnetic properties and efficiency of optimization processes are two important factors that affect the optimization effect of electromagnetic devices. A fast calculation method was proposed in this paper that aimed at electromagnetic devices with PM characteristics of strong non-linearity and low solving precision. The response surface method was used to derive the basis function, and the error between actual measured values and calculated values was modified by the Kriging method. On the basis of the calculation model, the distributed collaborative strategy was adopted to fulfill the parallel execution of the fast computation. Simultaneously, the updating strategy of particles was improved, and the robustness of the multi-objective particle swarm optimization algorithm in solving the optimization problem of electromagnetic devices with PM was enhanced. Effectiveness of the method proposed in this paper was verified by the case study of a specific type of electromagnetic relay with PM. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
19. Stabilization algorithm for the optimal transportation meshfree approximation scheme.
- Author
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Weißenfels, C. and Wriggers, P.
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ALGORITHMS , *MESHFREE methods , *COMPUTER-aided engineering , *APPROXIMATION theory , *DIRICHLET problem , *BOUNDARY value problems - Abstract
Meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility. Especially the tremendous progress in processor technology within recent years relativizes the increase in computation time due to the inherent search algorithm. Nevertheless meshfree approximation schemes are still faced with some challenges, like imposition of Dirichlet boundary conditions, robustness of the algorithm and accuracy. The recent developed Optimal Transportation Meshfree (OTM) method seemed to overcome most of these problems. In this paper the OTM solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation. However this scheme is not stable for some examples of application. Hence an investigation is conducted which shows that the reason for this instability is due to underintegration. Based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the Finite Element Method. In contrast to the original publication, the OTM algorithm is derived here from the principle of virtual work. Local maximum entropy shape functions are used which possess a weak Kronecker- δ property. This enables a direct imposition of Dirichlet boundary conditions if the boundary is convex. The limitations of this basis function are also addressed in this paper. Additionally, the search algorithm presented fulfills basic topological requirements. Several examples are investigated demonstrating the improved behavior of the stabilized OTM algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
20. Adaptive Randomized Dimension Reduction on Massive Data.
- Author
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Darnell, Gregory, Georgiev, Stoyan, Mukherjee, Sayan, and Engelhardt, Barbara E.
- Subjects
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ESTIMATION theory , *DIMENSIONAL reduction algorithms , *ALGORITHMS , *DIMENSION reduction (Statistics) , *APPROXIMATION theory , *PRINCIPAL components analysis , *GENOMICS - Abstract
The scalability of statistical estimators is of increasing importance in modern applications. One approach to implementing scalable algorithms is to compress data into a low dimensional latent space using dimension reduction methods. In this paper, we develop an approach for dimension reduction that exploits the assumption of low rank structure in high dimensional data to gain both computational and statistical advantages. We adapt recent randomized low-rank approximation algorithms to provide an efficient solution to principal component analysis (PCA), and we use this efficient solver to improve estimation in large-scale linear mixed models (LMM) for association mapping in statistical genomics. A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance by implicitly regularizing the covariance matrix estimate of the random effect in an LMM. These statistical and computational advantages are highlighted in our experiments on simulated data and large-scale genomic studies. [ABSTRACT FROM AUTHOR]
- Published
- 2017
21. Bounded perturbation resilience of extragradient-type methods and their applications.
- Author
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Dong, Q-L, Gibali, A, Jiang, D, and Tang, Y
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NUMERICAL solutions to equations , *MATHEMATICAL analysis , *APPROXIMATION theory , *HILBERT space , *ALGORITHMS - Abstract
In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of $O(1/t)$ . Numerical illustrations are given to demonstrate the performances of the algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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22. An efficient six-step method for the solution of the Schrödinger equation.
- Author
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Berg, Dmitriy and Simos, T.
- Subjects
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SCHRODINGER equation , *DERIVATIVES (Mathematics) , *APPROXIMATION theory , *ALGORITHMS , *PARTIAL differential equations - Abstract
In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are: This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
23. PACKING A KNAPSACK OF UNKNOWN CAPACITY.
- Author
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DISSER, YANN, KLIMM, MAX, MEGOW, NICOLE, and STILLER, SEBASTIAN
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KNAPSACK problems , *APPROXIMATION theory , *ALGORITHMS , *DENSITY , *GOLDEN ratio - Abstract
This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514{540]. In this sequel, we quantitatively improve several of the main results of the rst paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set A ⊂ ℝ: Ǝa ∈ A such that ∣A(A + a)∣ ≳ ∣A∣3/2+1/186, ∣A(A - A)∣ ≳ ∣A∣3/2+1/34, ∣A(A + A)∣ ≳ ∣A∣ 3/2+5/242, ∣{(a1 + a2 + a3 + a4)² + log a5 : ai ∈ A}∣ ⪢ ∣A∣²/log ∣A∣. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
24. Convergence of Markovian stochastic approximation for Markov random fields with hidden variables.
- Author
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Qi, Anna, Yang, Lihua, and Huang, Chao
- Subjects
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MONTE Carlo method , *MARKOV random fields , *STOCHASTIC approximation , *STOCHASTIC convergence , *ALGORITHMS , *POISSON processes , *APPROXIMATION theory - Abstract
This paper studies the convergence of the stochastic algorithm of the modified Robbins–Monro form for a Markov random field (MRF), in which some of the nodes are clamped to be observed variables while the others are hidden ones. Based on the theory of stochastic approximation, we propose proper assumptions to guarantee the Hölder regularity of both the update function and the solution of the Poisson equation. Under these assumptions, it is proved that the control parameter sequence is almost surely bounded and accordingly the algorithm converges to the stable point of the log-likelihood function with probability 1. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
25. Fast Cross-Validation for Kernel-Based Algorithms.
- Author
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Liu, Yong, Liao, Shizhong, Jiang, Shali, Ding, Lizhong, Lin, Hailun, and Wang, Weiping
- Subjects
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TAYLOR'S series , *APPROXIMATION theory , *ALGORITHMS , *SUPPORT vector machines - Abstract
Cross-validation (CV) is a widely adopted approach for selecting the optimal model. However, the computation of empirical cross-validation error (CVE) has high complexity due to multiple times of learner training. In this paper, we develop a novel approximation theory of CVE and present an approximate approach to CV based on the Bouligand influence function (BIF) for kernel-based algorithms. We first represent the BIF and higher order BIFs in Taylor expansions, and approximate CV via the Taylor expansions. We then derive an upper bound of the discrepancy between the original and approximate CV. Furthermore, we provide a novel computing method to calculate the BIF for general distribution, and evaluate BIF criterion for sample distribution to approximate CV. The proposed approximate CV requires training on the full data set only once and is suitable for a wide variety of kernel-based algorithms. Experimental results demonstrate that the proposed approximate CV is sound and effective. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
26. A constrained approach to multiscale stochastic simulation of chemically reacting systems.
- Author
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Cotter, Simon L., Zygalakis, Konstantinos C., Kevrekidis, Ioannis G., and Erban, Radek
- Subjects
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CHEMICAL reactions , *MULTISCALE modeling , *SIMULATION methods & models , *APPROXIMATION theory , *LANGEVIN equations , *ALGORITHMS , *MOLECULAR dynamics , *FOKKER-Planck equation - Abstract
Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper, we introduce a multiscale methodology suitable to address this problem, assuming that the evolution of the slow species in the system is well approximated by a Langevin process. It is based on the conditional stochastic simulation algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the constrained multiscale algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Fokker-Planck equation approximation, we can in turn approximate average switching times in stochastic chemical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2011
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27. Comparison of Brownian dynamics algorithms with hydrodynamic interaction.
- Author
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Schmidt, Ricardo Rodríguez, Cifre, José G. Hernández, and de la Torre, José García
- Subjects
- *
COMPARATIVE studies , *WIENER processes , *ALGORITHMS , *HYDRODYNAMICS , *SIMULATION methods & models , *NANOPARTICLES , *DILUTION , *MATHEMATICAL analysis , *APPROXIMATION theory , *MATHEMATICAL decomposition - Abstract
The hydrodynamic interaction is an essential effect to consider in Brownian dynamics simulations of polymer and nanoparticle dilute solutions. Several mathematical approaches can be used to build Brownian dynamics algorithms with hydrodynamic interaction, the most common of them being the exact but time demanding Cholesky decomposition and the Chebyshev polynomial expansion. Recently, Geyer and Winter [J. Chem. Phys. 130, 1149051 (2009)] have proposed a new approximation to treat the hydrodynamic interaction that seems quite efficient and is increasingly used. So far, a systematic comparison among those approaches has not been clearly made. In this paper, several features and the efficiency of typical implementations of those approaches are evaluated by using bead-and-spring chain models. The different sensitivity to the bead overlap detected for the different implementations may be of interest to select the suitable algorithm for a given simulation. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
28. Michaelis-Menten speeds up tau-leaping under a wide range of conditions.
- Author
-
Wu, Sheng, Fu, Jin, Cao, Yang, and Petzold, Linda
- Subjects
- *
ENZYME kinetics , *CHEMICAL reduction , *STOCHASTIC processes , *SIMULATION methods & models , *ALGORITHMS , *APPROXIMATION theory - Abstract
This paper examines the benefits of Michaelis-Menten model reduction techniques in stochastic tau-leaping simulations. Results show that although the conditions for the validity of the reductions for tau-leaping remain the same as those for the stochastic simulation algorithm (SSA), the reductions result in a substantial speed-up for tau-leaping under a different range of conditions than they do for SSA. The reason of this discrepancy is that the time steps for SSA and for tau-leaping are determined by different properties of system dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
29. Look before you leap: A confidence-based method for selecting species criticality while avoiding negative populations in τ-leaping.
- Author
-
Yates, Christian A. and Burrage, Kevin
- Subjects
- *
CHEMICAL kinetics , *STOCHASTIC processes , *SIMULATION methods & models , *ALGORITHMS , *RANDOM variables , *APPROXIMATION theory , *DISTRIBUTION (Probability theory) , *NUMERICAL analysis - Abstract
The stochastic simulation algorithm was introduced by Gillespie and in a different form by Kurtz. There have been many attempts at accelerating the algorithm without deviating from the behavior of the simulated system. The crux of the explicit τ-leaping procedure is the use of Poisson random variables to approximate the number of occurrences of each type of reaction event during a carefully selected time period, τ. This method is acceptable providing the leap condition, that no propensity function changes 'significantly' during any time-step, is met. Using this method there is a possibility that species numbers can, artificially, become negative. Several recent papers have demonstrated methods that avoid this situation. One such method classifies, as critical, those reactions in danger of sending species populations negative. At most, one of these critical reactions is allowed to occur in the next time-step. We argue that the criticality of a reactant species and its dependent reaction channels should be related to the probability of the species number becoming negative. This way only reactions that, if fired, produce a high probability of driving a reactant population negative are labeled critical. The number of firings of more reaction channels can be approximated using Poisson random variables thus speeding up the simulation while maintaining the accuracy. In implementing this revised method of criticality selection we make use of the probability distribution from which the random variable describing the change in species number is drawn. We give several numerical examples to demonstrate the effectiveness of our new method. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
30. Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization.
- Author
-
Petrenko, Taras, Kossmann, Simone, and Neese, Frank
- Subjects
- *
DENSITY functionals , *APPROXIMATION theory , *ALGORITHMS , *HYDROCARBONS , *GEOMETRY , *BASIS sets (Quantum mechanics) , *EQUATIONS - Abstract
In this paper, we present the implementation of efficient approximations to time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation (TDA) for hybrid density functionals. For the calculation of the TDDFT/TDA excitation energies and analytical gradients, we combine the resolution of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced 'chain of spheres exchange' (COSX) algorithm for the calculation of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approximation leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 molecules with the configuration interaction singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equilibrium geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calculated excited state normal coordinate displacements is of the order of 0.01, and relative error in the calculated excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approximation are, thus, insignificant compared to the errors related to the approximate nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calculations on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ∼26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ∼27-29 for 30 processors). The solution of the Z-vector equations shows a speedup of ∼24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ∼15-25 for 30 processors), but their contribution to the total calculation time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calculation on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calculation of the TDDFT excited state properties in the frozen-core approximation. The algorithms described in this work are implemented into the ORCA electronic structure system. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
31. Charge density identification in ion channels.
- Author
-
Wolansky, G. and Taflia, A.
- Subjects
- *
ION channels , *ION-permeable membranes , *PERMEABILITY , *ALGORITHMS , *POISSON'S equation , *APPROXIMATION theory , *MATHEMATICAL functions - Abstract
Biological channels permeate ions through cell membranes. Ion channels carry a permanent charge that has a significant role in determining channel's permeation properties such as selectivity to certain ions, current amplitude, etc. In this paper we deal with the determination of the permanent charge from current-voltage curves. The ion channel current behavior is modelled by Poisson-Nernst-Planck (PNP) equations system. Previous works on the fixed charge density identification problem contain several ill-posed steps and linearization of the nonlinear PNP system. We suggest here several methods to make these algorithms more stable and accurate. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
32. Superlinearly converging dimer method for transition state search.
- Author
-
Kästner, Johannes and Sherwood, Paul
- Subjects
- *
GERMANS , *ALGORITHMS , *APPROXIMATION theory , *OLIGOMERS , *NUMERICAL solutions to equations - Abstract
Algorithmic improvements of the dimer method [G. Henkelman and H. Jónsson, J. Chem. Phys. 111, 7010 (1999)] are described in this paper. Using the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer for the dimer translation greatly improves the convergence compared to the previously used conjugate gradient algorithm. It also saves one energy and gradient calculation per dimer iteration. Extrapolation of the gradient during repeated dimer rotations reduces the computational cost to one gradient calculation per dimer rotation. The L-BFGS algorithm also improves convergence of the rotation. Thus, three to four energy and gradient evaluations are needed per iteration at the beginning of a transition state search, while only two are required close to convergence. Moreover, we apply the dimer method in internal coordinates to reduce coupling between the degrees of freedom. Weighting the coordinates can be used to apply chemical knowledge about the system and restrict the transition state search to only part of the system while minimizing the remainder. These improvements led to an efficient method for the location of transition states without the need to calculate the Hessian. Thus, it is especially useful in large systems with expensive gradient evaluations. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
33. Rationalisation and Applications of Dupin's Cyclide as Canal Surface.
- Author
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Bittnerov, Daniela and Bimova, Daniela
- Subjects
- *
CYCLIDE , *PARAMETERIZATION , *APPROXIMATION theory , *GEOMETRIC surfaces , *ALGORITHMS - Abstract
The paper presents one important case of a canal surface called Dupin's cyclide. Non-degenerate forms of Dupin's cyclide are attractive for example for geometric design applications. In the paper, there is derived one possibility of a rational parameterization of the cyclide as a special case of a canal surface. An algorithm for that parameterization is also shown. The parameterization offers a good method how to approximate especially implicit blend canal surfaces. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
34. The slow-scale stochastic simulation algorithm.
- Author
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Cao, Yang, Gillespie, Daniel T., and Petzold, Linda R.
- Subjects
- *
ALGORITHMS , *SIMULATION methods & models , *STOCHASTIC analysis , *PHYSICAL sciences , *APPROXIMATION theory , *CHEMICAL reactions - Abstract
Reactions in real chemical systems often take place on vastly different time scales, with “fast” reaction channels firing very much more frequently than “slow” ones. These firings will be interdependent if, as is usually the case, the fast and slow reactions involve some of the same species. An exact stochastic simulation of such a system will necessarily spend most of its time simulating the more numerous fast reaction events. This is a frustratingly inefficient allocation of computational effort when dynamical stiffness is present, since in that case a fast reaction event will be of much less importance to the system’s evolution than will a slow reaction event. For such situations, this paper develops a systematic approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events. Developing an effective strategy to implement this theory poses some challenges, but as is illustrated here for two simple systems, when those challenges can be overcome, very substantial increases in simulation speed can be realized. © 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
35. FPTAS for Minimizing the Earth Mover's Distance Under Rigid Transformations and Related Problems.
- Author
-
Ding, Hu and Xu, Jinhui
- Subjects
- *
PHASE transitions , *APPROXIMATION theory , *ALGORITHMS , *SEQUENCE alignment , *BOUND-bound transitions - Abstract
In this paper, we consider the problem (denoted as EMDRT) of minimizing the earth mover's distance between two sets of weighted points A and B in $${\mathbb {R}}^{d}$$ under rigid transformation. EMDRT is an important problem in both theory and applications and has received considerable attentions in recent years. Previous research on this problem has resulted in only constant factor approximations and it has been an open problem for a long time to achieve PTAS solution. In this paper, we present the first FPTAS algorithm for EMDRT. Our algorithm runs roughly in $$O((nm)^{d+2}(\log nm)^{2d})$$ time (which is close to a lower bound on any PTAS for this problem), where n and m are the sizes of A and B, respectively. Our result is based on several new techniques, such as the Sequential Orthogonal Decomposition and Optimum Guided Base, and can be extended to several related problems, such as the problem of earth mover's distance under similarity transformation and the alignment problem, to achieve FPTAS for each of them. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. Improved Approximation Algorithms for Projection Games.
- Author
-
Manurangsi, Pasin and Moshkovitz, Dana
- Subjects
- *
APPROXIMATION theory , *FUNCTIONAL analysis , *ALGORITHMS , *DESCRIPTIVE geometry , *MATHEMATICAL programming - Abstract
The projection games (aka Label Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label Cover. In this paper we design several approximation algorithms for projection games: (1) A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar et al. (Algorithmica 61(1):190-206, 2011). (2) A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. (3) A polynomial-time approximation scheme (PTAS) for projection games on planar graphs and a tight running time lower bound for such approximation schemes. The conference version of this paper had only the PTAS but not the running time lower bound. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. Search space-based multi-objective optimization evolutionary algorithm.
- Author
-
Medhane, Darshan Vishwasrao and Sangaiah, Arun Kumar
- Subjects
- *
ALGORITHMS , *MATHEMATICAL optimization , *REAL-time computing , *APPROXIMATION theory , *MUTATION (Phonetics) - Abstract
Evolutionary multi-objective optimization (EMO) algorithms are actively used for answering optimization problems with multiple contradictory objectives and scheming interpretable and precise real-time applications. A majority of existing EMO algorithms performs better on two or three objectives non-dominated problems; however, they meet complications in managing and maintaining a set of optimal solutions to multi-objective optimization problems. This paper proposes a search space-based multi-objective evolutionary algorithm (SSMOEA) for multi-objective optimization problems. To accomplish the potential of the search space-based method for solving multi-objective optimization problems and to reinforce the selection procedure toward the ideal direction while sustaining an extensive and uniform distribution of solutions is our key objective. To the best of our knowledge, this paper is the first attempt to propose a search space-based multi-objective evolutionary algorithm for multi-objective optimization. The experimental setup used showed that the proposed algorithm is good and competitive in comparison to the existing EMO algorithms from the viewpoint of finding a scattered and estimated solution set in multi-objective optimization problems. SSMOEA can achieve a good trade-off between search space convergence and search space diversity in the appropriate experimental setup. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
38. On the analytical approximation of joint aggregate queue-length distributions for traffic networks: A stationary finite capacity Markovian network approach.
- Author
-
Osorio, Carolina and Wang, Carter
- Subjects
- *
TRAFFIC engineering , *APPROXIMATION theory , *FINITE capacity scheduling , *MARKOV processes , *TRAFFIC signs & signals , *ALGORITHMS - Abstract
This paper is motivated by recent results in the design of signal plans for Manhattan that highlight the importance of providing signal control algorithms with an analytical description of between-link dependencies. This is particularly important for congested networks prone to the occurrence of spillbacks. This paper formulates a probabilistic network model that proposes an aggregate description of the queue-length, and then approximates the joint aggregate queue-length distribution of subnetworks. The goal is to model between-queue dependencies beyond first-order moments, yet to do so in a tractable manner such that these techniques can be used for optimization purposes. This paper models an urban road network as a finite space capacity Markovian queueing network. Exact evaluation of the stationary joint queue-length distribution of such a network with arbitrary size and topology can be obtained numerically. Nonetheless, the main challenge to such an approach remains the dimensionality of the network state space, which is exponential in the number of queues. This paper proposes to address the dimensionality issue by: 1) describing the state of the network aggregately, and 2) decomposing the network into overlapping subnetworks. We propose an analytical approximation of the stationary aggregate joint queue-length distribution of a subnetwork. The model consists of a system of nonlinear equations with a dimension that is linear, instead of exponential, in the number of queues and that is independent of the space capacity of the individual queues. The method is derived for tandem Markovian finite capacity queueing networks. The proposed model is computationally tractable and scalable, it can be efficiently used for the higher-order distributional analysis of large-scale networks. The model is validated versus simulation estimates and versus other decomposition methods. We then use it to address an urban traffic control problem. We show the added value of accounting for higher-order spatial between-queue dependency information in the control of congested urban networks. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
39. An Improved Ensemble of Random Vector Functional Link Networks Based on Particle Swarm Optimization with Double Optimization Strategy.
- Author
-
Ling, Qing-Hua, Song, Yu-Qing, Han, Fei, Yang, Dan, and Huang, De-Shuang
- Subjects
- *
PARTICLE swarm optimization , *MACHINE learning , *STOCHASTIC convergence , *LEAST squares , *APPROXIMATION theory - Abstract
For ensemble learning, how to select and combine the candidate classifiers are two key issues which influence the performance of the ensemble system dramatically. Random vector functional link networks (RVFL) without direct input-to-output links is one of suitable base-classifiers for ensemble systems because of its fast learning speed, simple structure and good generalization performance. In this paper, to obtain a more compact ensemble system with improved convergence performance, an improved ensemble of RVFL based on attractive and repulsive particle swarm optimization (ARPSO) with double optimization strategy is proposed. In the proposed method, ARPSO is applied to select and combine the candidate RVFL. As for using ARPSO to select the optimal base RVFL, ARPSO considers both the convergence accuracy on the validation data and the diversity of the candidate ensemble system to build the RVFL ensembles. In the process of combining RVFL, the ensemble weights corresponding to the base RVFL are initialized by the minimum norm least-square method and then further optimized by ARPSO. Finally, a few redundant RVFL is pruned, and thus the more compact ensemble of RVFL is obtained. Moreover, in this paper, theoretical analysis and justification on how to prune the base classifiers on classification problem is presented, and a simple and practically feasible strategy for pruning redundant base classifiers on both classification and regression problems is proposed. Since the double optimization is performed on the basis of the single optimization, the ensemble of RVFL built by the proposed method outperforms that built by some single optimization methods. Experiment results on function approximation and classification problems verify that the proposed method could improve its convergence accuracy as well as reduce the complexity of the ensemble system. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
40. The Solution of Fully Fuzzy Quadratic Equations Based on Restricted Variation.
- Author
-
Moazam, L. Gerami
- Subjects
- *
QUADRATIC equations , *FUZZY systems , *PROBLEM solving , *APPROXIMATION theory , *ALGORITHMS - Abstract
Firstly, in this paper, we apply the Fuzzy Restricted Variation Method to achieve an analytical and approximate unsymmetrical fuzzy solution for Fully Fuzzy Quadratic Equation. In this application, after finding the real root of 1-cut of ÃX² + BX + C = D, initial guess is always chosen with possible unknown parameters that leads to highly accurate solution. This technique is applying to solve mentioned equation in four cases via the sign of coefficients and variable that there is not zero in support of them and we solve the problems to find positive or negative solution. This method has been shown to solve effectively, easily and accurately a large class of nonlinear quadratic equations with approximations converging rapidly to accurate solution. In this paper we present the solutions in four cases with formulas, that can be used to write the algorithm for this technique. Finally to illustrate easy application and rich behavior of this method, several examples are given. [ABSTRACT FROM AUTHOR]
- Published
- 2016
41. A geometrical stability condition for compressed sensing.
- Author
-
Flinth, Axel
- Subjects
- *
GEOMETRY , *STABILITY theory , *COMPRESSED sensing , *SIGNAL processing , *ALGORITHMS , *APPROXIMATION theory - Abstract
During the last decade, the paradigm of compressed sensing has gained significant importance in the signal processing community. While the original idea was to utilize sparsity assumptions to design powerful recovery algorithms of vectors x ∈ R d , the concept has been extended to cover many other types of problems. A noteable example is low-rank matrix recovery. Many methods used for recovery rely on solving convex programs. A particularly nice trait of compressed sensing is its geometrical intuition. In recent papers, a classical optimality condition has been used together with tools from convex geometry and probability theory to prove beautiful results concerning the recovery of signals from Gaussian measurements. In this paper, we aim to formulate a geometrical condition for stability and robustness, i.e. for the recovery of approximately structured signals from noisy measurements. We will investigate the connection between the new condition with the notion of restricted singular values , classical stability and robustness conditions in compressed sensing, and also to important geometrical concepts from complexity theory. We will also prove the maybe somewhat surprising fact that for many convex programs, exact recovery of a signal x 0 immediately implies some stability and robustness when recovering signals close to x 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
42. On the Optimal Pre-Computation of Window τNAF for Koblitz Curves.
- Author
-
Trost, William R. and Xu, Guangwu
- Subjects
- *
ELLIPTIC curve cryptography , *ALGORITHMS , *APPROXIMATION theory , *COMPUTATIONAL complexity , *COEFFICIENTS (Statistics) - Abstract
Koblitz curves have been an important subject of consideration for both theoretical and practical interests. The window τ-adic algorithm of Solinas (window τNAF) is the most powerful method for computing point multiplication for Koblitz curves. Pre-computation plays an important role in improving the performance of point multiplication. In this paper, the concept of optimal pre-computation for window τNAF is formulated. In this setting, an optimal pre-computation has some mathematically natural and clean forms, and requires 2w−2−1 point additions and two evaluations of the Frobenius map τ, where w is the window width. One of the main results of this paper is to construct an optimal pre-computation scheme for each window width w from 4 to 15 (more than practical needs).These pre-computations can be easily incorporated into implementations of window τNAF. The ideas in the paper can also be used to construct other suitable pre-computations. This paper includes a discussion of coefficient sets for window τNAF and the divisibility by powers of τ through different approaches. Some issues of implementation are also discussed. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
- Full Text
- View/download PDF
43. On nonlinear analysis by the multipoint meshless FDM.
- Author
-
Jaworska, Irena and Orkisz, Janusz
- Subjects
- *
NONLINEAR analysis , *APPROXIMATION theory , *ALGORITHMS , *DIFFERENTIAL equations , *FINITE element method , *DERIVATIVES (Mathematics) - Abstract
The main objective of this paper is to present an attempt of an application of the recently developed higher order multipoint meshless FDM in the analysis of nonlinear problems. The multipoint approach provides a higher order approximation and improves the precision of the solution. In addition to improved solution quality, the essential feature of the multipoint approach is its potentially wide ranging applicability. This is possible, because in both the multipoint and standard meshless FDM, the difference formulas are generated at once for the full set of derivatives. Using them, we may easily compose any required FD operator. It is worth mentioning that all derivative operators depend on the domain discretization rather than on the specific problem being analysed. Therefore, the solution of a wide class of problems including nonlinear ones, may be obtained with this method. The numerical algorithm of the multipoint method for nonlinear analysis is presented in this paper. Results of selected engineering benchmark problems – deflection of the ideal membrane and analysis of large deflection of plates using the von Karman theory – are considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
44. A novel approach for efficient updating approximations in dynamic ordered information systems.
- Author
-
Wang, Shu, Li, Tianrui, Luo, Chuan, Hu, Jie, Fujita, Hamido, and Huang, Tianqiang
- Subjects
- *
INFORMATION storage & retrieval systems , *APPROXIMATION theory , *ROUGH sets , *DATA mining , *ALGORITHMS - Abstract
Dynamic data in real-time application are typically updating in a multi-dimensional manner. In this paper, we introduce a novel approach based on Dominance-based Rough Set Approach (DRSA) to efficiently deal with the multi-dimensional variations of attribute set and attribute values in dynamic Ordered Information Systems (OIS). We improve the original notion of the P-generalized decision domains to make the feature value matrix be dominance symmetrical, and propose an efficient strategy based on the improved notion to obtain the dominance feature matrix. Then, we employ the dominance-feature-matrix-based incremental strategy to avoid repeated comparisons between original attributes, so that to efficiently update rough approximations of DRSA with the simultaneously increased attribute set and varied attribute values. In our approach, the steps based on these two combined strategies can work altogether or separately, not only efficiently dealing with the simultaneously increased attribute set and varied attribute values, but also efficiently dealing with the individually increased attribute set or varied attribute values in dynamic OIS. Efficient algorithm based on the updating strategies is designed and multiple groups of experiments are conducted. Experimental results on different real-world data sets show that the proposed algorithm is much faster than other algorithms for dealing with the multi-dimensional or the single-dimensional variations of attribute set and attribute values. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
45. A New Search Direction of DFP-CG Method for Solving Unconstrained Optimization Problems.
- Author
-
Wan Osman, Wan Farah Hanan, Hery Ibrahim, Mohd Asrul, and Mamat, Mustafa
- Subjects
- *
MATHEMATICAL optimization , *CONJUGATE gradient methods , *APPROXIMATION theory , *ALGORITHMS , *MATHEMATICAL analysis - Abstract
The conjugate gradient (CG) and Davidon, Fletcher and Powell (DFP) method are both well known solvers for solving unconstrained optimization problems. In this paper, we proposed a new hybrid DFP-CG method and compared with the ordinary DFP method in terms of number of iteration and CPU times. Numerical results show that the new algorithm is more efficient compared to the ordinary DFP method and proven to posses both sufficient descent and global convergence properties. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
46. High-precision numerical integration of equations in dynamics.
- Author
-
Alesova, I. M., Babadzanjanz, L. K., Pototskaya, I. Yu., Pupysheva, Yu. Yu., Saakyan, A. T., Kustova, Elena, Leonov, Gennady, Morosov, Nikita, Yushkov, Mikhail, and Mekhonoshina, Mariia
- Subjects
- *
EQUATIONS , *DYNAMICS , *ROBOTICS , *POLYNOMIALS , *APPROXIMATION theory , *ALGORITHMS - Abstract
An important requirement for the process of solving differential equations in Dynamics, such as the equations of the motion of celestial bodies and, in particular, the motion of cosmic robotic systems is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In this connection, we note that it is very advantageous to reduce the given equations of Dynamics to systems with polynomial (in unknowns) right-hand sides. This allows us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. In the paper, these questions are discussed and appropriate algorithms are considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
47. Deep stacked stochastic configuration networks for lifelong learning of non-stationary data streams.
- Author
-
Pratama, Mahardhika and Wang, Dianhui
- Subjects
- *
DEEP learning , *STOCHASTIC processes , *INFORMATION storage & retrieval systems , *ALGORITHMS , *APPROXIMATION theory - Abstract
The concept of SCN offers a fast framework with universal approximation guarantee for lifelong learning of non-stationary data streams. Its adaptive scope selection property enables for proper random generation of hidden unit parameters advancing conventional randomized approaches constrained with a fixed scope of random parameters. This paper proposes deep stacked stochastic configuration network (DSSCN) for continual learning of non-stationary data streams which contributes two major aspects: 1) DSSCN features a self-constructing methodology of deep stacked network structure where hidden unit and hidden layer are extracted automatically from continuously generated data streams; 2) the concept of SCN is developed to randomly assign inverse covariance matrix of multivariate Gaussian function in the hidden node addition step bypassing its computationally prohibitive tuning phase. Numerical evaluation and comparison with prominent data stream algorithms under two procedures: periodic hold-out and prequential test-then-train processes demonstrate the advantage of proposed methodology. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
48. On Randomized Algorithms for Numerical Solution of Applied Fredholm Integral Equations of the Second Kind.
- Author
-
Voytishek, Anton V. and Shipilov, Nikolay M.
- Subjects
- *
ALGORITHMS , *APPROXIMATION theory , *FREDHOLM equations , *MATHEMATICAL models , *BANACH spaces - Abstract
In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
49. Integrating Coflow and Circuit Scheduling for Optical Networks.
- Author
-
Wang, Haibo, Yu, Xiwen, Xu, Hongli, Fan, Jingyuan, Qiao, Chunming, and Huang, Liusheng
- Subjects
- *
OPTICAL switches , *OPTICAL fiber networks , *PACKET switching (Data transmission) , *ALGORITHMS , *APPROXIMATION theory - Abstract
There are more and more structured traffic flows (a.k.a coflow) in today's data center networks. Completing a coflow is extremely important for various applications, e.g., MapReduce. To reduce the coflow completion time or CCT, one may increase the link capacity by applying advanced optical circuit switches in data center networks. Due to special features of optical circuit switches, both traffic scheduling and circuit scheduling will influence the CCT. However, previous solutions have some significant limitations: they consider either coflow scheduling, or circuit scheduling for only one optical circuit switch, which are both insufficient. In this paper, we study the integrated coflow and circuit scheduling (GCCS) problem with the objective to minimize the CCT, and prove its NP-hardness. We present an integrated algorithm which includes two steps, coflow scheduling and circuit scheduling, respectively. We also analyze that the proposed algorithm can achieve the approximation ratio $O(h)$O(h) in most practical situations, where $h$h is the maximum number of ports among all lightpaths. Through large-scale simulations, we demonstrate that the integrated solution can significantly reduce the CCT by about 43-70 percent compared with the state-of-the-art coflow scheduler for optical networks. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
50. A shift and invert reorthogonalization Arnoldi algorithm for solving the chemical master equation.
- Author
-
Liu, Yong and Gu, Chuanqing
- Subjects
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TOEPLITZ matrices , *APPROXIMATION theory , *ALGORITHMS , *FINITE model theory , *EXPERIMENTS - Abstract
Abstract The shift and invert Arnoldi (SIA) method is a numerical algorithm for approximating the product of Toeplitz matrix exponential with a vector. In this paper, we extend the SIA method to chemical master equation (CME) and propose a SIA algorithm based on the strategy of reorthogonalization (SIRA). We establish a theoretical error of the resulting approximation of SIRA algorithm. Numerical experiments show that the SIRA algorithm is more efficient than the Krylov FSP algorithm in terms of finite models, and the error estimate can be used to determine whether this result obtained by SIRA algorithm is acceptable or not. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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