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Generalized Popoviciu Expansions for Bernstein Polynomials of a Rational Module

Journal of Mathematical Sciences (United States) 2022年第4期263卷 532-578页

作者： Tikhonov, I.V. Sherstyukov, V.B. Tsvetkovich, D.G. M. V. Lomonosov Moscow State University Moscow Russian Federation National Research Nuclear University MEPhI Moscow Russian Federation Moscow State Pedagogical University Moscow Russian Federation

We study Bernstein polynomials for simple nonsmooth rational module functions. We show that these polynomials can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called “generalized Popoviciu expansions.” To write generalized expansions, we develop a special formalism based on combinatorial calculations. Based on the formulas obtained, we propose a complete description of the character of convergence of the Bernstein polynomials studied in the complex plane. We also discuss the relationship of generalized Popoviciu expansions with the distribution of zeros of Bernstein polynomials. In the final part of the paper, we present additional new relations for Bernstein polynomials of the rational module function. © 2022, Springer Science+Business Media, LLC, part of Springer Nature.

关键词： 30C15 30E10 41A10 41A25 Bernstein polynomial distribution of zeros of a polynomial domain of convergence generalized Popoviciu expansion Kantorovich lemniscate piecewise linear function rational module

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Two Methods 'for Synthesis of State and Disturbance Observers for an Unmanned Aerial Vehicle

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AUTOMATION AND REMOTE CONTROL 2021年第8期82卷 1426-1441页

作者： Kokunko, Yu G. Krasnov, D., V Utkin, A., V Russian Acad Sci Trapeznikov Inst Control Sci Moscow 117997 Russia

As part of the synthesis of a tracking system for an unmanned aerial vehicle (UAV) exposed to uncontrolled external disturbances under the conditions of incomplete measurements of the state vector, we develop procedures for synthesizing novel low-order state and disturbance observers that do not require constructing dynamic models of external inputs. The observation subsystem includes two state observers. One of them estimates the velocities based on measurements of the UAV center of mass coordinates. The other observer uses the measurements of tracking errors to give estimates of mixed variables (state functions, external inputs, and their derivatives) from which the feedback is directly formed. It is noted that implementing the developed algorithms, which do not involve readjustment when external inputs change, will increase the UAV control system functionality and its reliability in the event of failure of measuring devices. The efficiency of our approach to tracking system synthesis is confirmed by numerical simulation results. We present the results of comparative analysis of closed-loop systems with static (under the assumption that all internal and external variables are measured) and dynamic feedback that uses two approaches to solving the problem of estimation under external disturbances-high-gain observers and observers with piecewise linear bounded corrective inputs. It is shown that, despite the simpler setup, it is expedient to use observers of the second type in linear feedback systems, while high-gain observers will be in demand in systems with a priori bounded controls.

关键词： unmanned aerial vehicle tracking block approach state and disturbance observer high gain piecewise linear function

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Efficient Hardware Implementation of DNN-Based Speech Enhancement Algorithm With Precise Sigmoid Activation function

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2021年第11期68卷 3461-3465页

作者： Chiluveru, Samba Raju Gyanendra Chunarkar, Snehit Tripathy, Manoj Kaushik, Brajesh Kumar Indian Inst Technol Roorkee Instrumentat & Signal Proc Grp Dept Elect Engn Roorkee 247667 Uttar Pradesh India Indian Inst Technol Roorkee Microelect & VLSI Grp Dept Elect & Commun Engn Roorkee 247667 Uttar Pradesh India

This brief presents the hardware implementation of deep neural network-based speech enhancement algorithm (DNN-SEA) with a precise sigmoid activation function. Further, an adaptive step-size-based slope and intercept method (AS-SIM) has been developed to approximate the sigmoid function that uses the maximum allowable error (epsilon) as an input parameter. The performance of the DNN-SEA is measured in terms of speech quality and intelligibility using a perceptual evaluation of speech quality (PESQ) and short-time objective intelligibility (STOI) parameters, respectively. Hardware implementation is performed by Synopsys Design Compiler with a TSMC 90-nm library. The performance comdparison has been conducted in terms of area, gate count, delay, and power consumption. Results confirm that the proposed AS-SIM approximation improves performance in terms of PESQ and STOI value and significantly reduces the area requirement and power consumption.

关键词： Approximation algorithms Hardware Speech enhancement Signal processing algorithms Noise measurement Neurons Training Speech enhancement sigmoid activation piecewise linear function deep neural network perceptual evaluation of speech quality short-time objective intelligibility

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Fully piecewise linear Vector Optimization Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021年第2期190卷 461-490页

作者： Zheng, Xi Yin Yang, Xiaoqi Yunnan Univ Dept Math Kunming 650091 Yunnan Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization problem with the objective and constraint functions being piecewise linear. To solve this problem, we divide it into some linear subproblems and structure a dimensional reduction method. Under some mild assumptions, we prove that its Pareto (resp., weak Pareto) solution set is the union of finitely many generalized polyhedra (resp., polyhedra), each of which is contained in a Pareto (resp., weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell's classical results on linear vector optimization problems in the framework of finite-dimensional spaces.

关键词： Polyhedron piecewise linear function Pareto solution Weak Pareto solution

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Radius of Robust Global Error Bound for piecewise linear Inequality Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021年第1期191卷 68-82页

作者： Thai Doan Chuong Ton Duc Thang Univ Optimizat & Applicat Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam

In this paper, we consider a subclass of uncertain convex inequality systems, called a class of uncertain piecewise linear systems, where the involved functions are piecewise linear. We define a concept of radius of robust global error bound for the piecewise linear inequality system under polytope uncertainty and provide formulas for calculating the radius of robust global error bound. This is achieved by employing a dual characterization for the existence of a robust global error bound of the uncertain piecewise linear system with polytope uncertainty.

关键词： Uncertain linear inequality Error bound system piecewise linear function Bounded data uncertainty

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A Computational Approach to Optimal Control Problems with Almost Smooth Controls

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2023年第1期40卷

作者： Zhang, Ying Zhang, Zhao Xu, Yingtao Zhejiang Normal Univ Dept Math Jinhua 321004 Zhejiang Peoples R China Zhejiang Normal Univ Xingzhi Coll Jinhua 321004 Zhejiang Peoples R China

In this paper, we consider a class of optimal control problems involving continuous control and state inequality constraints where the control is almost smooth. We first employ the control parametrization technique via approximating the control signal by a piecewise linear function. Then, we develop a time scaling transformation procedure for transforming the approximate problem into an equivalent problem that can be solved readily using conventional methods. On this basis, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. The gradient formulas and convergent properties ensure that the transformed unconstrained optimal parameter selection problems can be solved by existing optimization algorithms or software packages. Finally, an example is solved showing the effectiveness and applicability of the approach proposed.

关键词： Almost smooth control control parametrization piecewise linear function time scaling transformation exact penalty function

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piecewise linear programming approach to solve multi-objective matrix games with I-fuzzy goals

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Journal of Control and Decision 2021年第1期8卷 1-13页

作者： Sandeep Kumar Department of Mathematics Ch.Charan Singh UniversityMeerut 250004Uttar PradeshIndia

The intuitionistic fuzzy set(I-fuzzy set)plays an effective role in game theory when players face‘neither this nor that’situation to set their *** study presents a maxmin–minmax solution to multi-objective two person zero-sum matrix games with I-fuzzy *** this article,a class of piecewise linear membership and non-membership functions for I-fuzzy goals is *** functions are more effective in real games because marginal rate of increase(decrease)of such membership functions(non-membership functions)is different in different intervals of tolerance ***,one numerical example is given to examine the effectiveness and advantages of the proposed results.

关键词： Intuitionistic fuzzy set multi-objective matrix game I-fuzzy goals maxmin-minmax solution piecewise linear function

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Analytical solution to Windkessel models using piecewise linear aortic flow waveform

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PHYSIOLOGICAL MEASUREMENT 2023年第6期44卷 06NT01-06NT01页

作者： Gnudi, Gianni Alma Mater Studiorum Univ Bologna Dept Elect Elect & Informat Engn Guglielmo Marconi Cesena Italy

Objective. Deriving time-domain analytical solutions to two- three- and four-element Windkessel models, which are commonly used in teaching and research to analyse the behaviour of the arterial pressure-flow relationship. Approach. The governing (first-order, non-homogeneous, linear) differential equations are solved analytically, based on a piecewise linear function that can accurately approximate typical aortic flow waveforms. Main results. Closed-form expressions for arterial pressure are obtained both in transient conditions and in steady-state periodic regime. Significance. In most cases Windkessel models are studied in the frequency domain and when studied in the time domain, numerical methods are used. The main advantage of the proposed expressions is that they are an explicit, exact, and easily understood mathematical description of the model behaviour. Moreover, they avoid the use of Fourier analysis or numerical solvers to integrate the differential equations.

关键词： Windkessel models aortic flow piecewise linear function analytical solution

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The arithmetic of stepwise offer curves

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ENERGY 2022年 239卷

作者： Mestre, Guillermo Sanchez-Ubeda, Eugenio F. San Roque, Antonio Munoz Alonso, Estrella Univ Pontificia Comillas Escuela Tecn Super Ingn ICAI Inst Invest Tecnol Madrid Spain Univ Pontificia Comillas Escuela Tecn Super Ingn ICAI Madrid Spain

In liberalized electricity markets, aggregated stepwise supply and demand curves are at the core of many relevant processes. Efficient and meaningful representations of the offer curves is an essential procedure for agents participating in those markets. However, there is not a formal framework that allows operating with those offer curves using basic arithmetic operations. In this paper we first formalize the concept of stepwise offer curve by explicitly defining the standard True Offer Curve (TOC). To overcome the inherit difficulties of this non-continuous TOC, we propose the Encoded Offer Curve (EOC), a continuous piecewise version that approximates the steps of the TOC with high accuracy. We present fast and simple specialized algorithms to obtain both TOC and EOC models, as well as a formal framework to deal with elementary mathematical operations involving TOCs and EOCs. The proposed framework has been tested in the Italian electricity market, computing the residual demand curves of the producers in a particular Market Zone;and in the Iberian electricity market, quantifying the differences in the bidding behavior of market agents in different stages of the COVID-19 pandemic. (C) 2021 Elsevier Ltd. All rights reserved.

关键词： Electricity markets piecewise linear function Supply function Demand function Residual demand curve

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Consistent and Flexible Selectivity Estimation for High-Dimensional Data 21

Consistent and Flexible Selectivity Estimation for High-Dime...

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ACM SIGMOD International Conference on Management of Data (SIGMOD)

作者： Wang, Yaoshu Xiao, Chuan Qin, Jianbin Mao, Rui Onizuka, Makoto Wang, Wei Zhang, Rui Ishikawa, Yoshiharu Shenzhen Univ Shenzhen Inst Comp Sci Shenzhen Peoples R China Osaka Univ Suita Osaka Japan Nagoya Univ Nagoya Aichi Japan Dongguan Univ Technol Dongguan Peoples R China Univ New South Wales Kensington NSW Australia

ISBN: (纸本)9781450383431

Selectivity estimation aims at estimating the number of database objects that satisfy a selection criterion. Answering this problem accurately and efficiently is essential to many applications, such as density estimation, outlier detection, query optimization, and data integration. The estimation problem is especially challenging for large-scale high-dimensional data due to the curse of dimensionality, the large variance of selectivity across different queries, and the need to make the estimator consistent (i.e., the selectivity is non-decreasing in the threshold). We propose a new deep learning-based model that learns a query-dependent piecewise linear function as selectivity estimator, which is flexible to fit the selectivity curve of any distance function and query object, while guaranteeing that the output is non-decreasing in the threshold. To improve the accuracy for large datasets, we propose to partition the dataset into multiple disjoint subsets and build a local model on each of them. We perform experiments on real datasets and show that the proposed model consistently outperforms state-of-the-art models in accuracy in an efficient way and is useful for real applications.

关键词： selectivity estimation high-dimensional data piecewise linear function deep neural network

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