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ONE linear ANALYTIC approximation FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

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Acta Mathematica Scientia 2010年第4期30卷 1073-1085页

作者： Svetlana Jankovic Dejan Ilic University of Nis Department of Mathematics

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.

关键词： Stochastic integrodifferential equation linear approximation approximate solution L^p-convergence convergence with probability one

A Convex approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design

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Advances in Applied Mathematics and Mechanics 2023年第6期15卷 1540-1561页

作者： Mengyue Wu Jianhua Yuan Jianxin Zhang School of Science Beijing University of Post and TelecommunicationsBeijing 100876China

Based on a subspace method and a linear approximation method,a convex algorithm is designed to solve a kind of non-convex PDE constrained fractional optimization problem in this paper.This PDE constrained problem is an infinitedimensional Hermitian eigenvalue optimization problem with non-convex and low regularity.Usually,such a continuous optimization problem can be transformed into a large-scale discrete optimization problem by using the finite element methods.We use a subspace technique to reduce the scale of discrete problem,which is really effective to deal with the large-scale problem.To overcome the difficulties caused by the low regularity and non-convexity,we creatively introduce several new artificial variables to transform the non-convex problem into a convex linear semidefinite programming.By introducing linear approximation vectors,this linear semidefinite programming can be approximated by a very simple linear relaxation problem.Moreover,we theoretically prove this approximation.Our proposed algorithm is used to optimize the photonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals as an application.The results of numerical examples show the effectiveness of our proposed algorithm,while they also provide several optimized photonic crystal structures with a desired wide-band-gap.In addition,our proposed algorithm provides a technical way for solving a kind of PDE constrained fractional optimization problems with a generalized eigenvalue constraint.

关键词： PDE constrained optimization fractional programming linear approximation finite element method photonic band gap

Formulations and approximations of Branch Flow Model for General Power Networks

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Journal of Modern Power Systems and Clean Energy 2022年第5期10卷 1110-1126页

作者： Zhao Yuan Electrical Power Systems Laboratory(EPS-Lab) Department of Electrical and Computer EngineeringUniversity of IcelandReykjavikIceland IEEE

The formulations and approximations of the branch flow model for general(radial and mesh) power networks(General-BranchFlow) are given in this paper. Using different sets of the power flow equations, six formats of the exact General-BranchFlow model are listed. The six formats are mathematically equivalent with each other. linear approximation and second-order cone programming(SOCP) are then used to derive the six formats of the convex General-BranchFlow model. The branch ampacity constraints considering the shunt conductance and capacitance of the transmission line Π-model are derived. The key foundation of deriving the ampacity constraints is the correct interpretation of the physical meaning of the transmission line Π-model. An exact linear expression of the ampacity constraints of the power loss variable is derived. The applications of the General-BranchFlow model in deriving twelve formats of the exact optimal power flow(OPF) model and twelve formats of the approximate OPF model are formulated and analyzed. Using the Julia programming language, the extensive numerical investigations of all formats of the OPF models show the accuracy and computational efficiency of the General-BranchFlow model. A penalty function based approximation gap reduction method is finally proposed and numerically validated to improve the AC-feasibility of the approximate General-BranchFlow model.

关键词： Branch flow linear approximation second-order cone programming ampacity constraint optimal power flow radial network mesh network

Study on High-Speed Magnitude approximation for Complex Vectors

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Journal of Systems Engineering and Electronics 2003年第1期14卷 81-85页

作者：陈建春杨万海许少英 School of Electronic Engineering Xidian University Xi'an 710071 P. R. China

High-speed magnitude approximation algorithms for complex vectors are discussed intensively. The performance and the convergence speed of these approximation algorithms are analyzed. For the polygon fitting algorithms, the approximation formula under the least mean square error criterion is derived. For the iterative algorithms, a modified CORDIC (coordinate rotation digital computer) algorithm is developed. This modified CORDIC algorithm is proved to be with a maximum relative error about one half that of the original CORDIC algorithm. Finally, the effects of the finite register length on these algorithms are also concerned, which shows that 9 to 12-bit coefficients are sufficient for practical applications.

关键词： Modulus of complex number linear approximation Least mean square error criterion CORDIC.

Decentralized Robust Energy Management of Multi-area Integrated Electricity-gas Systems

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Journal of Modern Power Systems and Clean Energy 2021年第6期9卷 1478-1489页

作者： Nan Jia Cheng Wang Wei Wei Tianshu Bi IEEE State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power University Department of Electrical Engineering Tsinghua University

This paper proposes a fast and decentralized solution methodology for the robust operation of multi-area integrated electricity-gas systems(M-IEGSs).A deterministic reformulation is obtained for the two-stage robust model by applying the linear decision rule based electrical reserve utilization scheme as well as regulating the distributed gas storages.Two linear approximations are developed for the nonconvex Weymouth equation in the gas network to determine the gas flow directions.The penalty convex-concave procedure(P-CCP)is then adopted to refine a feasible local optimum for the nonconvex model with an acceleration strategy.The decentralized decision-making is enabled by the alternating direction multipliers method(ADMM).The convergence as well as computation performance of the overall solution procedure can be guaranteed as only convex optimizations are solved.Simulation results validate the effectiveness of the proposed methods as well as the benefits of the proposed convex programing based solution procedure.

关键词： Convex optimization decentralized robust optimization energy management linear approximation integrated electricity-gas system

Piecewise linear detection for direct superposition modulation

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Digital Communications and Networks 2018年第2期4卷 98-105页

作者： Martin Damrath Peter Adam Hoeher Gilbert J.M. Forkel Institute of Electrical and Information Engineering Faculty of Engineering University of Kiel Kaiserstr. 2 24143 Kiel Germany

Considering high-order digital modulation schemes, the bottleneck in consumer products is the detector rather than the modulator. The complexity of the optimal a posteriori probability (APP) detector increases exponentially with respect to the number of modulated bits per data symbol. Thus, it is necessary to develop low-complexity detection algorithms with an APP-like performance, especially when performing iterative detection, for example in conjunction with bit interleaved coded modulation. We show that a special case of superposition modulation, dubbed Direct Superposition Modulation (DSM), is particularly suitable for complexity reduction at the receiver side. As opposed to square QAM, DSM achieves capacity without active signal shaping. The main contribution is a low-cost detection algorithm for DSM, which enables iterative detection by taking a priori information into account. This algorithm exploits the approximate piecewise linear behavior of the soft outputs of an APP detector over the entire range of detector input values. A theoretical analysis and simulation results demonstrate that at least max-log APP performance can be reached, while the complexity is significantly reduced compared to classical APP detection.

关键词： Digital modulation Demodulation Detection algorithms linear approximation

Symbol detection based on Voronoi surfaces with emphasis on superposition modulation

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Digital Communications and Networks 2017年第3期3卷 141-149页

作者： Martin Damrath Peter Adam Hoeher Gilbert J.M. Forkel Institute of Electrical and Information Engineering Faculty of Engineering University of Kiel Kaiserstr. 2 24143 Kiel Germany

A challenging task when applying high-order digital modulation schemes is the complexity of the detector. Particularly, the complexity of the optimal a posteriori probability (APP) detector increases exponentially with respect to the number of bits per data symbol. This statement is also true for the Max-Log-APP detector, which is a common simplification of the APP detector. Thus it is important to design new detection algorithms which combine a sufficient performance with low complexity. In this contribution, a detection algorithm for two-dimensional digital modulation schemes which cannot be split-up into real and imaginary parts (like phase shift keying and phase-shifted superposition modulation (PSM)) is proposed with emphasis on PSM with equal power allocation. This algorithm exploits the relationship between Max-Log-APP detection and a Voronoi diagram to determine planar surfaces of the soft outputs over the entire range of detector input values. As opposed to state-of-the-art detectors based on Voronoi surfaces, a priori information is taken into account, enabling iterative processing. Since the algorithm achieves Max-Log-APP performance, even in the presence of a priori information, this implies a great potential for complexity reduction compared to the classical APP detection.

关键词： Digital modulation Demodulation Detection algorithms linear approximation

Optimal Perturbations Triggering Weather Regime Transitions:Onset of Blocking and Strong Zonal Flow

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Advances in Atmospheric Sciences 2011年第1期28卷 59-68页

作者：姜智娜穆穆王东海 State Key Laboratory of Severe Weather Chinese Academy of Meteorological Sciences Key Laboratory of Ocean Circulation and Wave Institute of OceanologyChinese Academy of Sciences State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of Sciences

In this paper, the approach proposed by Mu and Jiang （2008） to obtain the optimal perturbations for triggering blocking （BL） onset is generalized to seek the optimal perturbations triggering onset of the strong zonal flow （SZF） regime. The BL and SZF regimes are characterized by the same dipole-like anomaly pattern superposed on the climatological flow, but with opposite sign. The results show that this method is also superior at finding the initial optimal perturbations triggering onset of the SZF regime, especially in the medium range. Furthermore, by comparing the two kinds of conditional nonlinear optimal perturbations （CNOPs） trig-gering onset of BL and SZF regimes, we find that in the linear approximation, there is symmetry in the sensitivities for BL and SZF onset, and the perturbations that optimally trigger onset of BL and SZF regimes at times when linear approximation is valid are also characterized by the same spatial pattern but with opposite sign. Whereas when the optimization time is extended to 6 days, the two kinds of CNOPs lose their out-of-phase behavior. The nonlinearity results in an asymmetry between the sensitivity for BL and SZF onset. Additionally, we find that the optimal perturbations have one common property, which is that the second baroclinic mode contributes more to the initial perturbations while the barotropic mode dominates the final structures.

关键词： linear approximation symmetry nonlinearity asymmetry

Viewing the noise propagation mechanism in a unidirectional transition cascade from the perspective of stability

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Chinese Physics B 2021年第11期30卷 676-684页

作者：裴启明周彬倩周祎凡 Charles Omotomide Apata 蒋龙 School of Physics and Optoelectronic Engineering Yangtze UniversityJingzhou 434023China

Noise and noise propagation are inevitable and play a constructive role in various biological processes.The stability of cell homeostasis is also a critical issue.In the unidirectional transition cascade of colon cells,stem cells(SCs)are the source.They differentiate into transit-amplifying cells(TACs),and TACs differentiate into fully differentiated cells(FDCs).Two differentiation processes are irreversible.The stability factor is introduced so that the noise propagation mechanism from the perspective of stability is studied according to the noise propagation formulas.It is found that the value of the stability factor corresponding to the minimum noise in FDCs may be the best choice to enable colon cells to maintain high stability and low noise of the cascade.Moreover,for the source cell,the total noise only includes intrinsic noise;for the downstream cell with self-proliferation capability,the total noise mainly depends on its intrinsic noise and transmitted noise from upstream cells,and its intrinsic noise is dominant.For the downstream cell without self-proliferation capability,the total noise is mainly determined by transmitted noises from upstream cells,and there is a minimum value.This work provides a new approach for studying the mechanism of noise propagation while considering the stability of cell homeostasis in biological systems.

关键词： noise noise propagation stability linear approximation