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probability functions Generated by Set-Valued Mappings: A Study of First Order Information

SET-VALUED AND VARIATIONAL ANALYSIS 2024年第1期32卷 6-6页

作者： van Ackooij, Wim Perez-Aros, Pedro Soto, Claudia EDF Lab Paris Saclay OSIRIS 7 Blvd Gaspard Monge F-91120 Palaiseau France Univ Chile Dept Ingn Matemat Beauchef 851 Santiago Chile Univ Chile Ctr Modelamiento Matematico UMI2807 CNRS Beauchef 851 Santiago Chile Univ OHiggins Inst Ciencias Ingn Ave Libertador Bernardo OHiggins 611 Rancagua Chile

probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of that given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian. We also provide subgradient formul AE. The resulting formul AE are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application.

关键词： probability functions Spherical radial-like decomposition Set-valued mapping Lipschitz-like continuity Generalized differentiation

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Reconstruction of random two-phase polycrystalline solids using low-order probability functions and evaluation of mechanical behavior

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COMPUTATIONAL MATERIALS SCIENCE 2010年第4期49卷 705-719页

作者： Chung, Sang-Yeop Han, Tong-Seok Yonsei Univ Sch Civil & Environm Engn Seoul 120749 South Korea

The phase distribution in a multi-phase material can affect its material properties and mechanical behaviors significantly. Because multi-phase materials even with the same volume fraction can have different phase distributions, a method to describe the phase distribution is needed. For this purpose, contiguity and low-order probability functions are investigated for representing the phase distributions of microstructures. The virtual samples for evaluating the mechanical properties of the two-phase materials with random phase distribution are reconstructed using the low-order probability functions (two-point correlation and lineal-path functions), and the mechanical behaviors are evaluated using the finite element method based on the restricted slip system. Macro-scale mechanical response (stress-strain curve) and lattice strains for sets of crystal families, as well as characteristics of the probability functions, are almost the same between the original and reconstructed virtual samples. It is confirmed that the virtual microstructures of random isotropic and anisotropic phase distributions reconstructed from the low-order probability functions exhibit high potential for investigating the mechanical behavior such as lattice stress and strains through simulations, which can be used to supplement diffraction experiments. (C) 2010 Elsevier B.V.. All rights reserved.

关键词： Phase clustering Multi-phase Polycrystalline solids Contiguity probability functions Reconstruction Finite element method

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Effect of metastable atom reactions on the electron energy probability functions in afterglows

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PLASMA SOURCES SCIENCE & TECHNOLOGY 1998年第4期7卷 512-523页

作者： Overzet, LJ Kleber, J Univ Texas Plasma Applicat Lab Richardson TX 75083 USA

Time resolved electron densities, temperatures and energy probability functions (EEPFs) of modulated-power glow discharges through argon and helium in the Gaseous Electronics Conference reference reactor have been measured using an RF compensated Langmuir probe and a microwave interferometer. RF power was capacitively coupled to the glow and square wave amplitude modulated with a 50% duty cycle and 100% modulation depth. We found that a metastable-metastable ionization reaction can produce energetic electrons during the afterglow. This reaction can also cause the electron density to increase during the afterglow despite the RF excitation being off. The electron density as a function of time can be modelled and the result is an estimated metastable atom density as a function of time. The hot electrons in the EEPF can also be modelled, but the modelling result does not fit the experimental EEPF until the smoothing of the EEPF caused by the experimental method is taken into account. This smoothing of the EEPF can be accounted for using the Druyvesteyn method formula and indicates that accurate measurements of the EEPF in very low electron temperature plasmas can become difficult. In effect, one should have some knowledge of the shape of the EEPF before the experiment in order to obtain an accurate measurement. The electron density and EEPF results become self-consistent once the smoothing is taken into account. By moving the Langmuir probe along the diameter of the chamber it was determined that the electron density decreases more quickly between the electrodes than outside the electrode edges. This causes the plasma density profile in argon to becomes doughnut shaped during the afterglow and causes the glow to re-ignite from the edges into the centre. The electron temperature at re-ignition in helium discharges can become larger than that at steady state in the active glow. It quickly relaxes to the steady state value. This last effect is not nearly as pronounced in a

关键词： luminescence radiofrequency TIME functions electron energy Electron density afterglows smoothing probability functions Argon DUTY CYCLE electron temperature

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A Discussion of probability functions and Constraints from a Variational Perspective

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SET-VALUED AND VARIATIONAL ANALYSIS 2020年第4期28卷 585-609页

作者： van Ackooij, Wim EDF R&D OSIRIS 7 Blvd Gaspard Monge F-91120 Palaiseau France

probability constraints are a popular modelling mechanism in applications. They help to model feasible decisions when the latter are taken prior to observing uncertainty and both decisions and uncertainty are involved in a constraint structure of an optimization problem. The popularity of this paradigm is attested by a vast literature using probability constraints. In this work we try to provide, with variational analysis in mind, an introduction to the topic. We wish to highlight questions regarding the understanding of theoretical properties, such as continuity, (generalized) differentiability, convexity, but also regarding algorithms. We try to highlight open research avenues whenever possible.

关键词： probability constraints probability functions Set-Valued analysis Stochastic optimization

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GENERALIZED DIFFERENTIATION OF probability functions ACTING ON AN INFINITE SYSTEM OF CONSTRAINTS

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SIAM JOURNAL ON OPTIMIZATION 2019年第3期29卷 2179-2210页

作者： van Ackooij, Wim Perez-Aros, Pedro EDF R&D Dept OSIRIS 7 Blvd Gaspard Monge F-91120 Palaiseau France Univ OHiggins Inst Ciencias Ingn Av Libertador Bernardo OHiggins 611 Rancagua Chile

In decision-making problems under uncertainty, probability constraints are a valuable tool for expressing safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. When uncertainty results from two different sources, unequally known, it becomes intuitively appealing to consider the probability of the worst case with respect to the part of the uncertainty vector for which little information was available. This and other models lead to probability functions acting on infinite systems of constraints. In this paper we study generalized (sub)differentiation of such probability functions. We also develop explicit formulae for the subdifferentials that should prove useful in first-order methods or in formulating optimality conditions.

关键词： stochastic optimization probability functions probabilistic constraints generalized differentiation chance constraints

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Derivatives of probability functions: Unions of Polyhedra and Elliptical Distributions

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第2期30卷 487-519页

作者： van Ackooij, Wim Javal, Paul Perez-Aros, Pedro EDF R D OSIRIS 7 Blvd Gaspard Monge F-91120 Palaiseau France EDF R D MIRE 7 Blvd Gaspard Monge F-91120 Palaiseau France Univ OHiggins Inst Ciencias Ingn Rancagua Chile

In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonlinear programming methods, not only knowledge of differentiability, but also explicit formul AE for gradients are important. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by applications from energy management, wherein we face a variety of non-linear transforms of underlying Elliptical distributions, we investigate probability functions acting on decision dependent union of polyhedra. Union of polyhedra naturally occur as soon as one approaches the components of "difference-of-convex" (DC) functions with their respective cutting plane models. In this work, we will establish that the probability functions are locally Lipschitzian and exhibit explicit formul AE for "the" Clarke sub-gradients, under very mild conditions. We also highlight, on a numerical example, that the formul AE can be put to use "in practice".

关键词： probability functions Sub-differentiability Difference-of-convex

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Evaluation of effect of glass beads on thermal conductivity of insulating concrete using micro CT images and probability functions

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CEMENT & CONCRETE COMPOSITES 2016年 65卷 150-162页

作者： Chung, Sang-Yeop Han, Tong-Seok Kim, Se-Yun Kim, Jang-Ho Jay Youm, Kwang Soo Lim, Jae-Hong Yonsei Univ Dept Civil & Environm Engn Seoul 120749 South Korea GS E&C Res Inst Infra Struct Team Seoul 110130 South Korea Pohang Accelerator Lab Ind Technol Convergence Ctr Pohang 790834 Gyeongbuk South Korea

Insulating concrete is a type of concrete that is designed to reduce thermal conductivity. Insulating concrete contains numerous voids that play an important role in reducing heat conduction. Therefore, appropriate nondestructive methods are required to examine the spatial distribution of voids and constituents in a concrete specimen. In this study, an insulating concrete specimen containing hollow glass beads to increase the insulating effect is adopted. Then, micro computed tomography (CT) is used to investigate the spatial distribution of the voids in this specimen. By using a micro CT device, a series of cross-sectional images of the specimen at micrometer-order pixel size are generated by X-rays. To quantitatively describe the spatial distribution of voids in the specimen, probability functions such as two-point correlation, lineal-path, and two-point cluster functions are adopted. In addition, the thermal conductivity of the specimen is evaluated using finite element simulation. The results clarify the insulating effect of glass beads on the concrete specimen and reveal a strong relationship between the probabilistic characteristics of the void distribution and the material responses of insulating concrete. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Insulating concrete Glass bead Void distribution Micro CT image probability functions Thermal conductivity

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A novel methodology for modeling and analyzing thermal runaway propagation in lithium-ion battery modules using probability functions

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APPLIED THERMAL ENGINEERING 2024年 257卷

作者： Yu, Jin Guo, Chuanyu Yu, Jia-jia Chongqing Jiaotong Univ Sch Aeronaut Chongqing 400074 Peoples R China Chongqing Key Lab Green Aviat Energy & Power Chongqing 400074 Peoples R China Chongqing Univ Sch Energy & Power Engn Chongqing 400044 Peoples R China

To address the limitations of existing numerical simulation methods in accurately capturing the probabilistic changes in thermal runaway (TR) triggering temperatures in lithium-ion batteries (LIBs), this paper proposes a novel methodology for modeling and analyzing TR propagation using probability functions. By conducting a statistical analysis of the TR temperature ranges and frequency distributions from a large dataset of LIBs, the probability of TR triggering within each temperature range is computed. This probabilistic approach is integrated into the simulation process, with triggering probabilities determined based on random number-generated distributions. The proposed method is validated through experimental data, and a probabilistic trigger model (PTM) is developed to conduct numerical simulations of 18650-type LIBs from a probabilistic perspective. The TR propagation paths and probabilities in LIB modules, including configurations of four batteries in different arrangements, are examined. Notably, a new TR propagation mode, termed jumping propagation, is identified and simulated for the first time. Further analysis of an 8 x 8 module configuration shows that the TR propagation speed in the PTM closely matches that of the deterministic trigger model (DTM) when the fixed triggering temperature is set to 170 degrees C. Additionally, the use of probabilistic functions introduces fluctuations at the TR propagation front, resulting in an irregular square-shaped progression. The observed jumping propagation significantly reduces the TR time between adjacent cells. This methodology enhances the consistency between simulated and actual TR propagation, offering an effective tool for studying the probabilistic nature of TR processes in LIBs.

关键词： Jumping propagation Lithium-ion battery modules Probabilistic triggering probability functions Thermal runaway

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Development of Injury probability functions for the Flexible Pedestrian Legform Impactor

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SAE INTERNATIONAL JOURNAL OF PASSENGER CARS-MECHANICAL SYSTEMS 2012年第1期5卷 242-252页

作者： Takahashi, Yukou Matsuoka, Fumio Okuyama, Hiroyuki Imaizumi, Iwao Honda Res & Dev Co Ltd Haga Machi Japan Toyota Motor Co Ltd Toyota Japan Nissan Motor Co Ltd Yokohama Kanagawa Japan

The goal of this study was to develop injury probability functions for the leg bending moment and MCL (Medial Collateral Ligament) elongation of the Flexible Pedestrian Legform Impactor (Flex-PLI) based on human response data available from the literature. Data for the leg bending moment at fracture in dynamic 3-point bending were geometrically scaled to an average male using the standard lengths obtained from the anthropometric study, based on which the dimensions of the Flex-PLI were determined. Both male and female data were included since there was no statistically significant difference in bone material property. Since the data included both right censored and uncensored data, the Weibull Survival Model was used to develop a human leg fracture probability function. As for the MCL failure, since one of the two data sources does not provide tabulated data, two MCL failure probability functions as a function of the knee bending angle developed using the Weibull Survival Model were averaged over the knee bending angle. The functions developed for the human leg fracture and the MCL failure were converted to those for the Flex-PLI using the results of the previous study that investigated the correlation between human and Flex-PLI injury measures using human and Flex-PLI finite element (FE) models and simplified vehicle models. 10% increase in failure tolerance due to muscle tone estimated from the literature was taken into account when converting the function for the MCL. Since the conversion of the MCL failure probability function was made using the correlation function between different human and Flex-PLI injury measures, the correlation between the different measures both for human was investigated for validating the conversion. The injury thresholds proposed for the Flex-PLI were evaluated in terms of equivalence to the current injury thresholds for the TRL legform specified in the global technical regulation (gtr) on pedestrian safety by developing an injury pro

关键词： Data sources Material properties Human protection level probability functions Injuries impactors Pedestrian safety Lower Extremity Fracture MAXIMUM CONTAMINANT LEVELS transistor resistor logic Correlation function

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DERIVATIVES OF probability functions AND SOME APPLICATIONS

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ANNALS OF OPERATIONS RESEARCH 1995年第0期56卷 287-311页

作者： URYASEV, S BROOKHAVEN NATL LAB UPTONNY 11973 INT INST APPL SYST ANAL A-2361 LAXENBURGAUSTRIA

probability functions depending upon parameters are represented as integrals over sets given by inequalities. New derivative formulas for the intergrals over a volume are considered. Derivatives are presented as sums of integrals over a volume and over a surface. Two examples are discussed: probability functions with linear constraints (random right-hand sides), and a dynamical shut-down problem with sensors.

关键词： probability functions GRADIENT OF INTEGRAL SENSITIVITY ANALYSIS OPTIMIZATION DISCRETE EVENT DYNAMIC SYSTEMS SHUT DOWN PROBLEM PROBABILISTIC RISK ANALYSIS

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