Space Of Random Variables Research Articles

Short-term optimal scheduling of wind-photovoltaic-hydropower-thermal-pumped hydro storage coupled system based on a novel multi-objective priority stratification method

In the new power system with high proportion of uncertain renewable energy sources (RES), there is a defect of RES consumption at the expense of other power sources' operational efficiency. This paper proposes a short-term optimal scheduling model of wind-photovoltaic-hydropower-thermal-pumped hydro storage (WPHTPHS) coupled system, which realizes the multiple optimization objectives involving minimizing operation cost of thermal power units, maximizing clean energy power generation, minimizing net load fluctuation and thermal power regulation. First, to overcome the dimension disaster problem in the solution space of high-dimensional random variables, a method for pre-solving integer state variables is proposed. Then, a novel multi-objective solution strategy of priority stratification-coupled feedback combined with improved plant growth simulation algorithm is designed. Finally, the effectiveness and superiority of the proposed model and solution method are demonstrated by case studies, and the numerical results show that the number of startups and shutdowns, standard deviation of output and operating cost of thermal power units are reduced by 90.9 %, 65.34 %, and 14.01 % respectively, compared with traditional wind-photovoltaic-thermal strategy. This study contributes to resolving the relationship between conflicting objectives and highlighting the potential advantages of WPHTPHS coupled system to maximize overall performance from economic and stability perspectives.

Probabilistic Performance-Pattern Decomposition (PPPD): Analysis framework and applications to stochastic mechanical systems

Numerous research efforts have been devoted to developing quantitative solutions to stochastic mechanical systems. In general, the problem is perceived as “solved” when a complete or partial probabilistic description on quantities of interest (QoIs) is determined. However, in the presence of complex system behavior, there is a critical need to go beyond computing probabilities. In fact, to gain a better understanding of the system, it is crucial to extract physical characterizations from the probabilistic structure of the QoIs, especially when the QoIs are computed in a data-driven fashion. Motivated by this perspective, the paper proposes a framework to obtain structuralized characterizations on behaviors of stochastic systems. The framework is named Probabilistic Performance-Pattern Decomposition (PPPD). PPPD analysis aims to decompose complex response behaviors, conditional to a prescribed performance state, into meaningful patterns in the space of system responses, and to investigate how the patterns are triggered in the space of basic random variables. To illustrate the application of PPPD, the paper studies three numerical examples: (1) an illustrative example with hypothetical stochastic processes input and output; (2) a stochastic Lorenz system with periodic as well as chaotic behaviors; and (3) a simplified shear-building model subjected to a stochastic ground motion excitation.

Hybrid reliability analysis method for systems with random and non-parameterized p-boxes based on weight coefficients

This paper establishes a hybrid variable system failure probability optimization model based on sampling methods and weighting coefficients. By introducing auxiliary input variables, important sampling functions, and p-box, failure samples are mapped from the random variable space to the p-box variable space. The new weight coefficients are constructed, including important sampling weights and interval weights. Combining discretization methods and Monte Carlo simulation (MCS), the interval weights are transformed into variables, and constraints conforming to the p-box variable distribution are constructed. After calculating the weighting coefficients for all failure samples, the new failure probability optimization model is built. This model is independent of the performance functions and does not involve cyclic optimization, with computational complexity only related to the dimensions. Six cases are used for method comparison, validating that the new method exhibits higher efficiency and accuracy.

Law-invariant return and star-shaped risk measures

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Probabilistic design procedure for steel moment resisting frames equipped with FREEDAM connections

In this work, the Theory of Plastic Mechanism Control (TPMC) is combined with a probabilistic method to account for the influence of random material variability. Reference is made to steel Moment Resisting Frames (MRFs) equipped with FREEDAM connections. FREEDAM connections are beam-to-column connections equipped with friction dampers to dissipate the seismic input energy. TPMC is used to guarantee that in case of destructive seismic events the structural members such as beams and columns remain undamaged. To this scope, the structure is designed to assure a collapse mechanism characterized by the activation of all the friction dampers of the beam ends and the formation of plastic hinges at the base of the first storey columns only. From the probabilistic point of view, the random uncertainties are given by the static friction coefficient of the contact surfaces and the preloading of the bolts of the friction dampers as well as the yielding resistance of the steel members. The failure domain is related to all the possible failure events, where the term “failure” concerns the development of an undesired mechanism different from the global one. Generally, the design conditions to prevent undesired collapse mechanisms are stochastic events within the framework of the kinematic theorem of plastic collapse. The limit state function corresponding to each event can be represented by a hyperplane in the space of random variables. Consequently, the failure domain is a surface resulting from the intersection of the hyperplanes corresponding to the limit states of each single failure event. Since dissipative zones (member ends or friction dampers) in the frame members are common to many different mechanisms, the single limit state functions are correlated. Therefore, the probability of failure can be evaluated by means of the Bimodal or Ditlevsen bounds by assuming that the failure events are located in series. The output of the work is a simple relationship which provides the overstrength factor of FREEDAM connections to be considered in the column design phase to account for random material variability thus assuring a given level of reliability in the application of TPMC.

A sequential sampling-based Bayesian numerical method for reliability-based design optimization

For efficiently solving the Reliability-Based Design Optimization (RBDO) problem with multi-modal, highly nonlinear and expensive-to-evaluate limit state functions (LSFs), a sequential sampling-based Bayesian active learning method is developed in this work. The penalty function method is embedded to transform the constrained optimization problem into a non-constrained one to reduce the model complexity. The proposed method for solving RBDO problems starts by training a Gaussian process (GP) model, in the augmented space of random and design variables. It is then based on an efficient sampling scheme for simulating the GP model, the adaptive Bayesian optimization (BO) and Bayesian reliability analysis (BRA) procedures are combined in a collaborative way for sequentially producing the joint training points. BO driven by expected improvement (EI) function is used for inferring the global optimum in the design space with global convergence, and the BRA equipped with U function is implemented for inferring the failure probabilities at the identified design points with the desired accuracy. The superiority of the proposed method is demonstrated with two numerical and two real-world engineering examples.

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

Conditional aggregation operators defined by the Choquet integral and the Sugeno integral with respect to general fractal measures

Conditional aggregation operators are defined by the Choquet integral and the Sugeno integral with respect to a monotone set function that assesses positive measure of the conditioning set. General Hausdorff and packing measures are introduced and examples of infinite s-sets with positive and finite generalized Hausdorff and packing measures are constructed and their fractal dimensions are compared. Coherent upper conditional previsions on the linear space of all Choquet integrable random variables are defined by the Choquet integral with respect to the general Hausdorff and packing measures when the conditioning event has positive and finite generalized Hausdorff and packing measures in its respective fractal dimensions. Conditional aggregation operators are defined by the Sugeno integral with respect to general Hausdorff and packing measures on the class of all Sugeno integrable random variables. Actually, the general Hausdorff and packing dimensions are proven to be the Sugeno integral with respect to the Lebesgue measure of the general Hausdorff and packing measures respectively.

An elementary proof of the dual representation of Expected Shortfall

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.

Reliability of depleted cement–ground slab with waste granodiorite dust admixture on semi-saturated substrate

There are many types of pavements for roads. For technical roads, one of the most effective and economically advantageous pavement types is chemically stabilized soil pavement. When designing a pavement made of soil admixed with binders, it is necessary to meet the load-bearing conditions while considering the effects of environmental factors. The aims of this study were to optimize the strength parameters of pavement, minimize negative environmental effects, minimize the carbon footprint, and simultaneously dispose of mineral waste materials from the production of asphalt mixes for roads. After testing the strengths of samples admixed with cement or with reduced cement content and the addition of dust, questions arose as to how both materials affect the reliability of the pavement structure. Samples of soil doped with cement and dust were subjected to reliability analyses. Abaqus software with a three-dimensional subgrade model was used for a comparative analysis. The substrates under soil stabilization slabs were subjected to varying saturation states, which affected their vulnerability. In this study, the changes in the saturation characteristics of the subsoils associated with different types of climatic interactions were simulated. The Darcy–Buckingham law was used to describe the flow of liquids through the subsoil. Pavement reliability calculations were performed for a multidimensional space of random variables. The results served as an interesting starting point for recommendations to decrease the use of energy-intensive materials, increase the reuse of waste, and reduce the effects of the strengthening process on the environment.

Hamiltonian MCMC methods for estimating rare events probabilities in high-dimensional problems

Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The developed techniques in this paper are applicable from low- to high-dimensional stochastic spaces, and the basic idea is to construct a relevant target distribution by weighting the original random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.

Mini-Batch Risk Forms

Risk forms are real functionals of two arguments: a bounded measurable function on a Polish space and a probability measure on that space. They are convenient mathematical structures adapting the coherent risk measures to the situation of a variable reference probability measure. We introduce a new class of risk forms called mini-batch forms. We construct them by using a random empirical probability measure as the second argument and by post-composition with the expected value operator. We prove that coherent and law invariant risk forms generate mini-batch risk forms which are well defined on the space of integrable random variables, and we derive their dual representation. We demonstrate how unbiased stochastic subgradients of such risk forms can be constructed. Then, we consider pre-compositions of mini-batch risk forms with nonsmooth and nonconvex functions, which are differentiable in a generalized way, and we derive generalized subgradients and unbiased stochastic subgradients of such compositions. Finally, we study the dependence of risk forms and mini-batch risk forms on perturbation of the probability measure and establish quantitative stability in terms of optimal transport metrics. We obtain finite-sample expected error estimates for mini-batch risk forms involving functions on a finite-dimensional space.

ON PROPERTIES OF SPACES WITH MARTINGALE MIXED NORM WITH SUMMABILITY EXPONENTS CLOSE TO ONE

For spaces of random variables with martingale mixed norm the theorem of A. Pelczynski on the absence of an unconditional basis in the space $$L_1(0,1)$$ is generalized and (in terms of summability exponents) the cases are studied when these spaces coincide with the space $$L_1$$ . 2020 Mathematics subject classification: Primary 60G46; Secondary 60G42, 46E30.

Stochastic wave spectra estimation (SWSE) based on response surface methodology considering uncertainty in transfer functions of a ship

In this paper, a new wave spectra estimation method is proposed in which the frequency domain wave estimation method (FDWE) is extended into a probabilistic analytical framework in order to estimate the encountered sea states involving uncertainty in transfer functions of a ship. The proposed method, named the Stochastic Wave Spectra Estimation (SWSE), makes use of an Hermite polynomial chaos expansion (PCE) to represent the uncertainty in the transfer functions and the response surfaces. The method involves a mathematical formulation where an extension of the deterministic FDWE concept to the space of random variables is made. The proposed method can accurately and easily estimate the encountered wave spectra based on ship response measurements accounting for uncertainty in the transfer functions. In this paper, numerical and experimental investigations of the proposed SWSE are made, where the uncertainties in the transfer functions of heave and pitch motions of a containership are taken into account. The validity of the SWSE is demonstrated by comparison to results of uncertainty analyses through the Monte-Carlo simulations (MCS).

Кількісна оцінка рівня ризику збільшення витрат на розробку зразків космі-чної техніки

The goal of this work is to develop a methodological approach to quantitative estimation of the risk of an increase in the cost of space hardware prototyping. The paper considers a technology and mathematical models for quantitative estimation of the risk of an increase in the cost of a developmental work on space hardware prototyping. The main cause of the risk of development cost increase is that data used in expected cost estimation are incomplete and inaccurate. The risk level is estimated as the probability of the possible cost of an R&D project exceeding a critical (for the investor) value. The risk estimation technology is constructed on the basis of the Monte Carlo method embedded in a simulation model. The Monte Carlo method is based on an analytico-probabilistic model (a deterministic mathematical model and a probabilistic model with known distribution functions (laws)). The uniqueness, novelty, and technical complexity of space hardware prototypes do not allow one to construct any analytico-probabilistic model. This paper presents a mathematical model equivalent to an analytico-probabilistic one. The paper substantiates the appropriateness of a homomorphic mapping of a possibilistic space of random variables into a probabilistic space; i.e. in this case the proposed model is equivalent to an analytico-probabilistic one. The key component of the simulation model is the mathematical model of the development cost of a space hardware prototype. The cost model is based on a component-by-component analogy for relatively simple components of the space hardware prototype, moving (upward) along the weighted oriented tree graph that models the engineering structure of the space hardware prototype, and fuzzy methods. The proposed methodological approach may be used in the construction of a simulation model for quantitative estimation vc of the risk of a decrease in the efficiency of use of the prototype under development. To do this, it will be sufficient to replace the mathematical model of development cost with a mathematical model of expected efficiency.

A Look at the Primary Order Preserving Properties of Stochastic Orders: Theorems, Counterexamples and Applications in Cognitive Psychology

In this paper, we prove that for a set of ten univariate stochastic orders including the usual order, a univariate stochastic order preserves either both, one or none of additivity and multiplication properties over the vector space of real-valued random variables. Then, classifying participant’s quickness in a mental chronometry trial to “weakly faster” and “strongly faster”, we use the above results for the usual stochastic order to establish necessary and sufficient conditions for a participant to be strongly faster than the other in terms of the fitted Wald, Exponentially modified Wald(ExW), and Exponentially modified Gaussian(ExG) distributional parameters. This research field remains uncultivated for other univariate stochastic orders and in several directions.

Grid Feature-Based Weighted Simulation Method for Multi-Objective Reliability-Based Design Optimization

Reliability analysis and reliability-based design have attracted huge attention since their origin, especially in the engineering design optimization field. Integrating a novel grid feature (GF) mechanism into the weighted simulation method (WSM), a grid feature-based weighted simulation method (GFWSM) is proposed in this paper. By sorting the samples in the random variable space, the GF mechanism serializes the failure probability calculation process, which can reduce the number of performance function evaluations required by the WSM. The proposed GFWSM is integrated into an efficient evolutionary multi-objective optimizer, disruption-based multi-objective equilibrium optimization algorithm (DMOEOA), for solving multi-objective reliability-based design optimization (MORBDO) problems. Different reliability analysis examples are employed to verify the advantages of the proposed GFWSM. Various constrained multi-objective reliability-based design optimization problems have been selected to test the effectiveness of the GFWSM-DMOEOA algorithm. The simulation results indicate that GFWSM and GFWSM-DMOEOA algorithm have achieved a good balance between accuracy and efficiency.

Staircase modulation improvement to balance output power of stages of cascade H-bridge multilevel inverter

In this study, different methods are investigated to minimize the power imbalance between the stages of a cascade H-bridge (CHB) multilevel inverter (MLI). First-in-first-out (FIFO) and first-in-last-out (FILO) are two common staircase-modulation (SCM) methods. A CHB-MLI with FIFO modulation, even with the same conduction times, cannot distribute power equally between the stages. A new algorithm is presented in this study to balance the power between the inverter stages. This is important to increase the lifetime of the PV sources that feed the inverter. To implement the new algorithm, all possible switching patterns based on the angles obtained from the nearest level of control (NLC) method are defined as the sample spaces of random variables. Subsequently, a random vector is defined for a set of random variables, and an optimization problem with finite and discrete solutions is applied. This algorithm is effectively used to minimize the power imbalance between the stages, even in impedance loads, from the purely inductive to the pure capacitive range. In addition, with slight changes in the switching angles, the operation of the stages and the inverter efficiency increases. Finally, the simulation and experimental results confirm the validity and practicality of the proposed modulation.