Stochastic Dynamic Problem Research Articles

Inverse chance-constrained dynamic data envelopment analysis under natural and managerial disposability: Concerning renewable energy efficiency and potentials

The performance of dynamic systems with specific data has been examined in some data envelopment analysis (DEA) studies. However, in many real-life situations, dealing with dynamic systems becomes challenging because of random factors. So, in this paper, a chance-constrained dynamic DEA approach under managerial and natural disposability is first proposed to analyze the dynamic renewable energy efficacy of processes in the presence of random performance measures. Then, the potentials of some performance metrics are addressed for changes of others using the introduced inverse chance-constrained dynamic DEA models. The chance-constrained dynamic DEA techniques and their inverse dynamic stochastic problems are converted into linear problems. The introduced approaches are applied to probe the renewable energy efficiency and potentials of some OECD countries in a span of time. The results show stochastic dynamic DEA and inverse stochastic dynamic DEA provided are practical tools for making the best choices, when there is uncertainty.

Efficient reliability analysis of stochastic dynamic first-passage problems by probability density evolution analysis with subset supported point selection

This study introduces a novel point selection procedure for the Probability Density Evolution Method (PDEM) to estimate time-dependent reliability and failure probabilities in dynamic systems under first-passage failure conditions. The method integrates and modifies features of the Subset simulation procedure to adaptively generate dependent sample sets suitable for a reliability analysis by a direct probability integration approach levering PDEM. Performance function assessments are used as weighting factors in the Subset supported Point Selection (S-PS), enhancing the ultimate failure probability estimation accuracy. The presented approach effectively identifies samples in the failure region, particularly benefiting for dynamic systems under stochastic excitation, tested with random dimensions up to 60. It also offers a computationally efficient structural reliability estimation procedure by analyzing full time–history responses. The proposed method provides deeper insights into rare failure events and mechanisms through the visualization of intermediate results. This research presents an advanced framework for estimating structural reliability and understanding critical events in dynamic systems.

Efficient stochastic modal decomposition methods for structural stochastic static and dynamic analyses

AbstractThis article presents unified and efficient stochastic modal decomposition methods to solve stochastic structural static and dynamic problems. We extend the idea of deterministic modal decomposition method for structural dynamic analysis to stochastic cases. Standard/generalized stochastic eigenvalue equations are adopted to calculate the stochastic subspaces for stochastic static/dynamic problems and they are solved by an efficient reduced‐order method. The stochastic solutions of both static and dynamic equations are approximated by stochastic bases of the stochastic subspaces. Original stochastic static/dynamic equations are then transformed into a set of single‐degree‐of‐freedom (SDoF) stochastic static/dynamic equations, which are efficiently solved by the proposed non‐intrusive methods. Specifically, a non‐intrusive stochastic Newmark method is developed for the solution of SDoF stochastic dynamic equations, and the element‐wise division of sample vectors is used to solve the SDoF stochastic static equations. All of these methods have low computational effort and are weakly sensitive to the stochastic dimension, thus the proposed methods avoid the curse of dimensionality successfully. Two numerical examples, including two‐ and three‐dimensional spatial problems with low and high stochastic dimensions, are given to show the efficiency and accuracy of the proposed methods.

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes.

Analytical equivalent transformation method for nonlinear stochastic dynamics with multiple noises in high dimensions

In many practical applications, stochastic problems are high dimensionality and multiple noises. The study focuses on developing an analytical approach for processing multi-noise systems in high dimensions, which overcomes the challenge solving stochastic dynamics problems. Applying an analytical equivalent transformation and introducing a new partial differentiation, the author demonstrate a convenient method for obtaining the Fokker–Planck equation corresponding to Langevin dynamics equations for high-dimensional and multi-noise systems. This methodology is applied to analyze the cubic laser model subjected to white noises and the saturation laser model subjected to bounded sine Wiener noise. The results verify the correction, efficiency, and superiority of the method of processing multiple noises in high dimensions. This approach provides a novel framework for solving other practical dynamics problems for high-dimensional and multi-noise systems.

Probabilistic-learning-based stochastic surrogate model from small incomplete datasets for nonlinear dynamical systems

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the “identification observation” of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. We apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics.

Relaxation-based importance sampling for structural reliability analysis

This study presents an importance sampling formulation based on adaptively relaxing parameters from the indicator function and/or the probability density function. The formulation embodies the prevalent mathematical concept of relaxing a complex problem into a sequence of progressively easier sub-problems. Due to the flexibility in constructing relaxation parameters, relaxation-based importance sampling provides a unified framework for various existing variance reduction techniques, such as subset simulation, sequential importance sampling, and annealed importance sampling. More crucially, the framework lays the foundation for creating new importance sampling strategies, tailoring to specific applications. To demonstrate this potential, two importance sampling strategies are proposed. The first strategy couples annealed importance sampling with subset simulation, focusing on low-dimensional problems. The second strategy aims to solve high-dimensional problems by leveraging spherical sampling and scaling techniques. Both methods are desirable for fragility analysis in performance-based engineering, as they can produce the entire fragility surface in a single run of the sampling algorithm. Three numerical examples, including a 1000-dimensional stochastic dynamic problem, are studied to demonstrate the proposed methods.

Social Security safety net with rare event risk

AbstractThe US Social Security program was created in part as an explicit response to the Great Depression. We evaluate the performance of Social Security as a protective safety net against a rare episode of sudden and significant loss of private wealth such as the Great Depression. We construct a model in which a rare event causes a shock to wealth at an unknown time. How well does Social Security function as a safety net against such risk? The answer depends critically on whether households optimize in the face of this risk. If the household has full information on the distribution of rare event risk and solves a dynamic stochastic problem to hedge this risk, then Social Security is unnecessary along this dimension. Alternatively, if the household does not account for rare event risk in its financial planning, then Social Security can provide very large welfare gains as a safety net.

An Adaptively Filtered Precise Integration Method Considering Perturbation for Stochastic Dynamics Problems

An Adaptively Filtered Precise Integration Method Considering Perturbation for Stochastic Dynamics Problems

A simple but powerful simulated certainty equivalent approximation method for dynamic stochastic problems

We introduce a novel simulated certainty equivalent approximation (SCEQ) method for solving dynamic stochastic problems. Our examples show that SCEQ can quickly solve high‐dimensional finite‐ or infinite‐horizon, stationary or nonstationary dynamic stochastic problems with hundreds of state variables, a wide state space, and occasionally binding constraints. With the SCEQ method, a desktop computer will suffice for large problems, but it can also use parallel tools efficiently. The SCEQ method is simple, stable, and can utilize any solver, making it suitable for solving complex economic problems that cannot be solved by other algorithms.

A high-performance calculation scheme for stochastic dynamic problems

The goal of stochastic dynamic problems is to efficiently and accurately calculate statistical moments and random dynamic responses. Thus, a high-performance calculation scheme is proposed to achieve the goal. First, to enhance the accuracy of the statistical moments, we proposed an adaptive weights quasi-Monte Carlo (AWQMC) method. Compared to the integration weights of the traditional quasi-Monte Carlo method, the adaptive weights can better reflect the discrepancy of the sample points to obtain more accurate statistical moments. Additionally, the adaptive weights only depend on the sample set, which can be applied to other stochastic problems. Second, the reduced-order model technology is introduced to improve the calculation efficiency of random dynamic responses. Combining of the Galerkin method, a novel reduced-order model based on deterministic reduced bases is proposed for random dynamic system. Then, an efficient dynamic equation solver is constructed based on the deterministic reduced bases and the Newmark method. Finally, according to the three numerical examples, the high-performance calculation scheme combined the AWQMC method and the efficient dynamic equation solver can significantly improve the accuracy and efficiency of statistical moments for stochastic dynamic problems.

On-Demand Delivery from Stores: Dynamic Dispatching and Routing with Random Demand

Problem definition: On-demand delivery has become increasingly popular around the world. Motivated by a large grocery chain store who offers fast on-demand delivery services, we model and solve a stochastic dynamic driver dispatching and routing problem for last-mile delivery systems where on-time performance is the main target. The system operator needs to dispatch a set of drivers and specify their delivery routes facing random demand that arrives over a fixed number of periods. The resulting stochastic dynamic program is challenging to solve because of the curse of dimensionality. Methodology/results: We propose a novel structured approximation framework to approximate the value function via a parametrized dispatching and routing policy. We analyze the structural properties of the approximation framework and establish its performance guarantee under large-demand scenarios. We then develop efficient exact algorithms for the approximation problem based on Benders decomposition and column generation, which deliver verifiably optimal solutions within minutes. Managerial implications: The evaluation results on a real-world data set show that our framework outperforms the current policy of the company by 36.53% on average in terms of delivery time. We also perform several policy experiments to understand the value of dynamic dispatching and routing with varying fleet sizes and dispatch frequencies. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72222011 and 72171112], China Association for Science and Technology [Grant 2019QNRC001], and the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2022-04950]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.1171 .

A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

AbstractThe deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

Stochastic rotating waves

Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has recently received considerable attention are traveling wave patterns occurring in stochastic partial differential equations (SPDEs). Here, one is interested in how deterministic traveling waves behave under stochastic perturbations. In this paper, we start the mathematical study of a related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic partial differential equation (PDE) dynamics techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart.

A weak-intrusive stochastic finite element method for stochastic structural dynamics analysis

This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method.

Improving defensive air battle management by solving a stochastic dynamic assignment problem via approximate dynamic programming

Military air battle managers face several challenges when directing operations during quickly evolving combat scenarios. These scenarios require rapid assignment decisions to engage moving targets having dynamic flight paths. In defensive operations, the success of a sequence of air battle management decisions is reflected by the friendly force’s ability to maintain air superiority and defend friendly assets. We develop a Markov decision process (MDP) model of a stochastic dynamic assignment problem, named the Air Battle Management Problem (ABMP), wherein a set of unmanned combat aerial vehicles (UCAV) must defend an asset from cruise missiles arriving stochastically over time. Attaining an exact solution using traditional dynamic programming techniques is computationally intractable. Hence, we utilize an approximate dynamic programming (ADP) technique known as approximate policy iteration with least squares temporal differences (API-LSTD) learning to find high-quality solutions to the ABMP. We create a simulation environment in conjunction with a generic yet representative combat scenario to illustrate how the ADP solution compares in quality to a reasonable, closest-intercept benchmark policy. Our API-LSTD policy improves mean success rate by 2.8% compared to the benchmark policy and offers an 81.7% increase in the frequency with which the policy performs perfectly. Moreover, we find the increased success rate of the ADP policy is, on average, equivalent to the success rate attained by the benchmark policy when using a 20% faster UCAV. These results inform military force management and defense acquisition decisions and aid in the development of more effective tactics, techniques, and procedures.

Improved mode acceleration-based vibroacoustic coupling analysis of functionally graded shell under random excitation

This paper focuses on the vibroacoustic response of functionally graded material (FGM) shell structures under stationary random excitations. An eight-node finite element model of FGM shells is developed, which is valid for both flat and curved shells of FGM. Then, the finite element formulation of the acoustic-structural coupled system for FGM shells under random excitation is established. It is shown that the commonly used combined method of the pseudo excitation method (PEM) and mode displacement method (MDM) in previous work is highly efficient for stochastic dynamics problems, but its accuracy is reduced by the modal truncation error. To remedy this, the combined method of PEM and the conventional mode acceleration method (MAM) is an effective alternative, but not applicable to the coupled acoustic-structural system due to its singular stiffness matrix. To circumvent this, an efficient and accurate method integrating the PEM and the improved MAM (IMAM) is proposed for the vibroacoustic coupling analysis of FGM shell structures under stationary random excitation. Numerical examples are given to verify the effectiveness of the proposed method and to explore the random vibroacoustic properties of the FGM shells with different power-law distributions.

Reliability of dynamical systems with combined Gaussian and Poisson white noise via path integral method

Analyzing the reliability of a stochastic dynamical system represents one of the most challenging problems in stochastic dynamics. In this paper, the reliability of dynamical systems excited by combined Gaussian and Poisson white noise is investigated by adapting the path integral (PI) method. We derive the PI solutions for the reliability based on the reliability density function (RDF), where the short-time transition probability density function (TPDF) for the PI method is derived based on the Fourier transformation. And the impact of the combined noise on the reliability is analyzed theoretically for the one-dimensional case. We prove that the reliability for the system excited by combined noise will approach that for the system excited by Gaussian white noise when the mean arrival rate goes to infinite and jump amplitude for the Poisson white noise satisfies certain conditions. Then, one-dimensional and single-degree-of-freedom (SDOF) dynamical systems excited by combined Gaussian and Poisson white noise are worked out to show the application of the PI method on the reliability. Reliability under different system parameters is calculated and presented. The roles of jump amplitude with different distributions on the reliability are calculated and compared, which verifies the theoretical results. Moreover, it is discovered that the first passage of the system is more easily caused by the Gaussian white noise compared with the combined Gaussian and Poisson white noise when the noise intensity is the same.

Transfer matrices and solution of the problem of stochastic dynamics of aerosol clusters by Monte Carlo method

Abstract A Monte Carlo algorithm based on the use of transfer matrices is developed to describe the stochastic dynamics of the rotational--translational motion of aerosol clusters taking into account fluctuations in the molecular fluxes of the gas medium. In the general case, the cluster is immersed into a rarefied gas medium, the temperatures of its surfaces may differ from the temperature of the surrounding gas, for example, due to absorption of visible and infrared radiation. The motion of the cluster is described based on Langevin motion equations. The algorithm allows one to calculate parameters of the probability distribution of a six-dimensional vector consisting of components of the momentum and angular momentum vectors transmitted to the cluster by molecular flows. The numerical method allows one to apply preliminary analytical averaging modulo velocities of molecules for both the average values of the components of momentum and angular momentum and their correlation characteristics, which significantly reduces the calculation time.

Influence of Inherent Characteristic of PV Plants in Risk-Based Stochastic Dynamic Substation Expansion Planning Under MILP Framework

A suitable probabilistic scenario set of load demand and natural characteristics of renewable energy is becoming a crucial issue in power system planning studies. Properly addressing the impact of potentially thousands of residential PV plants on the resilience and reliability needs of substations necessitates the representation of inherent relations between photovoltaics and the load throughout the long-term planning period. The optimal planning of substation expansions is achievable through proper modeling of input parameters which describes the characteristics of the service areas. In this paper, the co-existence of PV plants and the load in a service area under three different states such as daytime with clear-sky and no-fault, daytime with abnormal events, and nighttime are incorporated into the stochastic dynamic optimization problem by using scenario-based approach. The scenario tree of the problem is branched from three different bases simultaneously instead of only one as in conventional approach. This paper also combines the risk-constrained stochastic dynamic SEP problem and Mixed Integer Linear Programming (MILP) framework under one roof. The comparison between integrating inherent characteristics of PV plants with and without considering abnormal events into the optimization is performed to show the impact of suitable probabilistic model on dynamic nature of investment decisions.