Number Of Random Parameters Research Articles

Devising genuine cross-spectral densities by randomization of coherent fields.

We propose a concept of devising the authentic cross spectral densities (CSDs) of partially coherent beam waves by averaging the second moment of a coherent field over a certain number of random parameters. Several CSD examples created by randomization are presented. We discuss application of randomization to generation of twisted Gaussian Schell model beams and show that compound randomization leads to a new, more general class of twisted beams. We revisit the necessary nonnegative definiteness condition of Opt. Lett.34, 1399 (2009)OPLEDP0146-959210.1364/OL.34.001399 and show that the two-dimensional space of parameters is necessary.

Scenario-based verification of uncertain parametric MDPs

We consider parametric Markov decision processes (pMDPs) that are augmented with unknown probability distributions over parameter values. The problem is to compute the probability to satisfy a temporal logic specification with any concrete MDP that corresponds to a sample from these distributions. As solving this problem precisely is infeasible, we resort to sampling techniques that exploit the so-called scenario approach. Based on a finite number of samples of the parameters, the proposed method yields high-confidence bounds on the probability of satisfying the specification. The number of samples required to obtain a high confidence on these bounds is independent of the number of states and the number of random parameters. Experiments on a large set of benchmarks show that several thousand samples suffice to obtain tight and high-confidence lower and upper bounds on the satisfaction probability.

On the Scenario-Tree Optimal-Value Error for Stochastic Programming Problems

Stochastic programming problems generally lead to large-scale programs if the number of random outcomes is large or if the problem has many stages. A way to tackle them is provided by scenario-tree generation methods, which construct approximate problems from a reduced subset of outcomes. However, it is well known that the number of scenarios required to keep the approximation error within a given tolerance grows rapidly with the number of random parameters and stages. For this reason, to limit the fast growth of complexity, scenario-tree generation methods tailored to problems must be developed. These will use more information about the problem than just the underlying probability distributions; namely, they will also take into account the objective function and the constraints. In this paper, we develop a general framework to build problem-driven scenario trees. We do so by studying how the optimal-value error arises as a sum of lower-level errors made at each node of the tree. We show how these small but numerous node errors depend on the specific features of the problem and how they can be controlled by designing scenario trees with appropriate branching structures and discretization points and weights. We illustrate our approach on two examples.

Data-driven uncertainty analysis of distribution networks including photovoltaic generation

This paper investigates residential distribution networks with uncertain loads and photovoltaic distributed generation. An original probabilistic modeling of consumer demand and photovoltaic generation is presented that is based on the analysis of large set of data measurements. It is shown how photovoltaic generation is described by complex non-standard distributions that can be described only numerically. Probabilistic analysis is performed using an enhanced version of the Polynomial Chaos technique that exploits a proper set of polynomial basis functions. It is described how such functions can be generated from the numerically available data. Compared to other approximate methods for probabilistic analysis, the novel technique has the advantages of modeling accurately truly nonlinear problems and of directly providing the detailed Probability Density Function of relevant observable quantities affecting the quality of service. Compared to standard Monte Carlo method, the proposed technique introduces a simulation speedup that depends on the number of random parameters. Numerical applications to radial and weakly meshed networks are presented where the method is employed to explore overvoltage, unbalance factor and power loss, as a function of photovoltaic penetration and/or network configuration.

A Perturbative Stochastic Galerkin Method for the Uncertainty Quantification of Linear Circuits

This paper presents an iterative and decoupled perturbative stochastic Galerkin (SG) method for the variability analysis of stochastic linear circuits with a large number of uncertain parameters. State-of-the-art implementations of polynomial chaos expansion and SG projection produce a large deterministic circuit that is fully coupled, thus becoming cumbersome to implement and inefficient to solve when the number of random parameters is large. In a perturbative approach, component variability is interpreted as a perturbation of its nominal value. The relaxation of the resulting equations and the application of a SG method lead to a decoupled system of equations, corresponding to a modified equivalent circuit in which each stochastic component is replaced by the nominal element equipped with a parallel current source accounting for the effect of variability. The solution of the perturbation problem is carried out in an iterative manner by suitably updating the equivalent current sources by means of Jacobi- or Gauss-Seidel strategies, until convergence is reached. A sparse implementation allows avoiding the refinement of negligible coefficients, yielding further efficiency improvement. Moreover, for time-invariant circuits, the iterations are effectively performed in post-processing after characterizing the circuit in time or frequency domain by means of a limited number of simulations. Several application examples are used to illustrate the proposed technique and highlight its performance and computational advantages.

Scenario-Based Verification of Uncertain MDPs.

We consider Markov decision processes (MDPs) in which the transition probabilities and rewards belong to an uncertainty set parametrized by a collection of random variables. The probability distributions for these random parameters are unknown. The problem is to compute the probability to satisfy a temporal logic specification within any MDP that corresponds to a sample from these unknown distributions. In general, this problem is undecidable, and we resort to techniques from so-called scenario optimization. Based on a finite number of samples of the uncertain parameters, each of which induces an MDP, the proposed method estimates the probability of satisfying the specification by solving a finite-dimensional convex optimization problem. The number of samples required to obtain a high confidence on this estimate is independent from the number of states and the number of random parameters. Experiments on a large set of benchmarks show that a few thousand samples suffice to obtain high-quality confidence bounds with a high probability.

Reliability analysis of Pole Kheshti historical arch bridge under service loads using SFEM

The main purpose of this paper is the probabilistic safety analysis of the historical masonry arch bridges (HMAB) and to calculate its reliability index (RI) using the “probabilistic design system” of the ANSYS software. In evaluating the reliability of bridge, the load-resistance model has been used to express the bridge failure functions. Calculating the RI requires the definition of loads effects on the structure and structure resistance. The load and resistance implicit functions are evaluated by stochastic finite element method and the Monte Carlo method has been used for laboratory simulation. The sampling method is the Latin hypercube sampling. The innovations in this paper is to use the functions dependent on parameters, modulus of elasticity, Poisson ratio, density of materials, and traffic load of bridge deck. The number of random parameters is 19. These random parameters are defined by the Log-normal distribution function. In this paper, the reliability status of bridge is investigated in the ultimate limit state under gravitational loading. The constitutive law of the bridge material is considered to be linear elastic. Three types of compressive, tensile, and allowable deflection are considered as limit states of the present research. The case study of the Pole Kheshti Langroud HMAB showed that the required safety is not provided for the ultimate limit state and the bridge is at risk of failure. The RI of bridge in the tensile limit state is lower than the target RI. The sensitivity analysis of random variables of the load and resistance implicit functions to the deflection and tensile responses is investigated, and random parameters with more impact are specified. In the stress limit state and deflection limit state, the modulus of elasticity and weight per unit volume of the sidewalls have the greatest impact on safety, respectively.

A Hierarchical Approach to Dimensionality Reduction and Nonparametric Problems in the Polynomial Chaos Simulation of Transmission Lines

Polynomial chaos (PC)-based techniques recently became popular alternatives for the stochastic analysis of electrical circuits, especially in the context of signal integrity and electromagnetic compatibility investigations. Among the challenging issues of PC, there are two longstanding limitations: curse of dimensionality (i.e., efficiency decrease as the number of random parameters is increased) and the inability to handle nonparametric variations (i.e., random variables (RVs) that cannot be unambiguously parametrized upfront). This paper aims at covering this gap by putting forward a suitable hierarchical approach, with specific emphasis on transmission-line analysis. First, the problem is characterized by estimating the distribution of the transmission line per-unit-length parameters. Second, the obtained distribution is fitted using a multivariate mixture of Gaussians (MoGs). The MoGs is a flexible model that is capable of taking into account the existing dependence between inductance and capacitance entries in multiconductor lines. Next, appropriate orthogonal basis functions are generated based on a recent framework for non-Gaussian correlated RVs. Finally, the technique is combined with a stochastic Galerkin method to generate a deterministic and SPICE-compatible model that can be simulated in time domain. The proposed approach enables both dimensionality (hence, model order) reduction and handling of nonparametric variations. Application examples concerning cables with random cross-section illustrate and validate the methodology.

An Optimization-Simulation Approach to Chance Constraint Programming

A stochastic programming problem with a number of random parameters in the set of constraints is considered. An iterative optimization-simulation approach is applied as solving method. It consists of two phases: optimization phase, which includes solving a deterministic counterpart of the original chance-constrained problem, and a simulation phase in which the original constraints are checked using Monte Carlo simulation. One iteration corresponds to one scenario. If the decision maker is not satisfied with results, new scenario is generated in which the deterministic values of stochastic parameters are changed in the direction that will provide a more robust solution. Deterministic counterpart in the new scenario is formulated depending on the result of previous iteration. For that purpose, different heuristics are considered. The main goal is to provide a good insight on the optimization problem under uncertainty by performing relatively small number of iterations. General approach and results of the proposed framework will be illustrated on an advertisement placement example.DOI: http://dx.doi.org/10.5755/j01.itc.47.2.18712

Numerical Procedures for Random Differential Equations

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

Stochastic finite integration technique for magnetostatic problems

AbstractBoth nonintrusive and intrusive stochastic approaches on the basis of polynomial chaos expansion are presented for the finite integration technique over generic polyhedral grids for 3D magnetostatic linear problems. Such algorithms outperform Monte Carlo methods (when the number of random parameters is small), both in accuracy and efficiency. A novel algorithm for the intrusive approach is also provided, by which the intrusive approach becomes less computationally expensive than the nonintrusive approach. Validation is performed by solving a magnetic circuit where the reluctivity is uncertain.

Uncertainty Quantification Using Sparse Approximation for Models With a High Number of Parameters: Application to a Magnetoelectric Sensor

To face the curse of dimensionality met in uncertainty quantification problems when the model has a high number of random parameters, methods based on sparse approximation, such as the least angle regression (LAR) method, should be used. In this paper, we propose an extension of the LAR method, and we apply it to quantify the impact of uncertainties on the characteristics of the material on the magnetoelectric sensor performance. The sensor response is represented by a 2-D finite-element model with ten random parameters. A global sensitivity analysis is carried out in order to determine the most influential parameters.

Efficient Statistical Extraction of the Per-Unit-Length Capacitance and Inductance Matrices of Cables with Random Parameters

Cable bundles often exhibit random parameter variations due to uncertain or uncontrollable physical properties and wire positioning. Efficient tools, based on the so-called polynomial chaos, exist to rapidly assess the impact of such variations on the per-unit-length capacitance and inductance matrices, and on the pertinent cable response. Nevertheless, the state-of-the-art method for the statistical extraction of the per-unit-length capacitance and inductance matrices of cables suffers from several inefficiencies that hinder its applicability to large problems, in terms of number of random parameters and/or conductors. This paper presents an improved methodology that overcomes the aforementioned limitations by exploiting a recently-published, alternative approach to generate the pertinent polynomial chaos system of equations. A sparse and decoupled system is obtained that provides remarkable benefits in terms of speed, memory consumption and problem size that can be dealt with. The technique is thoroughly validated through the statistical analysis of two canonical structures, i.e. a ribbon cable and a shielded cable with random geometry and position.

Hybrid Stochastic Finite Element Method for Mechanical Vibration Problems

We present and analyze a new hybrid stochastic finite element method for solving eigenmodes of structures with random geometry and random elastic modulus. The fundamental assumption is that the smallest eigenpair is well defined over the whole stochastic parameter space. The geometric uncertainty is resolved using collocation and random material models using Galerkin method at each collocation point. The response statistics, expectation and variance of the smallest eigenmode, are computed in numerical experiments. The hybrid approach is superior to alternatives in practical cases where the number of random parameters used to describe geometric uncertainty is much smaller than that of the material models.

Seismic Response Analysis of Reinforced Concrete Frame Structures Considering the Variation of Parameters

The orthogonal polynomial expansion method expression of stochastic structure was deduced. Then, based on orthogonal polynomial expansion method, taking a 20-storey reinforced concrete frame structure as an example, the impact of the randomness of structural parameters on time history response was researched. Meanwhile, in order to verify the correctness of analysis program, the calculation results of orthogonal polynomial expansion method were compared with the Monte-Carlo method which based on Newmark integral. The results show that it can get relatively accurate results when the number of terms of the orthogonal polynomial is 5. Structural mass and stiffness have a greater impact on the structural dynamic response. And the greater number of random parameters, the greater the impact on structural dynamic response.

Highly Efficient Designs to Handle the Incorrect Specification of Linear Mixed Models

We apply a maximin criterion to examine the relative efficiency of several D q -optimal designs for a family of linear mixed models. Incorrect specifications of the order of the polynomial, size of the autocorrelation parameter, number of random parameters, and the correlation between random intercept and random slope are investigated. The results of our study allow us to draw the following conclusions: (1) the maximin D q -optimal design encountered appears to be highly efficient; (2) the variation of the minimum relative efficiencies of D q -optimal designs of the family of linear mixed models that were studied decreases as the order of the polynomial increases; (3) the effect of the autocorrelation parameter on the relative efficiencies of D q -optimal designs is the largest for first-degree polynomials; and (4) the relative efficiency of the equidistant design is lower than that of the maximin value and also lower than the reference value 0.85.

The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems

Dynamic response analysis of nonlinear structures involving random parameters has for a long time been an important and challenging problem. In recent years, the probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) of the dynamic response and its evolution, has been proposed and developed for nonlinear stochastic dynamical systems. In the probability density evolution method, the strategy of selecting representative points is of critical importance to the efficiency especially when the number of random parameters is large. Enlightened by Cantor’s set theory, a strategy of dimension-reduction via mapping is proposed in the present paper. In the strategy, a two-dimensional domain is firstly considered and discretized such that the grid points are assigned with probabilities associated to the joint PDF. These points are then sorted and set on a virtual line according to a certain principle. Partitioning the sorted points on the virtual line into a certain number of intervals and selecting one single point in each interval, the two random variables can be transformed to a single comprehensive random variable. The associated probability of each point is simultaneously transformed accordingly. In the case of multiple random parameters, the above dimension-reduction procedure from two to one could be used recursively such that the random vector is finally transformed to one single comprehensive random variable. Numerical examples are investigated, showing that the proposed method is of high efficiency and fair accuracy.

Second-order scenario approximation and refinement in optimization under uncertainty

When solving scenario-based stochastic programming problems, it is imperative that the employed solution methodology be based on some form of problem decomposition: mathematical, stochastic, or scenario decomposition. In particular, the scenario decomposition resulting from scenario approximations has perhaps the least tendency to be computationally tedious due to increases in the number of scenarios. Scenario approximations discussed in this paper utilize the second-moment information of the given scenarios to iteratively construct a (relatively) small number of representative scenarios that are used to derive bounding approximations on the stochastic program. While the sizes of these approximations grow only linearly in the number of random parameters, their refinement is performed by exploiting the behavior of the value function in the most effective manner. The implementation SMART discussed here demonstrates the aptness of the scheme for solving two-stage stochastic programs described with a large number of scenarios.

Galerkin Sampling Method for Stochastic Mechanics Problems

A numerical method for solving stochastic mechanics problems by representing the solution using a small number of random parameters is presented. In essence, the method is a Galerkin approximation in the sample space. The associated projection of the solution into the space of simple random variables reduces the stochastic problem to a set of deterministic problems. Alternatively, this method can be viewed as a modified—for computational efficiency—stratified sampling method. Several examples are considered involving the use of the Loeve‐Karhunen expansion for a stochastic field approximation. The examples deal with the determination of the natural frequencies and of the seismic response of a beam with random rigidity.

Stochastic extensions to necessary conditions in the theory of the calculus of variations

A stochastic version of the modified Young's generalized necessary conditions in the calculus of variations is given in this paper. It is based on an extension of Minkowski's theorem on the existence of a flat support for a convex figure, and it generalizes the necessary conditions of Weierstrass and Euler in the classical theory of the calculus of variations to a class of admissible curves which are expressible in terms of a finite number of random parameters. The integrals which we consider here are in the general Denjoy sense, except those with respect to the random parameters, which exist in the Lebesgue sense defined on a probability space. The importance of our stochastic analysis lies in the completion that a minimum not attained in the classical sense may be, and frequently is, attained in the stochastic case.