First- and second-order reliability methods

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

Approximate Methods First and Second Order Reliability Methods (FORM and SORM). 

Examples of probabilistic response analysis using the mean-centred First-Order Second-Moment (FOSM) approximation, time-invariant (First- and Second-Order Reliability Methods, FORM and SORM) and time-variant (mean outcrossing rate computation) reliability analyses are provided to illustrate the methodology presented and its current capabilities and limitations. 

Der Kiureghian, A. 2005. First- and Second-Order Reliability Methods. In Nikolaidis, E., Ghiocel, D M. Singhal, S. (eds.) Engineering Design Reliability Handbook. CRC PressINC. 

Finally the expected value and variation of the performance must be evaluated. Some, as described below have developed the first- and second-order reliability methods for this task. The reliability index approach followed herein provides similar information but appears to be much more efficient. 

Equation 1 expresses the fact that the failure domain D is measured by means of probability measure. It is not easy to calculate Pf using Equation 1, therefore many techniques are developed in the literature. The well known approaches are the FORM/SORM (respectively, First Order Reliability Methods and Second Order Reliability Methods) that consists in using a transformation to change variables into an appropriate space where vector U = T X) is a Gaussian vector with uncorrelated components. In this space, the design point, , is determined. Around this point, Taylor expansion of the limit state function is performed at first order or second order respectively for FORM or SORM method (Madsen et al). In the case of FORM, the structure reliability index is calculated as ... 

In (Hasofer Lind 1974), Hasofer and Lind introduced the reliability index technique for calculating approximations of the desired integral with reduced computation costs. The reh-abUity index has been extensively used in the first and second order rehabiUty methods (FORM (Hohenbichler Rackwitz 1983) and SORM (Fiessler, Neumann, Rackwitz 1979)). 

These are generally called Second Order Reliability Methods, where the use of independent, near-Normal variables in reliability prediction generally come under the title First Order Reliability Methods (Kjerengtroen and Comer, 1996). For economy and speed in the calculation, however, the use of First Order Reliability Methods still dominates presently. 

Experience in a variety of applications of the C ASSCF method has shown it to be a valuable tool for obtaining good zeroth-order approximations to the wavefunctions. Attempts have been made to extend the treatment to include also the most important dynamical correlation effects. While this can be quite successful in some specific cases (see below for some examples), it is in general an impossible route. Dynamical correlation effects should preferably be included via multireference Cl calculations. It is then rarely necessary to perform very large CASSCF calculations. Degeneracy effects are most often described by a rather small set of active orbitals. On the other hand experience has also shown that it is important to use large basis sets including polarization functions in order to obtain reliable results. The CASSCF calculations will in such studies be dominated by the transformation step rather than by the Cl calculation. A mixture of first- and second-order procedures, as advocated above, is then probably the most economic alternative. 

An even more effective numerical method for calculating the magnetic properties (25), (28), (31), and (34), based on formal annihilation of the paramagnetic contribution to the current density all over the molecular domain (CTOCD-PZ), has been outlined in a series of papers. Extensive numerical tests document the reliability, simplicity, and accuracy of this numerical procedure. To build up the CTCXID-PZ computational scheme it is expedient to define a set of generalized transformation functions theoretical methods which have been examined in detail by Coriani et al. Unlike the DZ procedure, the CTOCD-PZ equations cannot, in general, be solved to obtain closed form expressions. Rather the functions employed to cancel the transverse paramagnetic contribution to current density are evaluated pointwise via conditions imposed on first- and second-order current densities (compare for equations 113 and 115) ... 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

In the past two decades, a significant effort has been made towards development of reliable computational techniques for calculations of nonlinear optical properties of molecules. This has been reflected in many methods for calculations of first-(jS) and second-order (y) hyperpolarizabilities implemented in widely available quantum-chemical packages. However, the purely resonant properties, like two-and three-photon absorptivities have been coded in only a few of them. Also the inclusion of the influence of environment on NLO properties made a significant step forward in comparisons of theoretical and experimental data for large organic systems. 

A very important point is that, contrary to methods based on a Hartree-Fock zero-order wave function, those rooted in the Kohn-Sham approach appear equally reliable for closed- and open-shell systems across the periodic table. Coupling the reliability of the results with the speed of computations and the availability of analytical first and second derivatives paves the route for the characterization of the most significant parts of complex potential energy surfaces retaining the cleaness and ease of interpretation of a single determinant formalism. This is at the heart of more dynamically based models of physico-chemical properties and reactivity. 

The evaluation of the integral in Eq. 1 can be computationally difficult some examples are as follows fx is often not well-defined because of the incompleteness of the statistical information available G(X) may have a nonlinear form the computation of the multifold integral can be very difficult if the number of tmcertain parameters is high. Various methods have been proposed for solving the integral form in Eq. 1. These approaches range from the classical moment methods for structural reliability (e.g., first-order second-moment reliability method) to the simulation-based approaches (i.e., Monte Carlo family of methods), and also the PEER approach, which is quite different compared to the other two techniques. In this entry, alternative methods for estimating the probability of failure are described. 

This section discusses a class of methods known as the first-order reliability methods to compute the probability of failure of structural systems. These methods are based on the first-order Taylor s series expansion of the performance function G(X). The first-method, known as the first-order second-moment (FOSM) method, focuses on approximating the mean and standard deviation of G and uses this information to compute Pf. Then, the FOSM method is extended to the advanced FOSM method in two steps first, the methodology is developed for the case where all the variables in X are Gaussian (normal) and, second, the methodology is extended to the general case of non-normal variables. 

The section discussed the use of first-order reliability methods in order to estimate the reliability of structures. First, the first-order second-moment (FOSM) method was presented and then extended to the advanced FOSM method. The concept of most probable point (MPP) was introduced. It was derived that the distance from the origin to the MPP, in standard normal space, is equal to the safety index or reliability index, denoted by ft. Information regarding the gradient at the MPP can be used to identify the sources of uncertainty that are significant contributors to the failure of the structure. 

This type of electrode is a particularly powerful analytical tool since by performing steady-state measurements alone, it can measure faster rate constants than any other method. For a second-order reaction, the RRDE can reliably and reproducibly determine rate constants as fast as 10 mol dm ) s, while the maximum first-order rate constant measurable with the RRDE is about 10 s . A further advantage of the RRDE is the way that steady-state currents are measured (see below), whereas other methods of determining such high values ofk require the measurement of transients. 

---