First- and second-order reliability

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. 

Der Kiureghian A, Dakessian T (1998) Multiple design points in first and second-order reliability. Struct Saf 20(1) 37 9... 

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. 

In order to ensure reliable convergence, we generally employ VB optimization procedures that require first and second derivatives with respect to all of the variational parameters. Expressions for these derivatives are most easily derived by considering the first- and second-order changes in Evb with respect to the VB parameters defined by O andc . These maybe generated from combinations of cf Eqs. (3) and (4)) and the analogous operator for the structure space, E/i. For the first-order variations in orbitals and structure coefficients, we find ... 

The available reliable information on the rate coefficient of reaction (xvi) depends almost entirely on fast flow-discharge studies, and, with the exception of one recent shock tube result, direct measurements are confined to near 300 K. Even here there is a factor of two or three disagreement between authors. Results are summarized in Table 39. Uncertainties arise from two major causes. First, the second order gas phase decay of OH is accompanied by a first order heterogeneous decay. Optimization of the separation of the observed decay into its first and second order components is difficult, and this may account for some of the reported discrepancies [222]. Secondly, all the investigations have used the H + NO2 reaction as the source of OH, with the NO2 added downstream of the discharge. The relevant elementary steps causing the growth and decay of OH are then... 

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. 

The rates calculated with eqn 3.3 are listed in the fourth column, and rate coefficients for first- and second-order reaction in the fifth and sixth columns, respectively. It may seem difficult to distinguish between first and second order on the basis of the coefficients alone. This is because all data points except the first are in a narrow range of low conversion. Distinction between orders hinges on linearities of plots, and, over narrow ranges, even curved plots are not far from linear. However, the results for 12 mL s-1 flow rate are the least reliable because the coefficients are calculated from a small difference between large concentration values. 

It is concluded in Ref. 217 that an analysis of the order of the herringbone transition entirely based on thermodynamic quantities might be very misleading, because it was shown that most of the data can be rationalized in terms of both first- and second-order transitions. Thus, an analysis along these lines would require systems which are orders of magnitude larger than those available in Refs. 56 and 217, but only this would allow to reliably estimate the latent heat and the order parameter jump at the transition. 

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 ... 

There have been a number of determinations of the self-reaction rate constant for 804. In recent work, a first-order component of the decay became apparent earlier work was generally not of sufficient precision to extract reliably any first-order process. The existence of both first- and second-order components of the decay is due to the simultaneous occurrence of the reactions (30) and (31) ... 

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)). 

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) ... 

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. 

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