Numerical techniques stochastic approach

There are basically two approaches to the numerical solution of stochastic optimization problems by using Monte Carlo sampliug techniques. One approach is known as the stochastic approximation method and originated in Robbins and Monro (1951). The other method was discovered and rediscovered by different researchers and is known imder vruious names. 

In this paper the estimation problem is solved from a point of view which is essentially different from the stochastic approach of the conventional Kalman filter. The time-variant state estimation problem is re-phrased into a time-invariant parameter estimation problem, and least-squaxes techniques, which is derived under deterministic framwork, are then used. The advantage of using least-squaxes approach is that it does not require a pHori knowledge of the noise statistics, the initial values of the estimated state, and its corresponding error covariance. Close relations are found between the Kalman filter and the least-squaxes Alter. Finally, a numerical example is provided to illustrate the feasibility of the proposed method. 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

Numerical integration (sometimes referred to as solving or simulation) of differential equations, ordinary or partial, involves using a computer to obtain an approximate and discrete (in time and/or space) solution. In chemical kinetics, these differential equations are typically the rate laws that describe the time evolution of the system. One obtains results for the mean concentrations, without any information about the (typically very small) fluctuations that are inevitably present. Continuation and sensitivity analysis techniques enable one to extrapolate from a numerically obtained solution at one set of parameters (e.g., rate constants or initial concentrations) to the behavior of the system at other parameter values, without having to carry out a full numerical integration each time the parameters are changed. Other approaches, sometimes referred to collectively as stochastic methods (Gardiner, 1990), can provide data about fluctuations, but these require considerably more computational labor and are often impractical for models that include more than a few variables. 

Today the spectral profiles can be simulated for any motional regime by a numerical integration of the stochastic Liouville equation, as discussed in Chapter 12 and in the references therein. The noticeable improvement in the techniques of calculation of the magnetic parameters and their dependence on the solvent, and of the minimum energy conformation of the molecules, have opened the possibility of an integrated computational approach. Since it gives calculated spectral profiles completely determined by the molecular and physical properties of the radical and of the solvent at a given temperature, this method is a step forward in the direction of a sound interpretation of complex spectra. 

ABSTRACT The implantation of simple engineering techniques in time-dependent survival probability predictions of structural and technical systems as stochastic systems of events is discussed. A possibility to avoid complicated multidimensional integrations in a probabilistic safety analysis of stochastic systems with perfectly ductile components is based on the approaches of Transformed Conditional Probabilities (TCP) and Conventional Correlation Vectors (CCV). Dynamic (time-dependent) autosystems of extreme events of particular single and mixed ductile elements are treated as correlated components of static stochastic systems with m random failure modes. The unsophisticated and fairly exact prediction of the probability-based reliability of stochastic systems of events is demonstrated by numerical examples and histograms of their reliability indices. 

The direct MCS examined in section Response Variability of Stochastic Systems becomes inefficient for the solutimi of reliability problems where a large number of low-probability realizations in the failure domain must be produced. In order to alleviate this problem without deteriorating the accuracy of the solutimi, numerous variants of this approach have been developed. An important class of improved MCS is variance reduction techniques where the generation of samples of the basic random variables is controlled in an efficient way. 

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