Second-order Taylor series

The Rational Function Optimization (RFO) expands the function in terms of a rational approximation instead of a straight second-order Taylor series (eq. (14.3)). 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

The function U = U(ft) can be approximated by the second-order Taylor series expansion at the point i.e.,... 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

We assume that the energy hypersurface E(x) is a unique single-valued function of the nuclear coordinates x. We further assume E(x) and its derivatives with respect to x are continuous. Under these conditions, the second order Taylor series for the energy hypersurface E(x) is well defined. 

A more efficient strategy would be to control both direction and distance. Consider a second-order Taylor series approximation to S(0) expanded around 0 ... 

It is important to remember that for nonlinear functions the function is first linearized using a Taylor series approximation and then the mean and variance are calculated based on the approximation. How good the approximation is depends on how nonlinear g is around 0 and on the size of the variance of 0. Better approximations to g(0) can be found through higher order approximations. For example, a second order Taylor series about 0 leads to... 

One type of approximation is the Laplacian approximation.1 Given a complex integral, Jf(x)dx,f(x) is reexpressed as exp [Ln(f(x)] = exp [g(x)]. g(x) can then be approximated using a second-order Taylor series approximation about the point x0... 

Hence, the second-order Taylor series approximation would be... 

Figure A.2 Plot of the function exp(xy) (top plot), its second-order Taylor series approximation (middle plot), and third-order Taylor-series approximation (bottom plot).

Furthermore, stresses were calculated as functions of strain and temperature. For each temperature, each component of stress was fit to a second order Taylor series expansion in terms of the strains, about the P = 1 atm reference volume Vq(T) at each specific temperature. Based on the stresses, the elastic moduli Cij and the Griineisen coefficients ji (i = 1,2,3) of the non-crystalline interlamellar phase were calculated using... 

In the partitioned rational function optimization (P-RFO) the energy is approximated by a rational function instead of a second-order Taylor series ... 

In this section, the URP (updated reference-point) method proposed originally for stochastic input is explained (Fujita and Takewaki 2011a). This method can be used as an efficient uncertainty analysis to obtain the robustness function a explained in the previous section. Since the URP method takes full advantage of an approximation of first- and second-order Taylor series expansion in the interval analysis, the formulation of Taylor series expansion in the interval analysis and the achievement of second-order Taylor series expansion proposed by Chen et al. (2009) are explained briefly. 

As a simple approximation, an objective function f using second-order Taylor series expansion with only diagonal elements can be rewritten as... 

From Eq. 7, the increment of the objective function can be evaluated by using first- and second-order Taylor series expansion as the sum of the increments of the objective function in each one-dimensional domain. The perturbation Afi(X) of the objective function by the variation of the -th interval parameter X, can be described as... 

Estimation of the Variation of the Objective Function by Second-Order Taylor Series Expansion... 

Second-order Taylor series approximation, (b) response surface method (RSM)... 

Application of the URP method to a base-isolated building model is shown in this section. The validity of the URP method using second-order Taylor series approximation (section Estimation of the Variation of the Objective Eunction by 2nd-Order... 

Figure 21 shows the upper bounds of the maximum drift of the base-isolation story compared for various methods (URP methods with second-order Taylor series approximation/with RSM and the Monte Carlo Simulation (MCS)). As explained before, the difference between the URP method with second-order Taylor series approximation and that with RSM is how to estimate the variation of the objective function. In the former one, the numerical sensitivities, i.e., the gradient vector and the Hessian matrix, of the objective function are needed. On the other hand, in the latter one, a kind of RSM is applied where appropriate response samplings are made and the gradient vector and the Hessian matrix are evaluated from the constmcted approximate function. 

Figure 23 presents the upper bounds of the maximum top-story floor acceleration by the interval analysis with various methods (URP methods with second-order Taylor series approx-imation/with RSM and the Monte Carlo simulation). It can be observed from Fig. 23 that the level of variability of the maximum value into the increasing side derived by the URP method with RSM is larger than that of second-order Taylor series approximation. In other words, while the URP method with RSM provides a definite upper bound for the Monte Carlo simulation, the URP method with second-order Taylor series approximation does not necessarily assure the upper bound. This may result from the difficulty... 

Fujita K, Takewaki I (2011a) An efficient methodology for robustness evaluation by advanced interval analysis using updated second-order Taylor series expansion. Eng Struct 33(12) 3299-3310... 

Another important search direction method is the Newton direction. This direction is derived from the second-order Taylor series. Methods that use the Newton direction have a fast rate of local convergence. Nevertheless, the main drawback is that it requires the explicit computation of the Hessian matrix (V /(ar)). 

Once Newton s method is close enough to the real solution for the second-order Taylor series approximation to be accurate, the sequence of estimates converges very rapidly (quadrati-caUy) to die solution. 

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