Response surface computer-developed

Recall that the developed response surfaces, or surrogate models as we refer to them in the context of model optimization, replace the solutions of the differential equations. Once the surrogate models are developed, via the design of computer experiments described in the preceding section, we can turn to optimization. The objective function now takes the form... 

Also, when using a standard FE solver, the computational cost to evaluate all three objective functions for all modes is equal to the computational cost to evaluate one objective function for one mode. Because of the very low computational cost of a single evaluation of the response surface approximation function, the computational cost of a global optimisation on the response surface is feasible for most practical applications. These properties can be used to develop a very efficient optimisation algorithm for the full fuzzy FE algorithm (Munck et al, 2006). 

Another approach for designing second-order models to be used for process optimisation, without using higher-order factorial design for optimisation, is response surface design. Many different approaches have been developed, many of which require the use of specialised computer software to obtain a tractable solution. 

Metamodel or response surface-based methods perhaps provide the best balance between computational intensity and information about the partial variances due to input parameter imcertainties. In many cases, the development of an accurate metamodel can be achieved using a far smaller sample size than that required by FAST or Sobol s basic method. The metamodel is then used for calculating global sensitivity indices. In common with the Sobol method, HDMR, for example, is based on the analysis of variance. Where higher-order terms (>2) in the HDMR expansion are weak, global sensitivity indices can be achieved using a relatively small quasi-random sample even for large parameter systems. 

I apply these computational methods to various aspects of the Earth system, including the responses of ocean and atmosphere to the combustion of fossil fuels, the influence of biological activity on the variation of seawater composition between ocean basins, the oxidation-reduction balance of the deep sea, perturbations of the climate system and their effect on surface temperatures, carbon isotopes and the influence of fossil fuel combustion, the effect of evaporation on the composition of seawater, and diagenesis in carbonate sediments. These applications have not been fully developed as research studies rather, they are presented as potentially interesting applications of the computational methods. 

Access to powerful computers and to commercial partial-differential-equation (PDE) solvers has facilitated modeling of the impedance response of electrodes exhibiting distributions of reactivity. Use of these tools, coupled with development of localized impedance measurements, has introduced a renewed emphasis on the study of heterogenous surfaces. This coupling provides a nice example for the integration of experiment, modeling, and error analysis described in Chapter 23. 

Other applications that were recently demonstrated with photorefractive polymers include homodyne detection of ultrasonic surface displacements using two-wave mixing [115]. With the development of new photorefractive polymers with response times in the millisecond range, numerous optical processing techniques can be performed at video rates. Image amplification and novelty filtering at video rates were demonstrated recently [116]. All-optical processing techniques compete with computational methods. Therefore, it is important that photorefractive polymers exhibit faster response times in the future. 

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