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Chemical Engineering Hand Written_Notes_Part_101

Chemical Engineering Hand Written_Notes_Part_101 - 204 5...

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204 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES The fi rst two approaches use the linear least square formulation as basis while the nonlinear programming approaches is a separate class of algorithms. 8.1. Weighted Least Square For Analytically Linearizable Models. Many nonlinear-in-parameter forms can be converted into linear-in-parameter forms by means of some linearizing transformations. In such cases, it appears that we can apply linear least square to obtain parameter estimates. However, this may not minimize e T We as explained below Consider linearizable form of model (8.7) b y = θ 0 [ f 1 ( x )] θ 1 [ f 2 ( x )] θ 2 · · · · · · [ f m ( x )] θ m which can be transformed as (8.8) ln b y = ln θ 1 + θ 2 ln [ f 1 ( x )] + θ 3 ln [ f 2 ( x )] · · · θ m ln [ f m ( x )] Now, parameter set that minimizes (8.9) e Ψ = N X i =1 ( ln y i ln b y i ) 2 = N X i =1 ( e i ) 2 may not minimize (8.10) Ψ = N X i =1 ( y i b y i ) 2 = N X i =1 ( e i ) 2 A rigorous approach to avoid this problem is to use nonlinear programming of Gauss-Newton approach. However, it is often possible to use weighted coe
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