Chemical Engineering Hand Written_Notes_Part_89

Chemical Engineering Hand Written_Notes_Part_89 - 180 5....

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180 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES where x R r and we have N experimental data sets { ( x ( i ) ,y i ): i =1 , ...... N } . De f ning the i th approximation error as (4.2) e i = y i b y i = y i m X j =1 θ j f j ( x ( i ) ) it is desired to choose a solution that minimizes the scalar quantity (4.3) min θ 1 ....θ m " f = N X i =1 e 2 i w i # where w i 0 are the weights associated with the individual measurements. These weights can be chosen to re F ect reliability of each experimental data. A relatively large weight w i can be selected for the experimental data set ( x ( i ) i ) that is more reliable and vice-versa. The above optimization problem can be expressed in compact vector-matrix notation as follows b y 1 = θ 1 f 1 ( x (1) )+ θ 2 f 2 ¡ x (1) ¢ + .......... + θ m f m ( x (1) ) (4.4) b y 2 = θ 1 f 1 ( x (2) θ 2 f 2 ¡ x (2) ¢ + .......... + θ m f m ( x (2) ) (4.5) ..... = ................................................. b y N = θ 1 f 1 ( x ( N ) θ 2 f 2 ¡ x ( N ) ¢ + .......... + θ m f m ( x ( N ) ) (4.6) f ning θ = h θ 1 θ 2 .... θ m i T R m (4.7) ϕ ( i ) = h f 1 ( x ( i ) ) f 2 ( x ( i ) ) .... f m ( x ( i ) ) i T R m (4.8) b y = h b y 1 b y 2 .... b y N i R N (4.9) and (4.10) Φ = f 1 ( x (1) ) ........ f m ( x (1) ) ...... ......... ......
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_89 - 180 5....

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