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CHE338_Lecture#09

For reading

Reference Chapter Page

Chapra 13,23 351-366,653-668

Optimization and

Numerical differentiation

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Objectives

To introduce numerical optimization.

To introduce numerical differentiation.

To introduce matlab functions, fminbnd, fminsearch,

diff, and gradient.

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Optimization

x

f(x)

To determine the minima and maxima of a function

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One-dimensional optimization

x

f(x) Global optima

Local optima

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Golden-section search

x

f(x)

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Parabolic interpolation

x

f(x) A parabolic function

x1

x2 x3

)]()()[()]()()[(

)]()([)()]()([)(

2

1

12323212

12

2

3232

2

1224

xfxfxxxfxfxx

xfxfxxxfxfxxxx

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Matlab function: fminbnd

[xmin,fval]=fminbnd(fun,x1,x2);

To determine the minimum cost of the following bioreactor system

[A] [B]

= 1

1 2

0.6

+ 61

0.6

[A] [B]

[A]

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Matlab function: fminsearch

[xmin,fval]=fminsearch(f,x1,x2)

f=@(x) 2+x(1)-x(2)+2*x(1)^2+2*x(1)*x(2)+x(2)^2

[xopt,optima]=fminsearch(f,[-0.5,0.5])

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Matlab function: fmincon

[xmin,fval] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

[xopt,optima]=fmincon('mainf',[1,1],[],[],[],[],[],[2;1],'tankdesign')

function f=mainf(x)

mass=8000*((x(1)*pi*((x(2)/2+0.03)^2-

(x(2)/2)^2)+2*pi*(x(2)/2+0.03)^2*0.03));

lw=4*pi*(x(2)+0.03);

f=4.5*mass+20*lw;

end

function [c,ceq]=tankdesign(x)

c=[];

ceq=pi*x(1)*x(2)^2/4-0.8;

end

To determine the dimensions of a cylindrical tank for a liquid with the

lowest cost.

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)(2

)(3)(4)()(

)(2

)()(2)()()()(

)()()(2)(

)(

)(2

)()()()(

2

)()()()(

212

2

2

121

2

12

21

2

1

hOh

xfxfxfxif

hOhh

xfxfxf

h

xfxfxif

hOh

xfxfxfxf

hOhxf

h

xfxfxif

hxf

hxfxfxf

iii

iiiii

iiii

iii

iiii

Forward differentiation

High Accuracy Differentiation

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Forward differentiation

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Backward differentiation

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Centered differentiation

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To improve derivative estimates

Decrease the step size Use a higher-order formula Use Richardson extrapolation

Accurate approximation

Richardson extrapolation, uses two derivative

estimates to compute a third.

For centered difference approximations with O(h2). The application of this

formula yield a new derivative estimate of O(h4).

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Example 9.1 (Example 23.1 and 23.2)

2.125.05.015.01.0)( 234 xxxxxf

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For unequally spaced data

iiiiii

i

iiii

iii

iiii

iii

xxxx

xxxxf

xxxx

xxxxf

xxxx

xxxxfxf

111

11

11

11

111

11

2)(

2)(

2)()(

Use a second-order Lagrange interpolating polynomial

to estimate the first derivative.

x is the value at which you want to estimate the derivative.

One can estimate the derivative anywhere within the range prescribed

by three points

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Example 9.2 (Example 23.3)

Heat flux on the surface

(z=0):

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Matlab function: diff and gradient

From the matlab help,

diff(X), for a vector X

[FX,FY] = gradient(F) returns the numerical gradient of the matrix F. FX corresponds to dF/dx, the differences in x

(horizontal) direction. FY corresponds to dF/dy

[FX,FY] = gradient(F,H), where H is a scalar, uses H as

the spacing between points in each direction.

f =@(x) 0.2+25*x-200*x.^2+675*x.^3-900*x.^4+400*x.^5

From 0 to 1.