Local algebraic approximations Variants on Taylor series LocalGlobal approximations Variants on fudge factor Local algebraic approximations Linear Taylor series Intervening variables Transformed approximation ID: 610984
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Slide1
Local Single- and Multi-fidelity Approximations
Local algebraic approximationsVariants on Taylor seriesLocal multi-fidelity approximationsVariants on “fudge factor”Slide2
Local algebraic approximations
Linear Taylor seriesIntervening variablesTransformed approximationMost common: y
i
=1/x
iSlide3
Beam example
Tip displacement
Intervening variables
y
i
=1/IiSlide4
Reciprocal approximation
It is often useful to write the reciprocal approximation in terms of the original variables x instead of the reciprocals ySlide5
Conservative-convex approximation
At times we benefit from conservative approximations
All second derivatives of
g
C
are non-negativeConvex linearization obtained by applying the approximation to both objective and constraintsSlide6
Three-bar truss exampleSlide7
Stress constraint on member C
Stress in terms of areasStress constraintUsing non-dimensional variables
What assumption on stress
?Slide8
Results around (1,1)
.
x
1
x
2
g
gLgR
gC0.750.75
0.36350.27830.3635
0.38501.000.75
0.4227
0.3426
0.4493
0.4493
1.25
0.75
0.4205
0.4070
0.5008
0.5137
0.75
1.00
-0.0856
-0.0417
-0.0631
-0.0417
1.25
1.00
0.0619
0.0870
0.0741
0.0871
0.75
1.25
-0.3786
-0.3617
-0.3191
-0.2977
1.00
1.25
-0.2440
-0.2974
-0.2334
-0.23341.251.25-0.1819-0.2330-0.1819-0.1690Slide9
Problems local approximations
What are intervening variables? There are also cases when we use “intervening function” in order to improve the accuracy of a Taylor series approximation. Can you give an example? AnswersWhat is conservative about the conservative approximation? Why is that a plus? Why is it useful that it is convex?
AnswersSlide10
Local Approximations pros and cons
Derivative based local approximations have several advantagesDerivatives are often computationally inexpensiveDerivatives are needed anyhow for optimization algorithms
These approximations allow rigorous convergence proofs
There are some disadvantages too
They can have very small region of acceptable accuracy
They do not work well with noisy functionsSlide11
Global approximations
Can be based on more approximate mathematical modelCan be based on same mathematical model with coarser discretizationCan be based on fitting a meta-model (surrogate, response surface) to a number of simulations
Current jargon: Low-fidelity model
Pro and cons complement those of local approximations: Wider range, noise tolerance, but more expensive, and less amenable to math proofsSlide12
Local multi-fidelity approximations
Can use derivatives to combine the two models
The local multi-fidelity approximation matches value and slope of the high-fidelity function at
x
0.
Originally called global-local approximationSlide13
Multi-dimensional form
Straight forward extension to n variablesSlide14
Example
Approximating the sine function as a quadratic polynomialSlide15
Absolute and relative comparison
.Slide16
Python code for previous slide
# uses Anaconda importing namespace from #Numpy and MatPlotLibz0=0.1
y0=sin(pi*z0)
dy0=pi*cos(pi*z0)
yLF0=4*z0*(1-z0)
dyLF0=4-8*z0print('z0=',z0, 'y(z0)=','%.4f'% y0,'y_LF(z0)=',yLF0)print('dy(z0)=','%.4f'% dy0,'dy_G(z0)=',dyLF0)#Generate Z vector from 0.006 to
0.6 to limit region of high 1/zZ=linspace(0.006,0.6,100)#generate function and approximations vectors
Y=sin(pi*Z)YL=y0+dy0*(Z-z0)YLF=4*Z*(1-Z)beta0=y0/yLF0dbeta0=dy0/yLF0-y0*dyLF0/yLF0**2BETA=beta0+(Z-z0)*dbeta0YMF=BETA*YLF
figure()plot(Z,Y,label='y',color='black')plot(
Z,YL,label=r'$y_L$',color='blue')plot(Z,YLF,label=r'$y_{LF}$', color='green')plot(Z,YMF,label=
r'$y_{MF}$', color='red')xlabel('x')legend(prop={'size': 20})#plot approximations divided by functionsRat_L=YL/Y
Rat_LF
=YLF/Y
Rat_MF
=YMF/Y
figure()
plot(
Z,Rat_L,label
=r'$
y_L
/
y$',color
='blue')
plot(
Z,Rat_LF,label
=
r'$y
_{LF}/y$', color='green')
plot(
Z,Rat_MF,label
=
r'$y
_{MF}/y$', color='red')
xlabel
('x')
legend(prop={'size': 20})Slide17
Problems Multi-fidelity
Given the function y=sinx, compare the linear, reciprocal, and local multi-fidelity approximation about x0=p
/3
, where the low-fidelity approximation is
yLF
=2x/p. Plot to compare the three approximations in the interval (p/6, p
/2) Solution