Local algebraic approximations Variants on Taylor series LocalGlobal approximations Variants on fudge factor Local algebraic approximations Linear Taylor series Intervening variables Transformed approximation ID: 463177
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Slide1
Local and Local-Global Approximations
Local algebraic approximationsVariants on Taylor seriesLocal-Global approximationsVariants on “fudge factor”Slide2
Local algebraic approximations
Linear Taylor seriesIntervening variablesTransformed approximation
Most common:
y
i=1/xiSlide3
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 constraint
Using non-dimensional variables
What assumption on stress?Slide8
Results around (1,1)
.
x
1
x
2ggL
gRgC
0.750.750.36350.27830.3635
0.38501.000.750.4227
0.34260.44930.44931.25
0.750.42050.40700.5008
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
-0.2334
1.25
1.25
-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? Answers
What 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 discretization
Can be based on fitting a meta-model (surrogate, response surface) to a number of simulations
Pro and cons complement those of local approximations: Wider range, noise tolerance, but more expensive, and less amenable to math proofsSlide12
Combining local and global approximations
Can use derivatives to combine the two models
The combined approximation matches the value and slope at
x
0.Slide13
Example
Approximating the sine function as a quadratic polynomialSlide14
Overall comparison
.Slide15
Without linear
.Slide16
Problems local-global
Given the function y=sinx, compare the linear, reciprocal, and global local approximation about x0=
p
/3
, where the global approximation is yS
=2x/p