/

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

Download Presentation from below link

Download Presentation The PPT/PDF document "Local and Local-Global Approximations" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

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