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Local and Local-Global Approximations

Local and Local-Global Approximations. Local algebraic approximations Variants on Taylor series Local-Global approximations Variants on “fudge factor”. Local algebraic approximations. Linear Taylor series Intervening variables Transformed approximation Most common: y i =1/x i.

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Local and Local-Global Approximations

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  1. Local and Local-Global Approximations • Local algebraic approximations • Variants on Taylor series • Local-Global approximations • Variants on “fudge factor”

  2. Local algebraic approximations • Linear Taylor series • Intervening variables • Transformed approximation • Most common: yi=1/xi

  3. Beam example • Tip displacement • Intervening variables yi=1/Ii

  4. Reciprocal approximation • It is often useful to write the reciprocal approximation in terms of the original variables x instead of the reciprocals y

  5. Conservative-convex approximation • At times we benefit from conservative approximations • All second derivatives of gCare non-negative • Convex linearization obtained by applying the approximation to both objective and constraints

  6. Three-bar truss example

  7. Stress constraint on member C • Stress in terms of areas • Stress constraint • Using non-dimensional variables • What assumption on stress?

  8. Results around (1,1) .

  9. 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? • What is conservative about the conservative approximation? Why is that a plus? Why is it useful that it is convex?

  10. Local Approximations pros and cons • Derivative based local approximations have several advantages • Derivatives are often computationally inexpensive • Derivatives 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 functions

  11. Global approximations • Can be based on more approximate mathematical model • Can 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 proofs

  12. Combining local and global approximations • Can use derivatives to combine the two models • The combined approximation matches the value and slope at x0.

  13. Example • Approximating the sine function as a quadratic polynomial

  14. Overall comparison .

  15. Without linear .

  16. Problems local-global • For cantilever beam (Ex. 6.1.1) use a global approximation of a uniform beam with average moment of inertia. The exact solution can be simplified to • Given the function y=sinx • What is the linear approximation about x=0? • What is the reciprocal approximation about x=pi/2? • What is the global-local approximation about x=0, given the global approximation y=2x/pi?

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