1 / 21

Approximation Theory

Simple Function. Complicated Function. Signal Image Solution to PDE. Polynomials Splines Rational Func. Metric:. Approximation Theory. 1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly

dyani
Download Presentation

Approximation Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Simple Function Complicated Function Signal Image Solution to PDE Polynomials Splines Rational Func Metric: Approximation Theory

  2. 1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly redundant dictionary. Functions g chosen from:

  3. Linear: , , , . . . . . . , n 2 1 Examples: (i) . -- Alg. poly. of degree (ii) -- Trig. poly. of degree . Splines -- piecewise poly. of degree r, pieces. (iii) 0 1 (iv) , span CONS

  4. 0 1 pieces I N Nonlinear : n dimensional manifold (i) . : Rational function (ii) free knots. Splines with (iii) - term approximation CONS

  5. arbitrary, , Bases B1, B2, . . . Bm, . . . best n-term Bj Highly Nonlinear choose best basis choose n-term approximation

  6. Characterize Main Question We shall restrict ourselves to approximation by piecewise constants in what follows.

  7. Linear Piecewise Constants 0 1/n 1 Theorem (DeVore-Richards) Fix . close to

  8. , , Theorem (DeVore-Richards) . for

  9. Nonlinear Linear . Noninear Theorem (Kahane) . Know (Petrushev)

  10. 1 0 I n - term 1 Haar Basis 1 0 -1 Dyadic Interval

  11. CONS

  12. Theorem (DeVore-Jawerth-Popov) known. Simple strategy: Choose n terms where largest.

  13. Nonlinear Linear

  14. Image Application Image Compression Piecewise constant function (Haar) Threshold Problem:Need to encode positions. Dominate Bits

  15. Cohen-Dahmen-Daubechies-DeVore: are almost the same requirements. Tree Approximation

  16. Generate tree as follows: 1) Threshold: 2) Complete to Tree: 1 1 3) Encode the subtree: 1 1 0 0 (Each bit tells whether the child is in the tree.) 1 1 0 0 0 0 0 0

  17. Progressive • Universal • Optimal • Burn In Features of Tree Encoder

  18. Position Bits of = P k B B P B B B P P B . . . 00 10 0 11 20 21 22 1 2 Encoder } B { bit b of = , jk j

  19. Cohen-Dahmen-DeVore Elliptic Equation Wavelet transform gives - positive definite. - has decay properties. CDDgives an adaptive algorithm

  20. , then the Theorem If adaptive algorithm produces : Theorem If , then using n computations the adaptive algorithm produces :

  21. residual Refinement:Let be the smallest Error: “Error Indicators”: set of indices such that . Define new set

More Related