Approximation Theory
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Transcript of Approximation Theory
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Approximation Theory
Metric:
Complicated Function
SignalImage
Solution to PDE
Simple Function
PolynomialsSplines
Rational Func
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1. Linear space of
dimension n.
2. Nonlinear manifold of
dimension n.
3. Highly nonlinear: Highly
redundant dictionary.
Functions g chosen from:
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Examples:
(i) -- Alg. poly. of degree .
(ii) -- Trig. poly. of degree .
(iii) Splines -- piecewise poly. of degree r, pieces.
(iv) span ,CONS
0 1
Linear: 1, 2 , . . . , n,. . .
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Nonlinear: n dimensional manifold
(i) : Rational function .
(ii) Splines with
(iii) - term approximation
CONS
free knots.
0 1
pieces
IN
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Highly Nonlinear
, arbitrary,
Bases B1, B2, . . . Bm, . . .
Bj best n-term
choose best basis choose n-term approximation
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Main Question
Characterize
We shall restrict ourselves toapproximation by piecewise constants in what follows.
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Linear
Theorem (DeVore-Richards) Fix
Piecewise Constants
0 11/n
.
close to
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Theorem (DeVore-Richards)
, ,
for .
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Noninear
Theorem (Kahane)
.
Linear Nonlinear
Know (Petrushev)
.
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n - term
Haar Basis0 1
1
-1
Dyadic Interval
I0 1
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CONS
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Theorem (DeVore-Jawerth-Popov)
known.
Simple strategy:
Choose n terms where
largest.
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Lin
ear
Nonlin
ear
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ApplicationImage Compression
Piecewise constant function
(Haar)
Threshold
Problem: Need to encode positions.
Dominate Bits
Image
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Tree Approximation
Cohen-Dahmen-Daubechies-DeVore:
are almost the same requirements.
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Generate tree as follows:
1) Threshold:
2) Complete to Tree:
3) Encode the subtree:
1
00 0 0
0
0
0
0 1
1 1
1 1
(Each bit tells whether the child is in the tree.)
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• Progressive
• Universal
• Optimal
• Burn In
Features of Tree Encoder
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Encoder
P BB B B B BP P0 00 1 10 11 2 20 21 22 . . .
Pk = Position Bits of
B { bit bjk = j of , }
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Cohen-Dahmen-DeVore
Elliptic Equation
Wavelet transform gives
- positive definite.
- has decay properties.
CDD gives an adaptive algorithm
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Theorem If , then
using n computations the adaptive algorithm produces :
Theorem If , then the
adaptive algorithm produces :
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Error:
“Error Indicators”:
Refinement: Let be the smallest
set of indices such that
residual
Define new set
.