What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Some Thoughts on Guaranteed Function
Approximation Satisfying Relative Error
Fred J. Hickernell
Department of Applied Mathematics, Illinois Institute of Technology
[email protected] mypages.iit.edu/~hickernell
Joint work with Yuhan Ding (IIT) and
Henryk Woźniakowski (Columbia U & U Warsaw)
October 16, 2014
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Function Approximation with Adaptive Linear Splines
Given data px0 , y0 q, . . . , pxn , yn q with y “ f pxi q for f : r0, 1s Ñ R,
Find An pf q “ φpy0 , . . . , yn q : r0, 1s Ñ R such that kf ´ An pf qk8 is small.
The linear spline is given by
„ ˆ ˙
ˆ
˙

i
i`1
pi ` 1 ´ nxq ` f
pnx ´ iq
An pf qpxq :“ f
n
n
i
i`1
for ď x ď
.
n
n
We know that
kf ´ An pf qk8 ď
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kf 2 k8
8n2
for f P W 2,8
Function Approximation w/ Relative Error
(Clancy et al., 2014).
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Satisfying Error Tolerances for Balls
kf 2 k8
for f P W 2,8 .
8n2
Absolute error tolerances: the computational cost is bounded by f 2 8 :
kf ´ An pf qk8 ď
c
mintn : kf ´ An pf qk8 ď εa @f P Bσ u “
σ
,
8εa
Bσ :“ tf P W 2,8 such that f 2 8 ď σu.
Hybrid error tolerances, the computational cost is the same as for the
absolute error tolerance:
c
σ
mintn : kf ´ An pf qk8 ď maxpεa , εr kf k8 q @f P Bσ u —
.
εa
Relative error tolerances, the computational cost is infinite:
mintn : kf ´ An pf qk8 ď εr kf k8 q @f P Bσ u “ 8
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Why a Relative Error Tolerance Doesn’t Help for Balls
Let Ãn be any algorithm using n data. Let ξ ď ζ be the consecutive data
sites spaced furthest apart. Define
1 “
4pζ ´ ξq2 ` p4x ´ 2ξ ´ 2ζq2
32
`p4x ´ ξ ´ 3ζq |4x ´ ξ ´ 3ζ| ´ p4x ´ 3ξ ´ ζq |4x ´ 3ξ ´ ζ|s
fbump pxq :“
Ãn pfbump q “ 0,
2 f
bump 8 “ 1,
kfbump k8 “
1
pζ ´ ξq2
ě
16
16pn ` 1q2
! )
min n : f ´ Ãn pf q ď maxpεa , εr kf k8 q @f P Bσ
8
! )
ě min n : σfbump ´ Ãn pσfbump q ď maxpεa , εr kσfbump k8 q
8
* c
"
σ
σ
ě min n :
ď εa —
2
16pn ` 1q
εa
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Satisfying Error Tolerances for Cones
Clancy et al. (2014) developed a way of choosing n based
on function data
to ensure that kf ´ An pf qk8 ď εa without knowing f 2 8 . Let
Cτ :“ tf P W 2,8 : f 2 8 ď τ f 1 ´ f p1q ` f p0q8 u.
By noting that for all f P W 2,8 ,
2
1
f ´ f p1q ` f p0q ´ An pf q1 ´ f p1q ` f p0q ď kf k8 ,
8
8
2n
it may be shown that
1
2
f ď τ kAn pf q ´ f p1q ` f p0qk8 ÐÝ data-based
8
1 ´ τ {p2nq
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Satisfying Error Tolerances for Cones
Clancy et al. (2014)’s algorithm chooses n to satisfy
kf ´ An pf qk8 ď
kf 2 k8
τ kAn pf q1 ´ f p1q ` f p0qk8
ď εa
ď
8n2
4np2n ´ τ q
looooooooooooooooomooooooooooooooooon
data-based
The computational cost for the absolute error tolerance is
c
mintn : kf ´ An pf qk8 ď εa @f P Cτ X Bσ u —
σ
.
εa
Hybrid error tolerances, the computational cost is unknown:
mintn : kf ´ An pf qk8 ď maxpεa , εr kf k8 q @f Cτ P Bσ u —?
What about relative error tolerances (εa “ 0)?
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Bounding Weaker Norms in Terms of Stronger Ones
Let g “ f ´
q, and note that g 1 is continuous. Let ξ be chosen such
1 pf
A
1
1
that g pξq “ g 8 . Also, define ζ “ ξ ` 1{τ or ζ “ ξ ´ 1{τ , whichever
falls inside r0, 1s. It follows from integration by parts and the triangle
inequality
ż ξ
1
2 kgk8 ě |gpξq ´ gpζq| “ g pxq dx
ζ
żξ
1
ξ
2
“ g pxqpx ´ ζq ζ ´
g pxqpx ´ ζq dx
ζ
1 ě g 1 pξq |ξ ´ ζ| ´ g 2 8 |ξ ´ ζ|2
2
1
1 1 1
1
ě g 8 ´ ˆ τ g 1 8 ˆ 2 “ g 1 8
τ
2
τ
2τ
Thus
kf k8 ě
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1
1 f 1 ´ A1 pf q1 ,
kf ´ A1 pf qk8 ě
8
2
8τ
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Adaptive Hybrid Error Tolerances
Now we choose n to satisfy
kf ´ An pf qk8 ď
kf 2 k8
τ kAn pf q1 ´ f p1q ` f p0qk8
ď
8n2
4np2n ´ τ q
looooooooooooooooomooooooooooooooooon
data-based
ˆ
εr kAn pf q1 ´ A1 pf q1 k8
ď max εa ,
8τ
˙
ˆ
˙
εr kf 1 ´ A1 pf q1 k8
ď max εa ,
8τ
ď maxpεa , εr kf k8 q
and get
mintn : kf ´ An pf qk8 ď maxpεa , εr kf k8 q @f P Cτ X Bσ u
d
ˆ
˙
σ 1
,
— min
εa εr
So either εa or εr positive gives bounded computational cost.
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
Next We Consider W r`1,8
Use piecewise rth degree polynomials to approximate function.
Maybe consider tensor product for d-variate functions.
This idea does not work for integration problems. Why?
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What We Know About Linear Splines
New Ideas for Relative Error Tolerances
Higher Order
References
References
Clancy, N., Y. Ding, C. Hamilton, F. J. H., and Y. Zhang. 2014. The cost of
deterministic, adaptive, automatic algorithms: Cones, not balls, J. Complexity 30,
21–45.
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