zachary j. grant [email protected] sidafa conde sconde ... · faced with discontinuous problems due...

Hybrid Trigonometric Polynomial Approximation


Zachary J. Grant [email protected] Conde [email protected]

Advisors: Dr. Sigal Gottlieb & Dr. Alfa HeryudonoUniversity of Massachusetts Dartmouth

Mathematics Department

June 11, 2010


Introduction

Abstract

Fourier Series approximations are well known for their spectralconvergence of data reconstructions on smooth and periodicfunctions. However, they fail to produce similar convergence whenfaced with discontinuous problems due to peculiar behavior nearthe discontinuities. Our work is to remedy this problem by using ahybrid method. In this method, polynomial approximation is usednear the discontinuity and Fourier approximations are used on theother regions. We present numerical differences between ourmethods and other previous methods applied to similar popularproblems.


Introduction

Outline

What is a Fourier Partial Sum?

What are some issues that we face when using this technique

What are Ideas to Resolve these issues.

Numerical Results

Conclusions/Summary


Introduction

Fourier Partial Sum

The Classic Fourier Sum says that we can take a function, f (x),and rewrite it as a summation of sine and cosine waves.

FN(x) =a0

2+

N�

k=1

ak cos(πkx) + bk sin(πkx)

ak =

�1

−1

f (x) cos(πkx)dx

bk =

�1

−1

f (x) sin(πkx)dx

This reprojection works out great given the conditions that F (x) isboth smooth and periodic, resulting in high rate of convergence.

maxa≤x≤b|f (x)− FN(x)| ≤ e−N


Introduction

Fourier Partial Sum

Fourier Partial Sum


Introduction

Ideas to Resolve Gibbs Phenomenon

Filtering

There are two stages for providing information about a function fora reconstruction:

Storage: The Projection into Fourier Space, ”the Coefficients”

Retrieval: Going from Fourier Space back into Real Space,”The Summation”

The coefficients f̂k decays at a faster rate for continuous/periodicproblems than the cases where Gibbs Phenomenon is present.

Continuous: f̂k ≈ Ae−bk

Discontinuous: f̂k ≈ 1

k


Introduction


With this idea of wanting to increase the convergence we can thinkto multiply our coefficients,f̂k , by a function σ( k

N ) such that itimposes the speedy convergence on the coefficients. This cansimply be done in the retrieval stage.

F σN(x) =

N�

k=−N

σ(k

N)f̂ke ikx


Introduction


Window Functions

Window functions works just as the name suggests, we only wantto look at only a certain part of a function and ignore all the otherdata thats not in our desired scope.Using the following idea:

f (x) =f (x) ∗ w(x)

w(x)

f (x) ≈ FN(x)

FN(x) =N�

k=−N

f̂ke ikx


Introduction


Window Cont.

Now from combining above we can have a windowed Fourierre-projection.

FwN (x) ≈ 1

w(x)

N�

k=−N

ˆf wk e ikx

ˆf wk =

1

2π

�2π

0

w(x) ∗ f (x)e−ikxdx

Where w(x) is our windowing function which we impose thefollowing conditions:

w(x) =

�1 : x ∈ [a, b] ⊂ [0, 2π]� : x /∈ [a, b]

window cont.Instead of just cutting off the subinterval [a, b] ⊂ [0, 2π] we wantto nicely dissect the interval.

w(x) = e(− (x−π)

π

−2λ)


Introduction


F versus Windowed F


Introduction


Window Cont.


Introduction


Hybrid Fourier Polynomial Approximation

We showed already that when using a subinterval [a, b] ⊂ [−1, 1]that is free of the discontinuity we can have an approximation thatis not affected by Gibbs Phenomenon . To try and gain even moreaccuracy instead of ignoring the area outside our subinterval wedecided to treat them as separate entities.

F (x) =

Pn(x) : −1 ≤ x < −SFN(x) :≤ x ≤ SPn(x) : S ≤ x < −1

We still used the Fourier Partial Sum over a majority of thedomain, however in our smaller subdomains which contain theshock we used a more flexible polynomial approximation. Thismethod is introduce to eliminate Gibbs Phenomenon indiscontinuous function.


Introduction



Conclusions/Future Work

Conclusion/Future Work

Many of the techniques in [1, 2, 5, 7, 8] requires knowing thelocation of the discontinuity for best result. We will look at someedge detection techniques and integrate it with the differentmethods and compare the results. Localization of the discontinuitycan be obtained from the coefficients f̂k [1]. We will explore thisidea to improve the hybrid method above and analyze the error.We also plan on projecting the Fourier reconstruction onto theGegenbauer polynomial as done in [1, 7]. We will also try toreproject the hybrid method, where no Gibbs Phenomenon exists,on the same polynomial and see what happens.We will study the Aliasing-Error and see how each method affectsit.



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zachary j. grant [email protected] sidafa conde sconde ... · faced with discontinuous problems due...

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