Modified Fourier and Spectral Method

7/30/2019 Modified Fourier and Spectral Method

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Modified Fourier Series and Spectral MethodsBen Adcock

[email protected]

DAMTP, University of Cambridge

Modified Fourier Series and Spectral Methods p.1/24


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Modified Fourier Series

Modified Fourier Basis (Iserles & Nrsett 2006):

S = {cos nx : n 0} {sin(n

1

2)x : n 1}.

The basis functions are eigenfunctions of the Laplace operator on

[1, 1] with zero Neumann boundary conditions.



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Modified Fourier Series

Modified Fourier Basis (Iserles & Nrsett 2006):

S = {cos nx : n 0} {sin(n

1

2)x : n 1}.

The basis functions are eigenfunctions of the Laplace operator on

[1, 1] with zero Neumann boundary conditions.

Orthonormal basis of L2[1, 1]:

FN[f](x) =1

2fC0 +

N

n=1

fCn cos nx + fSn sin(n

12)x f(x),

where f L2[1, 1] and

fCn = 1

1

f(x)cos nx dx, fSn = 1

1

f(x) sin(n 12)x dx.



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Asymptotic Formulae

Suppose f C[1, 1] is non-periodic.



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Asymptotic Formulae


Integrating by parts

fCn (1)n

k=0

(1)k

(n)2(k+1)

f(2k+1)(1) f(2k+1)(1)

,

fSn (1)n+1k=0

(1)k

((n 12 ))2(k+1)

f(2k+1)(1) + f(2k+1)(1)

.

In particular fCn , fSn O(n

2).



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Asymptotic Formulae


Integrating by parts

fCn (1)n

k=0

(1)k

(n)2(k+1)

f(2k+1)(1) f(2k+1)(1)

,

fSn (1)n+1k=0

(1)k

((n 12 ))2(k+1)

f(2k+1)(1) + f(2k+1)(1)

.

In particular fCn , fSn O(n

2).

Compare with conventional Fourier series

fDn = 1

1

f(x)sin nx dx O(n1).



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Convergence Properties

For f C2[1, 1] and f(2) of bounded variation

f(x) FN[f](x) O(N2

), x (, ) (1, 1),f(1) FN[f](1) O(N

1),

(S. Olver 2007).



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Convergence Properties

For f C2[1, 1] and f(2) of bounded variation

f(x) FN[f](x) O(N2

), x (, ) (1, 1),f(1) FN[f](1) O(N

1),

(S. Olver 2007).

If p P2r interpolates the first r odd derivatives of f at x = 1

then

FN[f p](x) +p(x) f(x)

at the increased rate of O(N2r2) for x (1, 1) and O(N2r1)

for x = 1 polynomial subtraction.



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Rapid Evaluation of Coefficients

Dont use the FFT; use Filon-type quadrature instead.



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Suppose 1 = c1 < c2 < ... < c = 1 are given quadrature nodes

and a polynomial such that

(2i)(ck) = f(2i+1)(ck), i = 0,...,mk 1, k = 1, 2,...,.

Then, if p(x) = f(0) +x0 (x

)dx

, we approximate the modifiedFourier coefficients by

fCn 1

1

p(x)cos nx dx, fSn 1

1

p(x) sin(n 1

2

)x dx

which may be calculated explicitly. The asymptotic error in doing

so is O(n2s2) where s = min{m1, m}.



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Suppose 1 = c1 < c2 < ... < c = 1 are given quadrature nodes

and a polynomial such that

(2i)(ck) = f(2i+1)(ck), i = 0,...,mk 1, k = 1, 2,...,.

Then, if p(x) = f(0) +x0 (x

)dx

, we approximate the modifiedFourier coefficients by

fCn 1

1

p(x)cos nx dx, fSn 1

1

p(x) sin(n 1

2

)x dx

which may be calculated explicitly. The asymptotic error in doing

so is O(n2s2) where s = min{m1, m}.

The first N coefficients may be calculated in O(N) operations, as

opposed to O(Nlog N) for the FFT.Modified Fourier Series and Spectral Methods p.5/24


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A Galerkin method for Neumann

boundary value problems

A model problem:

L[u] = uxx

+ aux

+ bu = f, x [1, 1], ux

(1) = ux

(1) = 0.



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A model problem:

L[u] = uxx

+ aux

+ bu = f, x [1, 1], ux

(1) = ux

(1) = 0.

We seek a solution uN SN of the form

uN(x) =1

2

aC0 +N

n=1

aCn cos nx + aSn sin(n

1

2

)x,

satisfying Galerkins equations

(L[uN], ) = (f, ) SN.



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A model problem:

L[u] = uxx

+ aux

+ bu = f, x [1, 1], ux

(1) = ux

(1) = 0.

We seek a solution uN SN of the form

uN(x) =1

2

aC0 +N

n=1

aCn cos nx + aSn sin(n

12 )x,

satisfying Galerkins equations

(L[uN], ) = (f, ) SN.

LaxMilgram error estimate:

u uNH1

inf

SNu H1 ,

where and are the constants of continuity and coercivity of L.


G


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FN[u] is the best approximation to u in L2[1, 1], hence also in

H1[1, 1], so the optimal estimate holds:

u uNH1

u FN[f]H1 O(N

5/2),

provided u Civ

[1, 1] and u(iv)

has bounded variation. Unsurprisingly, the convergence is not exponential.


A G l ki h d f N


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FN[u] is the best approximation to u in L2[1, 1], hence also in

H1[1, 1], so the optimal estimate holds:

u uNH1

u FN[f]H1 O(N

5/2),

provided u Civ

[1, 1] and u(iv)

has bounded variation. Unsurprisingly, the convergence is not exponential.

However, the constant is typically much lower than for Chebyshev

or Legendre spectral or collocation methods.



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Comparison

5 10 15 20N

-40

-30

-20

-10

log uniform error

uxx(x) + ux(x) + 2u(x) = e2x. Log uniform error log uN u for N = 1, ..., 20. Modified Fourier series in red, Chebyshev collocation in blue, Legendre spectral

(with weak imposition of the boundary conditions) in black.



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Uniform convergence

Numerical results show that the uniform error is O(N3). The

LaxMilgram theorem gives a nonoptimal estimate of O(N5/2).

20 40 60 80 100N

0.0049

0.0051

0.0052

0.0053

N3u

uN

uxx(x) + ux(x) + 2u(x) = x. Modified Fourier Galerkin approximation uN.

Scaled uniform error N

3

uN u

for N = 1, ..., 100.



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Uniform error estimates

The uniform convergence rate inferred from the LaxMilgram estimate

is non-optimal, however:

Theorem 1 The uniform erroru uN decays likeO(N3).

Sketch proof:

If eN = FN[u] uN we may show that eN is the Galerkin

approximation to L[v] = u FN[u] with zero Neumann boundaryconditions.



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Sketch proof:



The theorem is true provided we can show that the solution v to

L[v] = e

ix

with homogeneous Neumann boundary conditions isuniformly bounded and has uniformly bounded derivative for all .



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Sketch proof:




L[v] = e

ix


This can be shown by brute force or by some asymptotics.



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Sketch proof:




L[v] = e

ix


This can be shown by brute force or by some asymptotics.

Remark: unlike the case of function approximation, numericalresults indicate that the pointwise error is O(N3) for all x, not just

x = 1. Modified Fourier Series and Spectral Methods p.10/24


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Eigenvalues and condition number

Theorem 2 TheL2 condition number of the method isO(N2),

provided the operatorL is coercive.

This compares favourably with other methods: eg Chebyshev

methods have a condition number of O(N4).



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The eigenvalue ratio of the method is also O(N2) for large N.



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The eigenvalue ratio of the method is also O(N2) for large N. Unlike Chebyshev or Legendre methods, numerical results

indicate that there are no spurious large eigenvalues.



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Initial-boundary value problems

We consider the problem

ut + L[u] = f(x)g(t)

with homogeneous Neumann boundary conditions and initial condition

u(x, 0) = (x).

Galerkins equations gives a system of ODEs.



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u(x, 0) = (x).


A LaxMilgram estimate gives similar error estimates in finite time

intervals [0, T]:

T0

e(s)2H1 ds

1/2

CTN5/2

for some constant CT depending on T and u.



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u(x, 0) = (x).


A LaxMilgram estimate gives similar error estimates in finite time

intervals [0, T]:

T0

e(s)2H1 ds

1/2

CTN5/2

for some constant CT depending on T and u.

As in the stationary case we may derive an optimal L errorestimate.



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L2 stability

Theorem 3 The semidiscretization isL2 stable for allN, a andb.

Remarks:

The matrix A + A has a simple form:

1

2A + A =

DC aJ

aJT

DS

where DC and DS are diagonal and correspond to the discrete

operator uxx + bu on SN. J is the (N + 1) N matrix with

entries Jn,m = (1)m+n+1.



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L2 stability

Theorem 3 The semidiscretization isL2 stable for allN, a andb.

Remarks:

The matrix A + A has a simple form:

1

2A + A =

DC aJ

aJT

DS

where DC and DS are diagonal and correspond to the discrete

operator uxx + bu on SN. J is the (N + 1) N matrix with

entries Jn,m = (1)m+n+1.

The eigenvalues satisfy a simple relation:

( 0) = a

2

fN(), where fN() =

Nn=0

0

n

Nn=1

1

n .



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Implementation

Explicit timestepping routines require a less restrictive step size

than for other methods.



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Implementation



Semiimplicit schemes (dealing with the advection term uxexplicitly) are simple to implement because the matrix to be

inverted is diagonal. Such schemes are unconditionally stable

(Quarteroni & Valli 1994).



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Implementation



Semiimplicit schemes (dealing with the advection term uxexplicitly) are simple to implement because the matrix to be

inverted is diagonal. Such schemes are unconditionally stable

(Quarteroni & Valli 1994).

Waveform relaxation techniques may also be used: splitting the

Galerkin matrix A into diagonal and off-diagonal parts gives an

exponentially convergent iteration in finite time intervals provided

the operator L is coercive.



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Results

10 20 30 40t

410-7

510-7

610-7

710-7

810-7

910-7

110-6

relative uniform error

Modified Fourier Galerkin approximation for a = 1, b = 2 and N = 20 tou(x, t) = (1 + t)(sinhx x cosh 1) with f(x, t) given accordingly.

Relative uniform error 1t+1

u(, t) uN(, t) for 0 t 40.



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Results

10 20 30 40N

0.001

0.002

0.003

0.004

scaled uniform error

10 20 30 40N

0.001

0.002

0.003

0.004

scaled uniform error

Scaled uniform error N3u(, T) uN(, T) for the Galerkin approximation tothe problem ut + L[u] = f with solution

u(x, t) = sin(t + x) p(x, t),

where p(x, t) interpolates the Neumann boundary values of sin(t + x).

Left: T = 1 red, T = 2 blue.

Right: T = 5 red, T = 10 blue.



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Increased convergence

We seek to interpolate higher odd derivatives of the solution u at

x = 1 and apply the Galerkin method to an auxiliary problem where

the solution has vanishing higher derivatives. The LaxMilgram theorem guarantees a higher order of

convergence in that case.



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In general these derivatives are unknown.



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In general these derivatives are unknown. However, the relation L[u] = f gives a recurrence for the higher

derivatives of u in terms of u(1) and the derivatives of f. In

particular:

uxxx(1) = abu(1) (af(1) + fx(1)).

5xu(1) = ab(a2 + 2b)u(1)

(a3 + ab + b)f(1)

(a2

+ b)fx(1) + afxx(1)

.



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A PetrovGalerkin method

Suppose p0(x) and p1(x) obey homogeneous Neumann boundary

conditions.



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Suppose p0(x) and p1(x) obey homogeneous Neumann boundary

conditions.

We seek an approximation uN(x) of the form P(x) + vN(x), wherevN is a modified Fourier sum and

P(x) = A0p0(x) + A1p1(x)

that satisfies Galerkins equations

(L[uN], ) = (f, ), SN,

and the condition

(uN)xxx(1) = ab uN(1) (af(1) + fx(1)).



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This is a PetrovGalerkin method.



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Test functions are modified Fourier functions SN.



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The trial function uN XN where

XN = {v ZN : ab v(1) vxxx(1) = af(1) + fx(1)},

and ZN = SN + span{p0, p1}.



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However this method is sufficiently simple so that the LaxMilgram

type estimate (due to Necas and Babuka) reduces to

u uNH1 (1 + /) inf XN

u H1 .



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However this method is sufficiently simple so that the LaxMilgram

type estimate (due to Necas and Babuka) reduces to

u uNH1 (1 + /) inf XN

u H1 .

To prove H1 error estimates we need a projection HN : H1 XN.



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Optimal Projections

Theorem 4 Foru H1[1, 1] defineHN[u] byHN[u] XN and

FN[HN[u]] = FN[u],

thenu HN[u]H1 O(N9/2).

Remarks:

We require p0 (1)p1 (1) p0 (1)p1 (1) = 0.



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Optimal Projections


FN[HN[u]] = FN[u],


Remarks:

We require p0 (1)p1 (1) p0 (1)p1 (1) = 0.

Good choices are

p0(x) = cosh x 1

2x2 sinh1, p1(x) = sinh x x cosh1.



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Optimal Projections


FN[HN[u]] = FN[u],


Remarks:

We require p0 (1)p1 (1) p0 (1)p1 (1) = 0.

Good choices are

p0(x) = cosh x 1


Bad choices are p0(x) = cos(N + 1)x, p1(x) = sin(N +12)x.



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Optimal Projections


FN[HN[u]] = FN[u],


Remarks:

We require p0 (1)p1 (1) p0 (1)p1 (1) = 0.

Good choices are

p0(x) = cosh x 1


Bad choices are p0(x) = cos(N + 1)x, p1(x) = sin(N +12)x.

We cannot do any better than the above estimate.



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Increased Convergence

This method may be viewed as forcing the modified Fourier

coefficients of the residual L[uN] f to decay at an increased

rate.



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rate. However we write 2j+1x u in terms of u and f and its derivatives to

avoid the system becoming ill-conditioned for large N.



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Numerical results suggest the condition number remains O(N2

).



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). Using these methods we can approximate problems with other

boundary conditions. For the homogeneous Dirichlet problem we

have an error estimate of O(N3), and the eigenvalue ratio

remains O(N2).



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). Using these methods we can approximate problems with other

boundary conditions. For the homogeneous Dirichlet problem we

have an error estimate of O(N3), and the eigenvalue ratio

remains O(N2).

With a little care these methods can be adapted to

timedependent problems.



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Higher dimensions


L[u] = u + a.u + bu = f

on the unit square in R2 with zero Neumann boundary conditions.

The modified Fourier Galerkin method produces an error of

O(N3).



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Higher dimensions


L[u] = u + a.u + bu = f



O(N3).

Using a hyperbolic cross we only need O(Nlog N) terms.



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Higher dimensions


L[u] = u + a.u + bu = f



O(N3).


However we may construct a PetrovGalerkin method in a similar

manner as before to reduce the number of terms to O(N).



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Higher dimensions


L[u] = u + a.u + bu = f



O(N3).


However we may construct a PetrovGalerkin method in a similar

manner as before to reduce the number of terms to O(N). Unfortunately there is no obvious way to increase the

convergence rate further.



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Further problems and open areas

For general linear problems with a = a(x), b = b(x) these methods

are less attractive. There are pseudospectral methods based on

modified Fourier series, but the convergence rate is typicallyO(N2).



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There is no obvious technique to increase the convergence rate

arbitrarily in higher dimensions.



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On which other domains may we apply modified Fourier series?



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These methods perform worse than the standard spectral

methods for linear problems. What problems are they likely to

perform better for?



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These methods perform worse than the standard spectral

methods for linear problems. What problems are they likely to

perform better for?

Polyharmonic eigenfunctions offer a natural generalization of

modified Fourier series, (Iserles, Nrsett 2006). These may have

applications in 4th and higher order PDEs.



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References

References

[1] C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang, Spectral methods:

Fundamentals in single domains, Springer, 2006.

[2] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral methods for time-dependent

problems, CUP, 2007.

[3] A. Iserles & S. P. Nrsett, From high oscillation to rapid approximation I: Modified

Fourier Series, Technical report NA2006/05, DAMTP, University of Cambridge, 2006.[4] A. Iserles & S. P. Nrsett, From high oscillation to rapid approximation II: Expansions

in polyharmonic eigenfunctions, Technical report NA2006/07, DAMTP, University of

Cambridge, 2006.

[5] A. Iserles & S. P. Nrsett, From high oscillation to rapid approximation III: Multivariateexpansions, Technical report NA2007/01, DAMTP, University of Cambridge, 2007.

[6] S. Olver, On the convergence rate of modified Fourier series, Technical report

NA2007/02, DAMTP, University of Cambridge, 2007.

[7] A. Quarteroni & A. Valli, Numerical approximation of partial differential equations,S i V l 1994

Modified Fourier and Spectral Method

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