Adaptive wavelet collocation method for PDE's on the...

Transcript of Adaptive wavelet collocation method for PDE's on the...

Adaptive wavelet collocation method for PDEson the sphere

Mani Mehra

Department of Mathematics and Statistics

Mani Mehra McMaster University

Adaptive wavelet collocation method for PDEs on the sphere
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Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions

Collaborators

Nicholas Kevlahan (McMaster University)


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Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions


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Why general manifolds?

(e.g. Sphere)

Application of adaptive wavelet collocation method (AWCM)to the problems of geodesy, climatology, meteorology(Representative examples include forecasting the moisture andcloud water fields in numerical weather prediction).

Many PDEs arise from mean curvature flow, surface diffusionflow and Willmore flow on the sphere.


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(e.g. Sphere)




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(e.g. Sphere)




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(e.g. Sphere)




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Why wavelets?

High rate of data compression.

Fast O(N ) transform.

Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).

Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).


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Why wavelets?






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Why wavelets?






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Why wavelets?






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Why wavelets?






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Why wavelets on manifolds?

(e.g. spherical wavelets)

Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids

Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric

term which are unbounded near the poles.

To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.

Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.


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Wavelet multiresolution analysis of L2(S)

DefinitionMRA is characterized by the following axioms

V j V j+1 (subspaces are nested).

j=j= Vj = L2(S).

Each V j has a Riesz basis of scaling function {jk |k Kj}.

Define Wj = {jk(wavelets)|k M

j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .


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j=j= Vj = L2(S).





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j=j= Vj = L2(S).





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j=j= Vj = L2(S).





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j=j= Vj = L2(S).





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j=j= Vj = L2(S).





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Construction of spherical wavelets based on sphericaltriangular grids

The set of all vertices S j = {pjk S : pjk = p

j+12k |k K

j} andMj = Kj+1/Kj .

Level 0

Dyadic icosahedral triangulation of the sphereMani Mehra McMaster UniversityAdaptive wavelet collocation method for PDEs on the sphere
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Construction of spherical wavelets based on sphericaltriangular grids

The set of all vertices S j = {pjk S : pjk = p

j+12k |k K

j} andMj = Kj+1/Kj .

Level 1

Dyadic icosahedral triangulation of the sphereMani Mehra McMaster UniversityAdaptive wavelet collocation method for PDEs on the sphere
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Fast wavelet transform

e4 e3

e2e1

f1

f2

v1 v2

m

The members of the neighborhoodsused in wavelet bases (m Mj ,Km = {v1, v2, f1, f2, e1, e2, e3, e4}).

Analysis(j) :

m Mj : d jm = cj+1m

kKm

sjk,mc

jk ,

m Kj : c jk = cj+1k .

Synthesis(j) :

m Kj : c jk = cjk ,

m Mj : c j+1m = djm +

kKm

sjk,mc

jk .


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e4 e3

e2e1

f1

f2

v1 v2

m


Analysis(j) :

m Mj : d jm = cj+1m

kKm

sjk,mc

jk ,


Synthesis(j) :

m Kj : c jk = cjk ,


kKm

sjk,mc

jk .

For linear basis, sv1 = sv2 = 1/2


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e4 e3

e2e1

f1

f2

v1 v2

m


Analysis(j) :

m Mj : d jm = cj+1m

kKm

sjk,mc

jk ,


Synthesis(j) :

m Kj : c jk = cjk ,


kKm

sjk,mc

jk .

For Butterfly basis, sv1 = sv2 = 1/2,sf1 = sf2 = 1/8,se1 = se2 = se3 = 1/16


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e4 e3

e2e1

f1

f2

v1 v2

m


Analysis(j) :

m Mj : d jm = cj+1m

kKm

sjk,mc

jk ,

m Mj : c jk = cj+1k + s

jk,md

jm.

Synthesis(j) :

m Mj : c jk = cjk

kKm

sjk,md

jm,


kKm

sjk,mc

jk .


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj djm

jm(p)

Test function

1

0

1 10

11

0.5

0

0.5

1

yx

z

Wavelet locations xJk withoutcompression at J = 5, #K5 = 10242


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function

1

0

1 10

11

0.5

0

0.5

1

yx

zWavelet locations xJk at J = 5, = 105, N() = 995 and ratio

#K5

N() 10


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk withoutcompression at J = 6, #K6 = 40962


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk st J = 6, = 105, N() = 8175 and ratio

#K6

N() 5


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk withoutcompression at J = 7,

#K7 = 163842


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk at J = 7, = 105, N() = 20353 and ratio

#K7

N() 8


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk withoutcompression at J = 8,

#K8 = 655362


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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

djm

jm(p)

Test function Wavelet locations xJk at J = 8, = 105, N() = 64231 and ratio

#K8

N() 10


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Wavelet approximation estimates

Approximation error iscontrolled by the waveletthreshold ||uJ(p) uJ(p)|| c1

106

105

104

103

102

101

100

106

105

104

103

102

101

100

||u

J(p

)u

J (p

)||

LinearLinear liftedButterflyButterfly lifted

O()


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Wavelet approximation estimates

controls the total active gridpoints N() by the followingrelation N() c2

nk

Therefore, approximation erroris controlled by active gridpoints

||uJ(p) uJ(p)|| c3N() k

n

102

103

104

105

106

106

105

104

103

102

101

100

N()

||u

J(p

)u

J (p

)||


O(N()1)

O(N()2)


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Calculation of Laplace-Beltrami operator on adaptive grid

Su(pji ) =

1

AS(pji)

kN(i)cot i,k+cot i,k

2 [u(pjk) u(p

ji )]

where AS(pji ) is the area

of one-ringneighborhood regiongiven by AS(p

ji ) =

18

kN(i)(coti ,k +

cot i ,k)pjk p

ji

2.

pij

pkj

i,k

i,kp

k1j p

k+1j

AM

(pij)


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Su(pji ) =

1

AS(pji)


2 [u(pjk) u(p

ji )]

Convergence result for theLaplace-Beltrami operator ofuJ(p) for the test function

||SuJ(p) Su

J(p)|| c4

106

105

104

103

102

101

100

101

106

105

104

103

102

101

100

/||uJ(p)||

||

Su

J(p

)

Su

J (p

)||

/||

Su

J(p

)||

Butterfly

Butterfly lifted

O()


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Su(pji ) =

1

AS(pji)


2 [u(pjk) u(p

ji )]

Convergence result for theLaplace-Beltrami operator ofuJ(p) for the test function

||SuJ(p) Su

J(p)|| c4

c5N() k2

n

103

104

105

106

106

105

104

103

102

101

100

N()

||

Su

J(p

)

Su

J (p

)||

/||

Su

J(p

)||

ButterflyButterfly lifted

O(N()3/2)


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Calculation of other differential operators on adaptive grid

The flux term present in the spherical advection equation canbe expressed in the form of flux divergence and Jacobianoperators, .(Vu) = .(u) J(u, )

JS(u(pji ), (p

ji )) =

1

6AS(pji)

kN(i)(u(pjk) + u(p

ji ))((p

jk ) (p

ji ))

S .(u(pji ),S(p

ji )) =

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2

(u(pjk) + u(pji ))((p

jk ) (p

ji )).


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JS(u(pji ), (p

ji )) =

1

6AS(pji)

kN(i)(u(pjk) + u(p

ji ))((p

jk ) (p

ji ))

S .(u(pji ),S(p

ji )) =

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2


jk ) (p

ji )).


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JS(u(pji ), (p

ji )) =

1

6AS(pji)

kN(i)(u(pjk) + u(p

ji ))((p

jk ) (p

ji ))

S .(u(pji ),S(p

ji )) =

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2


jk ) (p

ji )).


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JS(u(pji ), (p

ji )) =

1

6AS(pji)

kN(i)(u(pjk) + u(p

ji ))((p

jk ) (p

ji ))

S .(u(pji ),S(p

ji )) =

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2


jk ) (p

ji )).


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Application of spherical wavelets to compress turbulence

Turbulence can be divided into two orthogonal parts: aorganized (coherent vortices), inhomogeneous, non-Gaussiancomponent and random noise (incoherent), homogeneous andGaussian component.

The coherent vortices must be resolved, but the noise may bemodeled (or neglected entirely).

The coherent vortices can be extracted using nonlinearwavelet filtering.

This is the concept of Coherent Vortex Simulation which wasdeveloped by Marie Farge, Kai Schneider and Nicholas Kevlahan inflat geometries.


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Vorticity function. Reconstructed with 40%significant wavelets.


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E (n, 0) = An/2

(n+n0)

100

101

102

107

106

105

104

103

102

k

E(k

)

E(k)E(k) u

Energy spectrum reconstructedwith 40% significant wavelets.

100

101

102

107

106

105

104

103

102

kE

(k)

E(k)E(k) u

Energy spectrum reconstructedwith 2% significant wavelets


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102

103

104

105

106

107

107

106

105

104

103

102

101

N()


Relation between and N() for thevorticity function.

20 40 60 80 10010

6

104

102

100

102

% of coefficients used

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f err

or


Rate of relative error as a function ofrate of significant coefficients

=N()100N(=0) .


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Diffusion equation

ut = Su + f

where f is localized source chosen such a way that the solution ofdiffusion equation is given by

u(, , t) = 2e

(0)2+(0)

2

(t+1)

Such equations arise in the application of Willmore flow, surfacediffusion flow etc.


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Diffusion Equation

Initial condition at t = 0 Adaptive grid at t = 0


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Diffusion Equation

Solution using AWCM att = 0.5

Adaptive grid at t = 0.5


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Diffusion Equation

The potential of adaptivealgorithm is measured bycompression coefficients

C =N( = 0)

N()

0 0.1 0.2 0.3 0.4 0.58

10

12

14

16

18

20

22

24

t

CThe compression coefficient C

as a function of time, = 105.


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Spherical advection equation

u

t+ V .Su = 0,

The nondivergent driving velocity field V = (v1, v2) for all times isgiven by

v1 = u0

(

cos() cos() + sin() cos(+3

2) sin()

)

,

v2 = u0 sin(+3

2) sin().


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u

t+ V .Su = 0,

The initial cosine bell test pattern to be advected is given by

u(, ) =

{

u02

(

1 + cos(rR

)

if r < R

0 if r R ,

where u0 = 1000m, radius R = a/3 and r = a arccos[sin(c) sin() + cos(c) cos() cos( c), which is the greatcircle distance between (, ) and the center (c , c) = (0, 0).Such equations arise typically in the context of numerical weatherforecast or in climatological studies, and they provide some of themost challenging and CPU time consuming problems in moderncomputational fluid dynamics.


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1 0.5 0 0.5 10.2

0

0.2

0.4

0.6

0.8

1

Analytic sol.AWCM

Solid body rotation of cosine bell using AWCM for = 105.


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Solution using AWCM Adapted grid for the solution.


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0 0.05 0.1 0.150

5

10

15

20

25

30

35

t

C

The compression coefficient Cas a function of time,

= 105.

0 0.05 0.1 0.1510

8

106

104

102

100

t

||u(p

,t)

uex(

p,t)|

| Time evolution of the

pointwise L error of thesolution.


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Conclusions

Adaptive wavelet collocation method:

Used for time evolution problems.

Dynamic adaptivity is necessary for atmospheric modeling.

Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.

Verified convergence result predicted in theory.


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Conclusions







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Conclusions







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Conclusions







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Conclusions







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Future directions

Implementation of wavelet bases based on optimal sphericaltriangulation.

Ref.:Discrete Laplace-Beltrami operator on sphere and optimalspherical triangulation, Int. J. Comp. Geometry Appl. 16 (1)(2006) 75-93.

Incorporation of appropriate time integration method inspherical domain which takes advantage of wavelet multileveldecomposition.

Ref.:An adaptive multilevel wavelet collocation method for ellipticproblems, J. Comp. Phys. 206 (2005) 412-431.

Application of AWCM to more realistic models.


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Future directions







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Future directions







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Future directions








MotivationAdaptive wavelet collocation method on the sphereNumerical simulationConclusions and future directions

Adaptive wavelet collocation method for PDE's on the...

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Transcript of Adaptive wavelet collocation method for PDE's on the...