Applications of Wavelets to Partial Differential Equationsweb.iitd.ac.in/~mmehra/TIFR_talk.pdf · Applications of Wavelets to Partial Di erential Equations Mani Mehra Department of

Applications of Wavelets to Partial DifferentialEquations

Mani Mehra

Department of Mathematics and Statistics

Mani Mehra Applications of Wavelets to Partial Differential Equations 1/51

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Part 1: Wavelet-Taylor Galerkin methodPart 2: Adaptive wavelet collocation method

1 Part 1: Wavelet-Taylor Galerkin method for PDEs on flat geometriesMotivationWavelet-Taylor Galerkin methodConclusion and future plan

2 Part 2: Adaptive wavelet collocation method for PDEs on manifoldsMotivationAdaptive wavelet collocation method on the sphereConclusions and future directions

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MotivationWavelet-Taylor Galerkin methodConclusion and future plan

Collaborator

Dr. B. V. Rathish Kumar (Indian Institute of Technology)

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Why Wavelets?

Efficient way to compress the smooth data except in localizedregion

Example

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

Initial function and approximate initial function with the waveletthreshold = 103, constructed with 13 retained wavelet

coefficients out of original 28 = 256

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Why Wavelets?

Good approximation properties

Sparse representation of Calderon-Zygmond type operators

Easy to control wavelet properties (e.g. smoothness, betteraccuracy near sharp gradients).

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Why Wavelets?

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Why Wavelets?

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Why Taylor-Galerkin method?

Higher order time accurate method

Improved stability properties

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Why Taylor-Galerkin method?

Higher order time accurate method

Improved stability properties

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Why wavelet-Taylor Galerkin method?

Successive powers of time iteration matrix become sparserwith increasing time

Example

0 1 2 3 4

x 104

0

2

4

6

8

10x 10

4

n

Num

ber

of n

on z

ero

coef

ficie

nts CN time stepping with FD

CN time stepping with wavelet

0 1 2 3 4

x 104

0

2

4

6

8

10x 10

4

n

Num

ber

of n

on z

ero

coef

ficie

nts TaylorGalerkin method,

WaveletTaylor Galerkin method

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Improved convergence and stability properties as compared towavelet Galerkin method and Taylor-Galerkin method .

Example

0 0.2 0.4 0.6 0.8 14

3

2

1

0

1

2

3

x

u(x

,t)

Wavelet Galerkin method

0 0.2 0.4 0.6 0.8 14

3

2

1

0

1

2

3

x

u(x

,t)

Taylor Galerkin method

0 0.2 0.4 0.6 0.8 11.5

1

0.5

0

0.5

1

x

u(x

,t)

Ref:M. Mehra, B.V. Rathish Kumar, Time-accurate solution ofadvection diffusion problems by wavelet-Taylor Galerkin method,Communications in Numerical Methods in Engineering, Vol.21 (2005)

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Improved convergence and stability properties as compared towavelet Galerkin method and Taylor-Galerkin method .

Example

0 0.2 0.4 0.6 0.8 14

3

2

1

0

1

2

3

x

u(x

,t)

Wavelet Galerkin method

0 0.2 0.4 0.6 0.8 14

3

2

1

0

1

2

3

x

u(x

,t)

Taylor Galerkin method

0 0.2 0.4 0.6 0.8 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

x

u(x

,t)

Ref:M. Mehra, B.V. Rathish Kumar, Time-accurate solution ofadvection diffusion problems by wavelet-Taylor Galerkin method,Communications in Numerical Methods in Engineering, Vol.21 (2005)

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Multiresolution Analysis of L2(R)

Definition

MRA is characterized by the following axioms

{0} V1 V0 V1 L2(R)j=

j= Vj = L2(R)

jZ Vj = {0}Invariance to dialations, i.e f Vj iff f (2(.)) Vj+1Invariance to translations, i.e scaling function{0k = (x k)|k Z} is an orthonormal basis for V0

Define Wj = {jm (wavelets) |m Mj} to be the complement ofVj in Vj+1, where Vj+1 = Vj + Wj .

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Each member of the wavelet family is determined by the set ofconstants ak from the dilation equation

(x) =

2

D1

k=0

ak(2x k)

and by the set of constants bk from the wavelet equation

(x) =

2D1

k=0

bk(2x k)

bk = (1)kaD1k , k = 0, 1, ,D 1.

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(x)dx = 1

j ,k(x)j ,l (x)dx = k,l

i ,k(x)j ,l (x)dx = i ,jk,l

i ,k(x)j ,l (x)dx = 0 j i

Orthogonal projections

PVj f (x) =

k=

cjkj ,k(x), x R

PWj f (x) =

k=

djkj ,k(x), x R

Wavelet decomposition

PVj f = PVJ0 f +J1

j=J0

PWj f

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Important family of wavelets

Daubechies wavelet with different compact support, Coiflet,Semi-orthogonal wavelet, Battle-Lemaries wavelets, biorthogonalwavelet, interpolats.Daubechies scaling and wavelet functions

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Scaling function for M = 3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.5

1

0.5

0

0.5

1

1.5

2

Wavelet function for M = 3

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Important family of wavelets

Daubechies wavelet with different compact support, Coiflet,Semi-orthogonal wavelet, Battle-Lemaries wavelets, biorthogonalwavelet, interpolats.Daubechies scaling and wavelet functions

0 1 2 3 4 5 6 70.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Scaling function for M = 4

0 1 2 3 4 5 6 71

0.5

0

0.5

1

1.5

Wavelet function for M = 4

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Wavelet Expansion

Since space VJ is decomposed into wavelet spaceVJ0 + WJ0 + WJ01 + + WJ1 Using this relation we obtain

PVJ f (x) =

k=

cJ0k J0,k(x) +

J1

j=J0

k=

djkj ,k(x).

Where J0 satisfy 0 J0 J.Where the coefficients c J0k and djk are

given by

cJ0k =

f (x)J0 ,k(x)dx

djk =

f (x)j ,k (x)dx , j = J0, , j 1

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An estimate for the decay of wavelet coefficients

The decay of wavelet coefficients is expressed in the followingtheorem.

Theorem

Let M = D/2 be the number of vanishing moments for a waveletj ,k and let f CM(R). Then the wavelet coefficients decay asfollows:

|d jk | CM2j(M+12)maxIj,k |f (M)()|.

Where CM is a constant independent of j , k and f and Ij ,k denotesthe support of j ,k .

f PVJ f =

j=J

k=

djkj ,k(x) = O(2JM)

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Our Model

Consider the equations of the form

ut = Au + Ng(u) + f (x , y)

with suitable initial and boundary conditions. Here A and N areconstant-coefficient differential operators that do not depend upontime t and the function g(u) is non-linear.Approximation in time

To obtain a second order method the Taylor series is taken as

un un1t

= un1t +t

2un1tt + O(t

2)

un1 unt

= unt +t

2untt + O(t

2)

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Combining the above expressions gives

un un1t

=1

2(un1t + u

nt ) +

t

4(un1tt untt) + O(t2)

Then using our model problem,

ut = Au, utt = A2u

the initial boundary value problem is converted in to a sequence ofboundary value problems

un = T un1

u(x , 0) = u0

t = t/N, tn = nt, n = 1, ,N + 1

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Approximation in space

The variational formulation of the wavelet Taylor-Galerkin schemeis

Given u0h VpNB(unh , vh)

t

2D(unh , vh) +

t2

4C(unh , vh) = B(un1h , vh) +

t

2D(un1h , vh)

+t2

4C(unh , vh)

Where the bilinear forms B, C and D are defined by

B, C,D :V V CB(u, v) = (u, v), C(u, v) = (Au,Av), D(u, v) = (Au, v)

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Advection Equation M. Holmstrom and J. Walden have appliedadaptive wavelet methods on such type of PDEs. But by ourapproach of FW-TGM method we are taking the advantage oftime accurate scheme as well as wavelet capabilities of compressionto produce fast algorithm based on fast matrix vector product interms of sparsity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

x

u(x,

t)

Solution at t = .3 using j = 8by F-WGM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

xu(

x,t)

Solution at t = .3 using j = 9by F-WGM

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Advection Equation M. Holmstrom and J. Walden have appliedadaptive wavelet methods on such type of PDEs. But by ourapproach of FW-TGM method we are taking the advantage oftime accurate scheme as well as wavelet capabilities of compressionto produce fast algorithm based on fast matrix vector product interms of sparsity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

x

u(x,

t)

Solution at t = .3, M = 0,v = 0 by FW-TGM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

xu(

x,t)

Solution at t = .3, M = 0,v = 10

6 by FW-TGMMani Mehra Applications of Wavelets to Partial Differential Equations 17/51

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Moreover, near the spike error in FW-TGM method is small. Fromthis we can conclude that near the sharp gradients we can take theadvantage of time accuracy and compression properties of waveletin FW-TGM method.

0 100 200 300 400 500 6000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

x

Err

or

D = 6 using F-WGM

0 100 200 300 400 500 6000.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

x

Err

or

D = 6 using FW-TGM

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Moreover, near the spike error in FW-TGM method is small. Fromthis we can conclude that near the sharp gradients we can take theadvantage of time accuracy and compression properties of waveletin FW-TGM method.

0 100 200 300 400 500 6000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

x

Err

or

D = 10 using F-WGM

0 100 200 300 400 500 6003

2

1

0

1

2

3

4x 10

3

x

Err

or

D = 10 using FW-TGM

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Inviscid Burgers equation (quasilinear hyperbolicconservation equation)

Case 1: Initial condition u(x , 0) = sin(2x). The solution due toFD scheme develops local oscillations while the solution due toFW-TGM continues to be smooth.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.5

0

0.5

1

1.5

x

u(x,

t)

Solutions at various time usingFD

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.5

0

0.5

1

1.5

xu(

x,t)

Solutions at various time usingFW-TGM

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Inviscid Burgers equation (quasilinear hyperbolicconservation equation)Case 2: Here we are using initial condition

u(x , 0) =

{

1 + x 1 x 01 x 0 x 1

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

u(x,

t)

Local oscillation near shockusing FW-TGM

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

u(x,

t)

Global oscillation usingFourier-Galerkin

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Numerical simulation of hill rotation

Consider the problem of a hill rotating around the origin governed

by tu + .(

yuxu

)

= u The solution is good agreement

with those presented by A. Smolianski and D. Kuzmin in 1999.

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t)

(a) Initial distribu-tion, min u = 0, maxu = 1

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t)

(b) min u = 3.9 106, max u =.9926, M = 0, v =0

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t)

(c) min u = 106,max u = .9926,M = 10

5, v =105

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Secondly same problem is considered with different initial conditionfor non zero value of = 103. Numerical scheme showsremarkable accuracy. In addition the final height of the cone wasfound 98 per cent of its original value, indicating a minimaldissipation error.

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t

)

(d) Initial distribu-tion, min u = 0, maxu = 1

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t

)

(e) min u =8.4 105,max u = .9887,M = 0, v = 0

1

0.5

0

0.5

1

0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy

u(x

,y,t

)(f) min u = 8.4 105, max u =.9887, M = 10

5,v = 10

5Mani Mehra Applications of Wavelets to Partial Differential Equations 22/51

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The Navier-Stokes equations

A two-dimensional incompressible viscous flow is described by theNavier-Stokes equations. In vorticity/stream-function formulation:

=

t+ J(, ) = + curlf

where is the vorticity field (curl of the non-divergent velocityfield), the stream function, f a forcing term andJ(, ) = yx xy the two-dimensional Jacobian operator.Rewriting these equations as follows:

=

t= + s, s = curlf J(, )

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Wavelet-Taylor Galerkin method for Navier-Stokes Split theproblem into three subproblems: solve a Poisson equation toobtain stream function from the vorticity, evaluate the non-linearterm, integrate the heat equation.Starting from n at time t = nt

Compute n by solving Poisson equation. Here the numericalsolution has been searched as the long time asymptoticsolution of the heat equation.

Perform inverse wavelet transform to obtain nodal values n

and n.

Compute the nonlinear r.h.s sn by collocation approach usinga second-order, energy enstrophy conserving, Arakawasscheme.

Compute the sn by sn.

Finally solve the heat equation using the wavelet-TaylorGalerkin schemes to obtain n+1.

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and n.

o

and n.

o

and n.

o

and n.

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Numerical simulation of turbulence

The numerical experiment we present studies the merging of twosame sign vortices. It concerns free decaying turbulence (no forcingterm). The initial condition for the simulation considered is

(x , y) =

i=3

i=1

Aiexp(((x xi )2 + (y yi)2)/2i )

with variables i = 1/, amplitudes A1 = A2 = 2A3 = , andpositions x1 = 3/4, x2 = x3 = 5/4,y1 = y2 = , y3 = (1 + 1/(92)). In fully developedtwo-dimensional turbulent flows the chance of vortex mergingincreases with the density of vortices. Here with only three vorticeswe need this specific configuration to ensure a rapid merger.

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In a 2 2 box, three vortices with a gaussian vorticity profile arepresent two are positive with the same intensity (), one isnegative with half the intensity of others.

Three vortex interaction: initial state (t=0)

Further parameters are J = 8, t = 2.5 103, = 5 105. Theturnover time of one of the positive vortices is initially T = 4.0,and the initial Reynolds number based on the circulation of one ofthe positive vortices is Re = 2 104. The thresholds used in thewavelet compression are v = 10

8, M = 108.

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t=10

t=30t=40

t=20

Three vortex interaction at different timesRef:B. V. Rathish Kumar, Mani Mehra A Time-Accuratepseudo-wavelet scheme for Parabolic and Hyperbolic PDEs,Nonlinear Analysis, Vol. 63 (2005) pp. e345-e356.

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A priori estimation for wavelet Taylor-Galerkin method

An abstract Cauchy problem for operator A

d

dtu+Au = 0, in [0 t],

u(0) = u0 at t = 0

R be a bounded domain with periodic boundary = u(t) is the solution of Cauchy problemun be its semi discrete approximation at time t = nt.T transient operator corresponding to FW-TGM.For an asymptotically stable scheme t such that |r(t)| 1

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EN = u(t) uN , the temporal approximation error

Temporal approximation error

The temporal error estimate is bounded by

EN cttmAm+1u0

for all u0 D(Am+1).

Eh = uN uNh , the spatial approximation error.

Spatial approximation error

Spatial approximation error is bounded by

Eh NLeBt

chsT I1u0Hs () + chsT Nu0Hs ()

for all u0 Hs()

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EN = u(t) uN , the temporal approximation error

Temporal approximation error

The temporal error estimate is bounded by

EN cttmAm+1u0

for all u0 D(Am+1).

Eh = uN uNh , the spatial approximation error.

Spatial approximation error

Spatial approximation error is bounded by

Eh NLeBt

chsT I1u0Hs () + chsT Nu0Hs ()

for all u0 Hs()

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ConclusionsWavelet-Taylor Galerkin method:

Space and time accurate method

Taking advantage of wavelet compression of operator as wellas localized function (e.g Gaussian) therefore computationallyefficient

Method shows Improved stability properties

Future plans

Incorporation of space-time adaptive features to the newlyproposed wavelet-Taylor Galerkin method

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Future plans

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Future plans

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Future plans

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Future plans

o

Future plans

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MotivationAdaptive wavelet collocation method on the sphereConclusions and future directions

Collaborator

Nicholas Kevlahan (McMaster University)

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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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Why wavelets?

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).

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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 metricterm which are unbounded near the poles.

To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods was an open problem.

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

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Wavelet multiresolution analysis of L2(S)

Definition

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 = {jm(wavelets)|m Mj} to be the complement of Vjin Vj+1, where Vj+1 = Vj Wj .

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Definition

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Definition

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Definition

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Definition

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Definition

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Construction of spherical wavelets based on sphericaltriangular gridsThe set of all vertices

S j = {pjk S : pjk = p

j+12k |k Kj} and Mj = Kj+1/Kj .

Level 0

Dyadic icosahedral triangulation of the sphere

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Construction of spherical wavelets based on sphericaltriangular gridsThe set of all vertices

S j = {pjk S : pjk = p

j+12k |k Kj} and Mj = Kj+1/Kj .

Level 1

Dyadic icosahedral triangulation of 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 = c j+1m

kKm

sjk,mc

jk ,

m Kj : c jk = cj+1k .

Synthesis(j) :

m Kj : c jk = cjk ,

m Mj : c j+1m = d jm +

kKm

sjk,mc

jk .

o

e4 e3

e2e1

f1

f2

v1 v2

m

Analysis(j) :

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) :

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) :

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 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|

d jmjm(p) +

mMj

|d jm|

d jmjm(p))

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

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Wavelet compression

uJ(p) =

kK0 cJ0k

J0k (p) +

j=J1j=J0

mMj

|d jm|

d jmjm(p)+Discarded term

Test function Wavelet locations x Jk 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 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|

Test function Wavelet locations x Jk 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 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|

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

#K8

N() 10

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Wavelet approximation estimates

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

nd

Therefore, approximation erroris controlled by active gridpoints

||uJ(p) uJ(p)|| c3N()dn

102

103

104

105

106

105

104

103

102

101

100

N()

||u(p

)u

(p)||

LinearLinear liftedButterflyButterfly lifted

O(N()1)

O(N()2)

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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(p

)u

(p)||

O()

Ref: M. Mehra, N.K.-R. Kevlahan, An adaptive waveletcollocation method for the solution of partial differential equationson the sphere, Under the revision in J. Comp. Phys. Available atwww.math.mcmaster.ca/kevla/pdf/articles/sphere_wlt.pdf

www.math.mcmaster.ca/kevla/pdf/articles/sphere_wlt.pdf

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Calculation of Laplace-Beltrami operator on adaptivegrid

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 pji 2.

pij

pkj

i,k

i,kp

k1j p

k+1j

AS(p

ij)

o

Su(pji ) =

1

AS(pji)

2 [u(pjk) u(p

ji )]

Convergence result for theLaplace-Beltrami operator ofuJ(p)

||SuJ(p) SuJ(p)|| c4

105

104

103

102

101

100

105

104

103

102

101

100

/||u(p)||

||

Su

(p)

Su

(p

)||

/||

Su(p

)||

ButterflyButterfly lifted

O()

o

Su(pji ) =

1

AS(pji)

2 [u(pjk) u(p

ji )]

Convergence result for theLaplace-Beltrami operator ofuJ(p)

||SuJ(p) SuJ(p)|| c4 c5N()

d2n

103

104

105

106

105

104

103

102

101

100

N()

||

Su

(p)

Su

(p

)||

/||

Su(p

)||

ButterflyButterfly lifted

O(N()1)

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Calculation of other differential operators on adaptivegrid

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(p

ji ), (p

ji )) =

1

6AS(pji)

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

ji ))((p

jk ) (p

ji ))

S .(u(pji ),S(pji )) =

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2

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

jk ) (p

ji )).

o

ji ), (p

ji )) =

1

6AS(pji)

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

ji ))((p

jk ) (p

ji ))

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2

jk ) (p

ji )).

o

ji ), (p

ji )) =

1

6AS(pji)

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

ji ))((p

jk ) (p

ji ))

1

2AS(pji )

kN(i)

coti ,k + cot i ,k2

jk ) (p

ji )).

o

Application of spherical wavelets to compressturbulence

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.

o

Vorticity function. Reconstructed with 40%significant wavelets.

o

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

o

102

103

104

105

106

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

% o

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

C

The compression coefficient Cas a function of time,

= 105.

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Spherical advection equation

u

t+ V .Su = 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.

o

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.

o

Solution using AWCM Adapted grid for the solution.

o

Poisson Equation

1

0

1 10

1

0.5

0

0.5

1

yx

z

Solutions

Grids

m=5 m=7 m=9

o

Poisson Equation

1 2 3 4 5 6 7 8 9 10 11 1210

4

103

102

101

100

101

Local iteration

Re

sid

ua

l err

or

Residual error for threshold = 104 and different choice ofsmoothing parameters 1 and 2

Ref: M. Mehra, N.K.-R. Kevlahan, An adaptive multilevel waveletsolver for elliptic equations on an optimal spherical geodesic grid,Submitted to SIAM J. Sci. Comp. (2007). Available atwww.math.mcmaster.ca/kevla/pdf/articles/sphere_mg.pdf

www.math.mcmaster.ca/kevla/pdf/articles/sphere_mg.pdf

o

Conclusions

Adaptive wavelet collocation method:

Used for time evolution problems (e.g diffusion and advectionequations)

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.

Adaptive multilevel wavelet solver:

Used for elliptic problems (e.g Poisson equation).

The improved truncation error and efficiency of solver on anoptimal spherical geodesic grid

o

Conclusions

o

Conclusions

o

Conclusions

o

Conclusions

o

Conclusions

o

Conclusions

o

Conclusions

o

Future plans

Application of AWCM and adaptive multilevel solver to morerealistic models (e.g. two-dimensional turbulence).

Application of diffusion wavelets to partial differentialequationsRef: Biorthogonal diffusion wavelets for multiscalerepresentations on manifolds and graphs Proc. SPIE Vol.5914, 5914M Wavelets XI; Manos Papadakis, Andrew F.Laine, Michael A. Unser; Eds., 2005

o

Future plans

o

Future plans

o

Thank you

http://www.math.mcmaster.ca/vmmehra

Main TalkPart 1: Wavelet-Taylor Galerkin method for PDE's on flat geometriesMotivationWavelet-Taylor Galerkin methodConclusion and future plan

Part 2: Adaptive wavelet collocation method for PDEs on manifoldsMotivationAdaptive wavelet collocation method on the sphereConclusions and future directions

Documents

Applications of Wavelets to Partial Differential Equationsweb.iitd.ac.in/~mmehra/TIFR_talk.pdf · Applications of Wavelets to Partial Di erential Equations Mani Mehra Department of