Adaptive wavelet collocation method for PDE's on the sphere

Adaptive wavelet collocation method for PDE’s on the sphere Mani Mehra Department of Mathematics and Statistics

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Collaborators

◮

Nicholas Kevlahan (McMaster University)

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 and cloud water fields in numerical weather prediction).

◮

Many PDE’s arise from mean curvature flow, surface diffusion flow and Willmore flow on the sphere.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 and cloud water fields in numerical weather prediction).

◮

Many PDE’s arise from mean curvature flow, surface diffusion flow and Willmore flow on the sphere.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 and cloud water fields in numerical weather prediction).

◮

Many PDE’s arise from mean curvature flow, surface diffusion flow and Willmore flow on the sphere.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 and cloud water fields in numerical weather prediction).

◮

Many PDE’s arise from mean curvature flow, surface diffusion flow and Willmore flow on the sphere.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets?

◮

High rate of data compression.

◮

Fast O(N ) transform.

◮

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

◮

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets?

◮

High rate of data compression.

◮

Fast O(N ) transform.

◮

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

◮

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets?

◮

High rate of data compression.

◮

Fast O(N ) transform.

◮

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

◮

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets?

◮

High rate of data compression.

◮

Fast O(N ) transform.

◮

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

◮

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets?

◮

High rate of data compression.

◮

Fast O(N ) transform.

◮

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

◮

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

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Why wavelets on manifolds? (e.g. spherical wavelets) ◮

Spherical triangular grids (quasi uniform triangulations) avoid the 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 PDE’s efficiently using adaptivity on general manifold by wavelet methods.

◮

Past application of wavelets to turbulence have been mainly restricted to flat geometries which severely limiting for geophysical applications. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet multiresolution analysis of L2 (S)

Definition MRA is characterized by the following axioms ◮ ◮ ◮

V j ⊂ V j+1 (subspaces are nested). Sj=∞ j=−∞ Vj = L2 (S).

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

Define Wj = {ψkj (wavelets)|k ∈ Mj } to be the complement of Vj in Vj+1 , where Vj+1 = Vj + Wj .

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Construction of spherical wavelets based on spherical triangular grids j+1 The set of all vertices S j = {pkj ∈ S : pkj = p2k |k ∈ Kj } and j j+1 j M = K /K .

Level 0

o

Mani Mehra

Dyadic icosahedral triangulation of the sphere

Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Construction of spherical wavelets based on spherical triangular grids j+1 The set of all vertices S j = {pkj ∈ S : pkj = p2k |k ∈ Kj } and j j+1 j M = K /K .

Level 1

o

Mani Mehra

Dyadic icosahedral triangulation of the sphere

Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Fast wavelet transform Analysis(j) : j j+1 ∀m ∈ Mj : dm = cm −

f1 e3

e4

j ˜sk,m ckj ,

k∈Km j

∀m ∈ K :

m

X

ckj

=

ckj+1 .

v2

v1

Synthesis(j) : e1

e2 f2

∀m ∈ Kj : ckj = ckj , j+1 j ∀m ∈ Mj : cm = dm +

X

j ˜sk,m ckj

k∈Km

The members of the neighborhoods used in wavelet bases (m ∈ Mj , Km = {v1 , v2 , f1 , f2 , e1 , e2 , e3 , e4 }). Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

o

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Fast wavelet transform Analysis(j) : j j+1 ∀m ∈ Mj : dm = cm −

f1 e3

e4

j ˜sk,m ckj ,

k∈Km j

∀m ∈ K :

m

X

ckj

=

ckj+1 .

v2

v1

Synthesis(j) : e1

e2 f2

∀m ∈ Kj : ckj = ckj , j+1 j ∀m ∈ Mj : cm = dm +

X

j ˜sk,m ckj

k∈Km

The members of the neighborhoods used in wavelet bases (m ∈ Mj , Km = {v1 , v2 , f1 , f2 , e1 , e2 , e3 , e4 }). Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

For linear basis, ˜sv1 = ˜sv2 = 1/2 o

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Fast wavelet transform Analysis(j) : j j+1 ∀m ∈ Mj : dm = cm −

f1 e3

e4

j ˜sk,m ckj ,

k∈Km j

∀m ∈ K :

m

X

ckj

=

ckj+1 .

v2

v1

Synthesis(j) : e1

e2 f2

∀m ∈ Kj : ckj = ckj , j+1 j ∀m ∈ Mj : cm = dm +

X

j ˜sk,m ckj

k∈Km

The members of the neighborhoods used in wavelet bases (m ∈ Mj , Km = {v1 , v2 , f1 , f2 , e1 , e2 , e3 , e4 }). Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

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

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Fast wavelet transform Analysis(j) : j j+1 ∀m ∈ Mj : dm = cm −

f1 e3

e4

j

∀m ∈ M :

m

ckj

=

ckj+1

+

X

j ˜sk,m ckj ,

k∈Km j j sk,m dm .

v2

v1

Synthesis(j) : e1

e2 f2

∀m ∈ Mj : ckj = ckj −

X

j j sk,m dm ,

k∈Km j

∀m ∈ M : The members of the neighborhoods used in wavelet bases (m ∈ Mj , Km = {v1 , v2 , f1 , f2 , e1 , e2 , e3 , e4 }). Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j+1 cm

=

j dm

+

X

j ˜sk,m ckj

k∈Km

o

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression u J (p) =

J0 J0 k∈K0 ck φk (p)

P

+

Pj=J−1 P

m∈Mj

j=J0

j j ψm (p) dm

1

z

0.5 0 −0.5 −1 −1

1 0

0 1 −1 x

Test function

y

Wavelet locations xkJ without compression at J = 5, #K5 = 10242 o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

m∈Mj j |dm |≥ǫ

j=J0

j j dm ψm (p)

1

z

0.5 0 −0.5 −1 −1

1 0

0 1 −1 x

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

y

Wavelet locations xkJ at J = 5, ǫ = 10−5 , N(ǫ) = 995 and ratio #K5 N(ǫ) ≈ 10 McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

j=J0

m∈Mj j |≥ǫ |dm

j j dm ψm (p)

Wavelet locations xkJ without compression at J = 6, #K6 = 40962 o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j=J0

m∈Mj j |dm |≥ǫ

j j dm ψm (p)

Wavelet locations xkJ st J = 6, ǫ = 10−5 , N(ǫ) = 8175 and ratio #K6 N(ǫ) ≈ 5 McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j=J0

m∈Mj j |≥ǫ |dm

j j dm ψm (p)

Wavelet locations xkJ without compression at J = 7, #K7 = 163842 McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j=J0

m∈Mj j |dm |≥ǫ

j j dm ψm (p)

Wavelet locations xkJ at J = 7, ǫ = 10−5 , N(ǫ) = 20353 and ratio #K7 N(ǫ) ≈ 8 McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j=J0

m∈Mj j |≥ǫ |dm

j j dm ψm (p)

Wavelet locations xkJ without compression at J = 8, #K8 = 655362 McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet compression J (p) = u≥

J0 J0 k∈K0 ck φk (p) +

P

Pj=J−1 P

Test function

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

j=J0

m∈Mj j |dm |≥ǫ

j j dm ψm (p)

Wavelet locations xkJ at J = 8, ǫ = 10−5 , N(ǫ) = 64231 and ratio #K8 o N(ǫ) ≈ 10 McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet approximation estimates

0

10

−1

∞

10

≥

Approximation error is controlled by the wavelet threshold ǫ J (p)|| ||u J (p) − u≥ ∞ ≤ c1 ǫ

||uJ(p)−uJ (p)||

◮

−2

10

Linear Linear lifted Butterfly Butterfly lifted

−3

10

−4

10

O(ε) −5

10

−6

10 −6 10

−5

10

−4

10

−3

10 ε

−2

10

−1

10

0

10

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Wavelet approximation estimates

ǫ controls the total active grid points N(ǫ) by the following n relation N(ǫ) ≤ c2 ǫ− k

0

10

≥

Therefore, approximation error is controlled by active grid points −k J (p)|| ||u J (p) − u≥ ∞ ≤ c3 N(ǫ) n

||uJ(p)−uJ (p)||

◮

−1

O(N(ε) )

−1

10 ∞

◮

−2

10

−3

Linear Linear lifted Butterfly Butterfly lifted

−2

10

O(N(ε) )

−4

10

−5

10

−6

10

2

10

3

10

4

10 N(ε)

5

10

6

10

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Calculation of Laplace-Beltrami operator on adaptive grid ∆S u(pij ) =

1 AS (pij )

P

k∈N(i )

where AS (pij ) is the area of one-ring neighborhood region given by AS (pij ) = 1P k∈N(i ) (cot αi ,k + 8 cot βi ,k )kpkj − pij k2 .

cot αi ,k +cot βi ,k [u(pkj ) 2

− u(pij )]

j

pi

j

pk−1

αi,k

j

AM(pi)

βi,k

j

pk+1

j

pk o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Calculation of Laplace-Beltrami operator on adaptive grid ∆S u(pij ) =

1 AS (pij )

P

k∈N(i )

cot αi ,k +cot βi ,k [u(pkj ) 2

− u(pij )]

0

S

J ||∆S u J (p) − ∆S u≥ (p)||∞ ≤ c4 ǫ

−1

Butterfly

−2

Butterfly lifted

10

S

∞ S ≥

Convergence result for the Laplace-Beltrami operator of J (p) for the test function u≥

||∆ uJ(p)−∆ uJ (p)|| /||∆ uJ(p)||

10

◮

10

−3

10

−4

10

−5

O(ε)

10

−6

10 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 J ε/||u (p)||

1

10

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Calculation of Laplace-Beltrami operator on adaptive grid ∆S u(pij ) =

1 AS (pij )

P

k∈N(i )

cot αi ,k +cot βi ,k [u(pkj ) 2

− u(pij )]

0

10

S

−1

10

−2

10

∞ S ≥

k−2 n

S

J ||∆S u J (p) − ∆S u≥ (p)||∞ ≤ c4 ǫ

≤ c5 N(ǫ)−

Butterfly Butterfly lifted

∞

Convergence result for the Laplace-Beltrami operator of J (p) for the test function u≥

||∆ uJ(p)−∆ uJ (p)|| /||∆ uJ(p)||

◮

−3

10 O(N(ε)−3/2) −4

10

−5

10

−6

10

3

10

4

5

10

10

6

10

N(ε) o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Numerical simulation

Adaptive wavelet collocation method on the sphere

Conclusions and future directions

Calculation of other differential operators on adaptive grid ◮

◮

The flux term present in the spherical advection equation can be expressed in the form of flux divergence and Jacobian operators, ∇.(Vu) = ∇.(u∇χ) − J(u, ψ) JS (u(pij ), ψ(pij )) = P j j j j 1 j k∈N(i ) (u(pk ) + u(pi ))(ψ(pk ) − ψ(pi )) 6AS (pi )

◮

∇S .(u(pij ), ∇S χ(pij )) =

X cot αi ,k + cot βi ,k 2 2AS (pij ) k∈N(i ) 1

(u(pkj ) + u(pij ))(ψ(pkj ) − ψ(pij )).

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Numerical simulation

Adaptive wavelet collocation method on the sphere

Conclusions and future directions

Calculation of other differential operators on adaptive grid ◮

◮

The flux term present in the spherical advection equation can be expressed in the form of flux divergence and Jacobian operators, ∇.(Vu) = ∇.(u∇χ) − J(u, ψ) JS (u(pij ), ψ(pij )) = P j j j j 1 j k∈N(i ) (u(pk ) + u(pi ))(ψ(pk ) − ψ(pi )) 6AS (pi )

◮

∇S .(u(pij ), ∇S χ(pij )) =

X cot αi ,k + cot βi ,k 2 2AS (pij ) k∈N(i ) 1

(u(pkj ) + u(pij ))(ψ(pkj ) − ψ(pij )).

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Numerical simulation

Adaptive wavelet collocation method on the sphere

Conclusions and future directions

Calculation of other differential operators on adaptive grid ◮

◮

The flux term present in the spherical advection equation can be expressed in the form of flux divergence and Jacobian operators, ∇.(Vu) = ∇.(u∇χ) − J(u, ψ) JS (u(pij ), ψ(pij )) = P j j j j 1 j k∈N(i ) (u(pk ) + u(pi ))(ψ(pk ) − ψ(pi )) 6AS (pi )

◮

∇S .(u(pij ), ∇S χ(pij )) =

X cot αi ,k + cot βi ,k 2 2AS (pij ) k∈N(i ) 1

(u(pkj ) + u(pij ))(ψ(pkj ) − ψ(pij )).

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Numerical simulation

Adaptive wavelet collocation method on the sphere

Conclusions and future directions

Calculation of other differential operators on adaptive grid ◮

◮

The flux term present in the spherical advection equation can be expressed in the form of flux divergence and Jacobian operators, ∇.(Vu) = ∇.(u∇χ) − J(u, ψ) JS (u(pij ), ψ(pij )) = P j j j j 1 j k∈N(i ) (u(pk ) + u(pi ))(ψ(pk ) − ψ(pi )) 6AS (pi )

◮

∇S .(u(pij ), ∇S χ(pij )) =

X cot αi ,k + cot βi ,k 2 2AS (pij ) k∈N(i ) 1

(u(pkj ) + u(pij ))(ψ(pkj ) − ψ(pij )).

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence ◮

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

◮

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

◮

The coherent vortices can be extracted using nonlinear wavelet filtering.

This is the concept of Coherent Vortex Simulation which was developed by Marie Farge, Kai Schneider and Nicholas Kevlahan in flat geometries. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence ◮

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

◮

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

◮

The coherent vortices can be extracted using nonlinear wavelet filtering.

This is the concept of Coherent Vortex Simulation which was developed by Marie Farge, Kai Schneider and Nicholas Kevlahan in flat geometries. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence ◮

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

◮

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

◮

The coherent vortices can be extracted using nonlinear wavelet filtering.

This is the concept of Coherent Vortex Simulation which was developed by Marie Farge, Kai Schneider and Nicholas Kevlahan in flat geometries. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence ◮

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

◮

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

◮

The coherent vortices can be extracted using nonlinear wavelet filtering.

This is the concept of Coherent Vortex Simulation which was developed by Marie Farge, Kai Schneider and Nicholas Kevlahan in flat geometries. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence

Vorticity function. Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

Reconstructed with 40% significant wavelets. McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence E (n, 0) =

Anγ/2 (n+n0 )γ

−2

−2

10

10 E(k) E(k) u

≥

−3

10

−4

10 E(k)

E(k)

≥

−4

10

−5

10

−5

10

−6

−6

10

10

−7

10

E(k) E(k) u

−3

10

−7

0

10

1

10 k

2

10

Energy spectrum reconstructed with 40% significant wavelets. Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

10

0

10

1

10 k

2

10

Energy spectrum reconstructed with 2% significant wavelets McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Application of spherical wavelets to compress turbulence 2

−1

10

10

−2

10

0

10

ε

% of error

−3

10

Linear Linear lifted Butterfly Butterfly lifted

−4

10

−5

10

Linear Linear lifted Butterfly Butterfly lifted

−2

10

−4

10

−6

10

−6

10 −7

10

2

10

3

10

4

5

10

10

6

10

7

20

40 60 80 % of coefficients used

100

10

N(ε)

Relation between ǫ and N(ǫ) for the vorticity function. Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

Rate of relative error as a function of rate of significant coefficients = N(ǫ)×100 o N(ǫ=0) . McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Diffusion equation

ut = ν∆S u + f where f is localized source chosen such a way that the solution of diffusion equation is given by u(θ, φ, t) = 2e

−

(θ−θ0 )2 +(φ−φ0 )2 ν(t+1)

Such equations arise in the application of Willmore flow, surface diffusion flow etc. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Diffusion Equation

Initial condition at t = 0

Adaptive grid at t = 0 o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Diffusion Equation

Solution using AWCM at t = 0.5

Adaptive grid at t = 0.5 o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Diffusion Equation

24

N(ǫ = 0) C= N(ǫ)

22 20 18 C

The potential of adaptive algorithm is measured by compression coefficients

16 14 12 10 8 0

0.1

0.2

0.3

0.4

0.5

t

The compression coefficient C as a function of time, ǫ = 10−5 . Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Spherical advection equation

∂u + V .∇S u = 0, ∂t The nondivergent driving velocity field V = (v1 , v2 ) for all times is given by   3π ) sin(α) , v1 = u0 cos(θ) cos(α) + sin(θ) cos(φ + 2 3π ) sin(α). v2 = −u0 sin(φ + 2

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Spherical advection equation ∂u + V .∇S u = 0, ∂t The initial cosine bell test pattern to be advected is given by (
u0 πr 1 + cos( if r < R R u(θ, φ) = 2 0 if r ≥ R, where u0 = 1000m, radius R = a/3 and r = a arc cos[sin(θc ) sin(θ) + cos(θc ) cos(θ) cos(φ − φc ), which is the great circle distance between (φ, θ) and the center (φc , θc ) = (0, 0). Such equations arise typically in the context of numerical weather forecast or in climatological studies, and they provide some of the most challenging and CPU time consuming problems in modern computational fluid dynamics. Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Spherical advection equation

1

Analytic sol. AWCM

0.8 0.6 0.4 0.2 0 −0.2 −1

−0.5

0

0.5

1

Solid body rotation of cosine bell using AWCM for ǫ = 10−5 . o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Spherical advection equation

Solution using AWCM

Adapted grid for the solution. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Spherical advection equation

0

35

10

30 −2

||u(p,t)−uex(p,t)||∞

25

C

20 15 10

10

−4

10

−6

10

5 0 0

−8

0.05

0.1

0.15

t

The compression coefficient C as a function of time, ǫ = 10−5 . Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

10

0

0.05

0.1

0.15

t

Time evolution of the pointwise L∞ error of the solution. McMaster University

o

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 finite difference schemes over triangulated surface for the differential operators is used.

◮

Verified convergence result predicted in theory.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 finite difference schemes over triangulated surface for the differential operators is used.

◮

Verified convergence result predicted in theory.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 finite difference schemes over triangulated surface for the differential operators is used.

◮

Verified convergence result predicted in theory.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 finite difference schemes over triangulated surface for the differential operators is used.

◮

Verified convergence result predicted in theory.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

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 finite difference schemes over triangulated surface for the differential operators is used.

◮

Verified convergence result predicted in theory.

o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Future directions ◮

Implementation of wavelet bases based on optimal spherical triangulation.

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

Incorporation of appropriate time integration method in spherical domain which takes advantage of wavelet multilevel decomposition.

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

Application of AWCM to more realistic models. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Future directions ◮

Implementation of wavelet bases based on optimal spherical triangulation.

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

Incorporation of appropriate time integration method in spherical domain which takes advantage of wavelet multilevel decomposition.

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

Application of AWCM to more realistic models. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Future directions ◮

Implementation of wavelet bases based on optimal spherical triangulation.

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

Incorporation of appropriate time integration method in spherical domain which takes advantage of wavelet multilevel decomposition.

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

Application of AWCM to more realistic models. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University

Motivation

Adaptive wavelet collocation method on the sphere

Numerical simulation

Conclusions and future directions

Future directions ◮

Implementation of wavelet bases based on optimal spherical triangulation.

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

Incorporation of appropriate time integration method in spherical domain which takes advantage of wavelet multilevel decomposition.

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

Application of AWCM to more realistic models. o

Mani Mehra Adaptive wavelet collocation method for PDE’s on the sphere

McMaster University