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Introduction PowellSabin splines Numerical simulation with PS splines References PowellSabin spline based multilevel preconditioners for 4th order elliptic equations on the sphere Jan Maes, Adhemar Bultheel Department of Computer

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Powell–Sabin spline based multilevel preconditioners for 4th order elliptic equations

  • n the sphere

Jan Maes, Adhemar Bultheel

Department of Computer Science Katholieke Universiteit Leuven

Bremen, November 9, 2006

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Outline

1

Introduction

2

Powell–Sabin splines The space of Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

3

Numerical simulation with PS splines The biharmonic equation in the plane The hierarchical basis preconditioner Optimal multilevel preconditioners The biharmonic equation on the sphere

4

References

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Introduction

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The biharmonic equation: plate bending problems

∇4u(x, y) = ∂2 ∂x2 + ∂2 ∂y2 ∂2 ∂x2 + ∂2 ∂y2

  • u(x, y) = f(x, y)

u(x, y) is the vertical displacement due to an external force. boundary conditions u = 0

∂u ∂n = 0

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Design of car windscreens

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Design of aircrafts

Courtesy of University of Bath

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Comparison with a classical method

Error ×1000

  • Classical method

Optimal PS spline based multilevel preconditioner

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Plate bending in a spherical geometry

Physical geodesy Oceanography Meteorology Earth dynamics

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The pole problem

Spherical coordinates give rise to the “pole problem” Therefore we will explore a different approach based on homogeneous polynomials in R3.

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Powell–Sabin splines

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Splines as mathematical building blocks

                                            

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Bernstein–Bézier representation

= ⇒

Pierre Étienne Bézier (1910-1999)

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Stitching together Bézier triangles

= ⇒ No C1 continuity at the red curve

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C1 continuity with Powell–Sabin splines

Conformal triangulation ∆ PS 6-split ∆PS S1

2(∆PS) = space of PS splines

M.J.D. Powell M.A. Sabin

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C1 continuity with Powell–Sabin splines

Conformal triangulation ∆ PS 6-split ∆PS S1

2(∆PS) = space of PS splines

M.J.D. Powell M.A. Sabin

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C1 continuity with Powell–Sabin splines

Conformal triangulation ∆ PS 6-split ∆PS S1

2(∆PS) = space of PS splines

M.J.D. Powell M.A. Sabin

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The dimension of S1

2(∆PS)?

There is exactly one solution s ∈ S1

2(∆PS) to the

Hermite interpolation problem s(Vi) = αi, Dxs(Vi) = βi, ∀Vi ∈ ∆, i = 1, . . . , N. Dys(Vi) = γi, The dimension of S1

2(∆PS) is 3N. Therefore we need 3N basis

functions.

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The dimension of S1

2(∆PS)?

There is exactly one solution s ∈ S1

2(∆PS) to the

Hermite interpolation problem s(Vi) = αi, Dxs(Vi) = βi, ∀Vi ∈ ∆, i = 1, . . . , N. Dys(Vi) = γi, The dimension of S1

2(∆PS) is 3N. Therefore we need 3N basis

functions.

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Powell–Sabin B-splines with control triangles

s(x, y) =

N

  • i=1

3

  • j=1

cijBij(x, y) Bij is the unique solution to [Bij(Vk), DxBij(Vk), DyBij(Vk)] = [0, 0, 0] for all k = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=1 Bij(x, y) = 1,

Bij(x, y) ≥ 0

(Paul Dierckx, 1997)

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Powell–Sabin B-splines with control triangles

s(x, y) =

N

  • i=1

3

  • j=1

cijBij(x, y) Bij is the unique solution to [Bij(Vk), DxBij(Vk), DyBij(Vk)] = [0, 0, 0] for all k = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=1 Bij(x, y) = 1,

Bij(x, y) ≥ 0

(Paul Dierckx, 1997)

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Powell–Sabin B-splines with control triangles

s(x, y) =

N

  • i=1

3

  • j=1

cijBij(x, y) Bij is the unique solution to [Bij(Vk), DxBij(Vk), DyBij(Vk)] = [0, 0, 0] for all k = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=1 Bij(x, y) = 1,

Bij(x, y) ≥ 0

(Paul Dierckx, 1997)

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Powell–Sabin B-splines with control triangles

Three locally supported basis functions per vertex

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Powell–Sabin B-splines with control triangles

The control triangle is tangent to the PS spline surface.

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Powell–Sabin B-splines with control triangles

It ‘controls’ the local shape of the spline surface.

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Spherical spline spaces

P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d: f(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 \ {0} ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of the unit sphere S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(τ), τ ∈ ∆}

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Spherical Powell–Sabin splines

s(vi) = fi, Dgis(vi) = fgi, Dhis(vi) = fhi, ∀vi ∈ ∆ has a unique solution in S1

2(∆PS)

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1 − 1 connection with bivariate PS splines

⇒ |v|2Bij( v |v|) ⇒ ← − Spherical PS B- spline Bij(v) piecewise trivari- ate polynomial of degree 2 that is homogeneous of degree 2 Restriction to the plane tangent to S at vi ∈ ∆

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1 − 1 connection with bivariate PS splines

Let Ti be the plane tangent to S at vertex vi Radial projection: Riv := v := v |v| ∈ S, v ∈ Ti Define ∆i as the star of vi in ∆, and let ∆PS

i

⊂ ∆PS be its PS 6-split. Theorem Let s ∈ S1

2(∆PS i

). Let s be the restriction of |v|2s(v/|v|) to Ti. Then s is in S1

2(R−1 i

∆PS

i

) and s(vi) = s(vi), Dgis(vi) = Dgis(vi), Dhis(vi) = Dhis(vi).

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1 − 1 connection with bivariate PS splines

Let Ti be the plane tangent to S at vertex vi Radial projection: Riv := v := v |v| ∈ S, v ∈ Ti Define ∆i as the star of vi in ∆, and let ∆PS

i

⊂ ∆PS be its PS 6-split. Theorem Let s ∈ S1

2(∆PS i

). Let s be the restriction of |v|2s(v/|v|) to Ti. Then s is in S1

2(R−1 i

∆PS

i

) and s(vi) = s(vi), Dgis(vi) = Dgis(vi), Dhis(vi) = Dhis(vi).

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Spherical B-splines with control triangles

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Applications on a spherical domain

Approximation of a mesh: consider the triangles of the original triangular mesh as control triangles of a PS spline.

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Applications on a spherical domain

Compression by smoothing: Decimate a given triangular mesh and approximate the decimated mesh.

triangular mesh reduced mesh (40000 triangles) (5000 triangles)

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Applications on a spherical domain

Compression by smoothing: Decimate a given triangular mesh and approximate the decimated mesh.

triangular mesh Powell–Sabin spline (40000 triangles) (5000 control triangles)

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Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

A nested sequence of subspaces S0 ⊂ S1 ⊂ S2 ⊂ · · · ⊂ Sℓ ⊂ · · ·

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Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

Complement spaces Wℓ Sℓ+1 = Sℓ ⊕ Wℓ

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Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

A stable basis for the complement space Wℓ Wℓ = span{ψℓ,i}

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Multiresolution analysis

Refine the triangulation ∆ and its PS 6-split ∆PS.

Vi Rki Vk Rjk Vj Rij Zijk Vi Rki Vk Rjk Vj Rij Zijk

dyadic refinement triadic refinement

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Multiresolution analysis

Refine the triangulation ∆ and its PS 6-split ∆PS.

Vi Vki Vk Vjk Vj Vij Zijk Vi Rki Vk Rjk Vj Rij Vik Vki Vkj Vjk Vji Vij Vijk

dyadic refinement triadic refinement

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Multiresolution analysis

Refine the triangulation ∆ and its PS 6-split ∆PS.

Vi Vki Vk Vjk Vj Vij Zijk Vi Rki Vk Rjk Vj Rij Vik Vki Vkj Vjk Vji Vij Vijk

dyadic refinement triadic refinement

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Multiresolution analysis with √ 3-refinement

∆PS ⊂ ∆PS

1

⊂ · · · ⊂ ∆PS

ℓ

⊂ · · · S1

2(∆PS 0 ) ⊂ S1 2(∆PS 1 ) ⊂ · · · ⊂ S1 2(∆PS ℓ ) ⊂ · · ·

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Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 0

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Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 1

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Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 1

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The hierarchical basis

{Vi ∈ ∆ℓ} ⊂ {Vi ∈ ∆ℓ+1} ∆PS

ℓ

⊂ ∆PS

ℓ+1

Sℓ := S1

2(∆PS ℓ ),

Sℓ ⊂ Sℓ+1 S2 = S0 ⊕ W0 ⊕ W1 Largest triangles control S0 Medium triangles control W0 Smallest triangles control W1

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The hierarchical basis

Basis functions: Sℓ = span{φℓ,k : k = 1, . . . , 3Nℓ} sℓ(x, y) = φℓcℓ =

Nℓ

  • i=1

3

  • j=1

Bijℓ(x, y)cijℓ φℓ+1 = [φo

ℓ+1 φn ℓ+1],

φo

ℓ+1 correspond to old reused vertices of level ℓ

φn

ℓ+1 correspond to the new vertices of level ℓ + 1

The set of splines φ0 ∪

m

  • ℓ=1

φn

ℓ

forms a hierarchical basis for Sm.

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A wavelet-type application

(a) Coarse part of (c) (b) Coarse part of (d) (c) Original surface (d) Coarse level edit of (c)

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Numerical simulation with PS splines

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A weak formulation

The biharmonic equation ∇4u = f on Ω ⊂ R2, u = ∂nu = 0 on ∂Ω We rewrite this difficult problem into an easier to solve problem. The weak formulation ∇2u, ∇2v = f, v, v ∈ H2

0(Ω)

Any solution of the biharmonic equation is also a solution of the weak formulation.

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The Galerkin approach

The idea is to approximate the real solution u by a Powell–Sabin spline uℓ ∈ Sℓ.

Boris Grigoryevich Galerkin (1871 - 1945)

∇2uℓ, ∇2v = f, v, v ∈ Sℓ Recall that uℓ(x, y) =

Nℓ

  • i=1

3

  • j=1

Bijℓ(x, y)cijℓ ⇓ ∇2

Nℓ

  • i=1

3

  • j=1

Bijℓcijℓ, ∇2Bpqℓ = f, Bpqℓ, p = 1, . . . , Nℓ, q = 1, 2, 3.

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The Galerkin approach

The idea is to approximate the real solution u by a Powell–Sabin spline uℓ ∈ Sℓ.

Boris Grigoryevich Galerkin (1871 - 1945)

∇2uℓ, ∇2v = f, v, v ∈ Sℓ Recall that uℓ(x, y) =

Nℓ

  • i=1

3

  • j=1

Bijℓ(x, y)cijℓ ⇓ ∇2

Nℓ

  • i=1

3

  • j=1

Bijℓcijℓ, ∇2Bpqℓ = f, Bpqℓ, p = 1, . . . , Nℓ, q = 1, 2, 3.

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A large linear system

Solve the 3Nℓ × 3Nℓ linear system Ac = b with A3p+q,3i+j := ∇2Bpqℓ, ∇2Bijℓ, b3p+q := f, Bpqℓ. For a large resolution level ℓ we find a good approximation uℓ to u. As ℓ increases the system becomes very large. Use an iterative method to solve the linear system such as the conjugate gradient (CG) method.

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A large linear system

Solve the 3Nℓ × 3Nℓ linear system Ac = b with A3p+q,3i+j := ∇2Bpqℓ, ∇2Bijℓ, b3p+q := f, Bpqℓ. For a large resolution level ℓ we find a good approximation uℓ to u. As ℓ increases the system becomes very large. Use an iterative method to solve the linear system such as the conjugate gradient (CG) method.

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Convergence rate of the CG method

The error at the k-th iteration is bounded by cte

  • κ(A) − 1
  • κ(A) + 1

k with κ(A) the condition number of the matrix A. We want κ(A) to be small κ(A) depends heavily on the choice of the basis functions for Sℓ. We made a bad choice: κ(A) = O( √ 3

4ℓ).

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Performance of the nodal basis functions

ℓ κ(A) residual # iterations 2 501 0.0000 4 3 3809 0.0016 40 4 37376 0.0011 155 5 281877 0.0005 452 6 2987048 0.0186 1000 Terrible!! → Preconditioning

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Performance of the nodal basis functions

ℓ κ(A) residual # iterations 2 501 0.0000 4 3 3809 0.0016 40 4 37376 0.0011 155 5 281877 0.0005 452 6 2987048 0.0186 1000 Terrible!! → Preconditioning

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Preconditioning through a change of basis

Use the hierarchical basis φ0 ∪ ℓ

j=1 φn j

(φn

m is a row vector holding the new basis functions at

level m)

Harry Yserentant

uℓ(x, y) = φ0c0 +

ℓ

  • j=1

φn

j cj

⇓ ∇2(φ0c0 +

ℓ

  • j=1

φn

j cj), ∇2φn m = f, φn m,

m = 0, . . . , ℓ, φn

0 := φ0

⇓ A new linear system Ac = b

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Performance of the hierarchical basis

Nodal basis                Hierarchical                    ℓ κ(A) residual # iterations 2 501 0.0000 4 3 3809 0.0016 40 4 37376 0.0011 155 5 281877 0.0005 452 6 2987048 0.0186 1000 ℓ κ(A) residual # iterations 3 29.2 5.4108e-4 11 4 74.8 8.0190e-4 19 5 118.6 4.8789e-4 25 6 197.5 3.5488e-4 35 7 287.8 2.1068e-4 35 8 406.9 1.1353e-4 43

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Stability is the key

Let {ψj : j = 1, . . . , 3Nℓ} be a basis for Sℓ. Suppose that k1

3Nℓ

  • j=1

|dj|2 ≤

  • 3Nℓ
  • j=1

ψjdj

  • 2

E

≤ k2

3Nℓ

  • j=1

|dj|2 with · E the energy norm of the PDE, then κ(A) = O(k2 k1 ) where Ai,j := ∇2ψi, ∇2ψj. ( · 2

E = ∇2·, ∇2· ∼

= · 2

H2)

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Stability investigation

The nodal basis functions:

Nℓ

  • i=1

3

  • j=1

|cijℓ|2

  • Nℓ
  • i=1

3

  • j=1

Bijℓcijℓ

  • 2

E

N2

ℓ Nℓ

  • i=1

3

  • j=1

|cijℓ|2 ⇒ κ(A) = O(N2

ℓ ) = O(

√ 3

4ℓ)

The hierarchical basis:

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φn

j cj

  • 2

E

ℓ2

ℓ

  • j=0

cj2

ℓ2

⇒ κ(A) = O(ℓ2)

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Optimal multilevel preconditioners

Wavelets HB of Lagrange type BPX-frame

Bramble–Pasciak–Xu

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The BPX-frame

The hierarchical basis:

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φn

j cj

  • 2

E

ℓ2

ℓ

  • j=0

cj2

ℓ2

⇒ κ(A) = O(ℓ2) The BPX-frame:

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φjcj

  • 2

E

  • ℓ
  • j=0

cj2

ℓ2

P . Oswald

⇒ κ(A) :=

λmax λmin=0 = O(1)

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Performance of the PS spline based BPX-frame

PS 6-split                PS 12-split                    ℓ dim κ(A) residual # iterations 3 84 15.8 8.2446e-4 9 4 276 28.1 7.5456e-4 15 5 954 36.1 4.9938e-4 16 6 2982 43.9 2.8602e-4 18 7 9384 50.8 1.9821e-4 20 8 28584

  • 9.2147e-4

23 9 87150

  • 6.4875e-5

23 ℓ dim κ(A) residual # iterations 2 79 45.0 2.1042e-3 11 3 402 80.1 1.0721e-3 21 4 1813 107.6 5.3133e-4 26 5 7704 128.1 2.6602e-4 27 6 31771 143.7 1.4577e-4 27 7 129054

  • 5.8638e-5

28

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The biharmonic equation on the sphere

Tangential gradient ∇Su := ∇u − (n · ∇u)n, n outward normal to S The Laplace–Beltrami operator on the 2-sphere S ∆S := ∇S · ∇S We are interested in ∆2

Su = f on S

∆Su, ∆Sv = f, v for all v ∈ H2(S) For every f ∈ L2(S) with

  • S f dω = 0 there exists a weak

solution u ∈ H2(S) and u is unique up to a constant. ‘Boundary condition’ →

  • S u dω = 0
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The biharmonic equation on the sphere

Tangential gradient ∇Su := ∇u − (n · ∇u)n, n outward normal to S The Laplace–Beltrami operator on the 2-sphere S ∆S := ∇S · ∇S We are interested in ∆2

Su = f on S

∆Su, ∆Sv = f, v for all v ∈ H2(S) For every f ∈ L2(S) with

  • S f dω = 0 there exists a weak

solution u ∈ H2(S) and u is unique up to a constant. ‘Boundary condition’ →

  • S u dω = 0
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A stable frame for H2(S)

Our aim A BPX-type preconditioner for the biharmonic equation on the sphere We need to prove that

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φjcj

  • 2

H2(S)

  • ℓ
  • j=0

cj2

ℓ2

hence κ(A) = O(1) where A(j1,k1),(j2,k2) := ∆Sφj1,k1, ∆Sφj2,k2. ·2

H2(S) ∼

= ∆S·, ∆S·

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

A stable frame for H2(S)

Our aim A BPX-type preconditioner for the biharmonic equation on the sphere We need to prove that

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φjcj

  • 2

H2(S)

  • ℓ
  • j=0

cj2

ℓ2

hence κ(A) = O(1) where A(j1,k1),(j2,k2) := ∆Sφj1,k1, ∆Sφj2,k2. ·2

H2(S) ∼

= ∆S·, ∆S·

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Proof sketch

uH2(S) :=

  • i
  • (αiu) ◦ ϕ−1

i

  • H2(Ti)

C∞ mappings ϕi : Bi → Ti with C∞ inverses ϕ−1

i

. Bi is an open subset of S and

i Bi covers S

Ti is an open subset of R2 {αi} is a partition of unity on {Bi}, i.e. αi vanishes outside Bi and

i αi = 1.

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Proof sketch

Recall the following theorem: Let Bi be a polygonal domain on S with PS-refinement BPS

i

Let Ti be the plane tangent to S at some vertex vi in the interior of BPS

i

Radial projection: ϕ−1

i

v := Riv := v := v |v| ∈ Bi ⊂ S, v ∈ Ti Theorem Let s ∈ S1

2(BPS i

). Let s be the restriction of |v|2s(v/|v|) to Ti. Then s is in S1

2(R−1 i

BPS

j

) and s(vi) = s(vi), Dgis(vi) = Dgis(vi), Dhis(vi) = Dhis(vi).

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Proof sketch

We can now prove that sL2(Bi) ∼ sL2(Ti) , sH2(Bi) ∼ sH2(Ti) , for s ∈ S1

2(BPS i

) Suppose that s = ℓ

j=0 sj, then s = ℓ j=0 sj. It is well-known

that s2

H2(Ti) ∼ inf ℓ

  • j=0

√ 3

4j

sj

  • 2

L2(Ti)

⇓ s2

H2(Bi) ∼ inf ℓ

  • j=0

√ 3

4j

sj

  • 2

L2(Bi)

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Proof sketch

If we take a bounded number of subsets Bi that cover S we find s2

H2(S) ∼ inf ℓ

  • j=0

√ 3

4j

sj

  • 2

L2(S)

Standard arguments now give the stability result

ℓ

  • j=0

cj2

ℓ2

  • φ0c0 +

ℓ

  • j=1

φjcj

  • 2

H2(S)

  • ℓ
  • j=0

cj2

ℓ2

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Introduction Powell–Sabin splines Numerical simulation with PS splines References

Performance of the PS preconditioners on the sphere

∆2

Su = 36xy

  • n

S, (u = xy)

BPX HB dim ℓ κ residual #iter κ residual #iter 96 1 51.0 3.0555e-11 10 60.7 1.5555e-03 5 276 2 65.7 8.7509e-04 7 82.0 4.9572e-04 9 816 3 79.5 3.9398e-04 8 103.7 5.4025e-04 15 2436 4 88.4 2.6345e-04 11 123.6 2.5576e-04 20 7296 5 96.8 1.7065e-04 11 152.0 1.8184e-04 26 21876 6 103.4 8.9656e-05 13 192.1 1.0873e-04 31 65616 7 107.7 5.0634e-05 13 237.0 6.3035e-05 36 196836 8 110.2 3.2635e-05 14 310.7 3.5946e-05 44

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References

P . Alfeld, M. Neamtu, and L. L. Schumaker. Bernstein–Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided

  • Geom. Design, 13:333–349, 1996.

P . Dierckx. On calculating normalized Powell–Sabin B-splines. Comput. Aided Geom. Design, 15(1), 61–78, 1997.

  • J. Maes and A. Bultheel. Modeling sphere-like manifolds with spherical

Powell-Sabin B-splines. Comput. Aided Geom. Design, to appear.

  • J. Maes, A. Kunoth and A. Bultheel. BPX-type preconditioners for 2nd

and 4th order elliptic problems on the sphere. SIAM J. Num. Anal., to appear.

  • M. Neamtu and L. L. Schumaker. On the approximation order of splines
  • n spherical triangulations. Adv. in Comp. Math., 21:3–20, 2004.

P . Oswald. Multilevel finite element approximation: Theory and

  • applications. B. G. Teubner, Stuttgart, 1994.