Solving Semilinear Elliptic PDEs on Manifolds

ContentsMotivation

Overview of GNGAPreliminary Experiments

Overview of the Closest Point Method (CPM)Future Work

Solutions of Semilinear Elliptic PDEs on

ManifoldsJeff Springer, Northern Arizona University

Springer Solving Semilinear Elliptic PDEs on Manifolds

ContentsMotivation



1 Motivation

2 Overview of GNGA

3 Preliminary Experiments

4 Overview of the Closest Point Method (CPM)

5 Future Work


ContentsMotivation



What is a PDE?

Definition

A Partial Differential Equation (PDE) is a relation involving anunknown function of several independent variables and their partialderivatives with respect to those variables.


ContentsMotivation



PDEs on Surfaces

For today’s talk we will be ultimately concerned with solvingPDEs (in some fashion) on surfaces.

Many applications of solving PDEs on surfaces: imageprocessing, flow and transport in earth’s oceans, etc.

We are primarily interested in solving surface eigenvalueproblems which has many specific applications.

Eigenvalue problems on surfaces lead to applications in designoptimization, shape recognition, and quantum billiards.


ContentsMotivation



Introduction

We are interested in solving the problem:

∆Su + f (u) = 0

In the sequel we choose the particular nonlinearityf (u) = su + u3.

We solve this PDE on a manifold W ⊆ Rn, where ∆S abovedenotes the Laplace-Beltrami operator.

Much of the well-known theory for the PDE ∆u + f (u) = 0extends nicely to solving this problem on manifolds.


ContentsMotivation



Introduction II

Taking a variational approach, we introduce the actionfunctional: J : H(1,2)(W )→ R.

The action functional J satisfies the properties:

J(u) =∫W

12 |∇u|2 − F (u)ds

J’(u)(v) = 〈∇J(u), v〉 =∫W ∇u · ∇v − f (u)vds

J”(u)(v,w) =∫W ∇v · ∇w − f ′(u)vwds


ContentsMotivation



A theorem

∇J(u) = 0 if and only if ,u is a classical solution of∆u + f (u) = 0. Hence, the solutions to this PDE are criticalpoints of the action functional

First, we solve the eigenvalue problem: ∆ψi = λiψi on W.This is done with known closed-form eigenfunctions or CPM.

Next we let u =∑

aiψi .


ContentsMotivation



The ψ matrix

Ψ =

ψ1( 1n ) ψ2( 1n ) · · · ψM( 1n )ψ1( 2n ) ψ2( 2n ) · · · ψM( 2n )

......

......

ψ1(n−1n ) ψ2(n−1n ) · · · ψM(n−1n )


ContentsMotivation



GNGA

The Gradient Newton Galerkin Algorithm (GNGA) is an Mdimensional Newtons method applied to an approximated gradientfunction g : RM → RM defined by

g =

J ′(u)(ψ1)

...J ′(u)(ψM)

=

∫W (∇u · ∇ψ1 − suψ1 − u3ψ1)dx

...∫W (∇u · ∇ψM − suψM − u3ψM)dx


ContentsMotivation



Derivation of Gradient

We define the gradient, as on the previous slide, as gi = J ′(u)(ψi ).Thus gi ∈ RM . The gradient can be simplified as:

J ′(u)(ψi )) =

∫W

(∇u · ∇ψi − suψi − u3ψi )dx

=

∫W

(∇Σakψk · ∇ψi − suψi − u3ψi )dx

= (λi − s)ai −∫W

u3ψi )dx

Note that (Ψ′ ∗ u3

n ≈∫W u3ψi )dx


ContentsMotivation



The Hessian matrix

We use a Jacobian of g as the derivative in the algorithm. Wecompute this approximated Hessian h as:

hij = ((λi − s)δij − 3∫W u2ψiψjdx)

where i , j = 1, 2, . . . ,M


ContentsMotivation



Derivation of Hessian

To obtain the simplified version of the Hessian on the previousslide we use the definition of the Hessian:

hij = J ′′(u)(ψi , ψj)

=

∫W

(∇ψi · ∇ψj − sψiψj − 3u2ψiψj)dx

= (λi − s)δij −∫W

(3u2ψiψj)dx


ContentsMotivation



GNGA Algorithm

1 Choose your initial coefficients a = a0 = akMk=1, setu = u0 =

∑akψk , and set n = 0

2 Loop over:

1 Calculate g = gn+1 = (J ′(u)(ψk))Mk=1 ∈ RM (gradient).2 Calculate A = An+1 = (J ′′(u)(ψj , ψk))Mj,k=1 (Hessian).

3 Compute χ = χn+1 = A−1g by solving the system.4 Set a = an+1 = an − δχ and update u = un+1 =

∑akψk .

5 Increment counter n6 Calculate sig(A(a)),if desired.7 STOP when

√g · g = ‖PG∇J(u)‖ < TOL .


ContentsMotivation



tGNGA

Create tangent vector T = (Pcur−Pold)‖Pcur−Pold‖ . Define initial guess

Pguess = Pcur + δT .

We then define the constraint κ = (P − Pguess) · T .

Forcing κ = 0 ensures that the vector made by our solution Pand Pguess is perpendicular to T .

Note that p = (ai , s) ∈ RM+1


ContentsMotivation



Illustration of tGNGA

S1 2 3 4

‖a‖

1

2

3

4

0

pinitial

pguess

pfinal

δ

1


ContentsMotivation



cGNGA

Given a solution p∗ where the kth eigenvalue is zero, let ek bethe corresponding eigenfunction of the Hessian.

We define the constraint κ = ||Pek u||2 − δ2.

Forcing this to be zero ensures that the projection of oursolution u onto the kth eigenvector is some positive amount δ.

If this is symmetry breaking then u not on the mother branch.


ContentsMotivation



The ring: x2 + y 2 = 1

∆Su + f (u) = 0 on the unit ring:

7.5cm


ContentsMotivation



Psi matrix for the ring

The Psi matrix for the ring is given by:

Ψ =

...

......

1√2π

cos(mθi )√π

sin(mθi )√π

......

...


ContentsMotivation



Eigenfunctions on the ring

Here is a picture of the seventh eigenfunction on the ring:


ContentsMotivation



Constant branch on the ring

Here is a picture of the constant bifurcation branch for the ring:


ContentsMotivation



First four nontrivial branches on the ring

Here is a picture of the first four non-trivial bifurcation branchesfor the ring:


ContentsMotivation



Example of Bifurcation Cone on the ring

Since any rotation of a solution is also a solution to the PDE weobtain a bifurcation cone for each branch:

1

Figure: Bifurcation cone for the 4th branch on the ring


ContentsMotivation



The Sphere

Our next experiment is to solve the PDE on the sphere.

Spherical harmonics are the well-known eigenfunctions of ∆S

on the sphere.

The spherical harmonics look like:Y ml (ψ, θ) = Pm

l (cos(φ)(acos(mθ) + b sin(mθ)), where Pml ,

|m| ≤ l is the associated Legendre function.


ContentsMotivation



Spherical Harmonics

Below is a picture illustrating the real part of several lowerorder/degree spherical harmonics:

Figure: Low order/degree real spherical harmonicsSpringer Solving Semilinear Elliptic PDEs on Manifolds

ContentsMotivation



Next steps for Sphere

Currently writing code to populate ψ matrix for sphere.

Next step to implement tGNGA/cGNGA code on the sphere

Symmetry analysis of the solutions on the sphere


ContentsMotivation



No closed form eigenfunctions?

What to do if we do not know closed form for eigenfunctions?

Use the Closest Point Method!

The Closest Point Method (CPM) is a recent embeddingmethod for solving time-dependent PDE’s on surfaces, whichcan also easily be utilized to solve eigenvalue problems onsurfaces.

With CPM we can solve our PDE of interest on generalmanifolds, even those without a well defined inside/outside!


ContentsMotivation



Building blocks of CPM

Closest Point Method uses three simple and fundamental ”buildingblocks” of numerical analysis:

1 Interpolation

2 Finite differences

3 Time stepping

These are combined in a straightforward way to solve surfaceeigenvalue problems.


ContentsMotivation



Details of the CPM

Formally, if S is a smooth surface in Rd then the closest pointextension of v : S → R is a function u : Ω→ R, defined in aneighborhood Ω ⊂ Rd of S ,as u(x) = v(cp(x)). We say thatu is a closest point extensionof v .

We use CPM to solve the problem: Given a surface Wdetermine a surface eigenfunction u : S → R and eigenvalue λsuch that:

−∆S(u(x)) = λu(x)


ContentsMotivation



An embedded eigenvalue problem

We first would like to solve the problem: Determine theeigenfunctions v : Ω ⊂ Rd → R and eigenvalues λ satisfying:

−∆(v(cp(x))) = λv(x)

This eigenvalue problem is ill-posed. Why?

The set of null-eigenfunctions of the embedded eigenvalueproblem is much larger than the set of null-eigenfunctions forthe Laplace-Beltrami eigenvalue problem.


ContentsMotivation



An eigenfunction on the Pig using CPM

Here is a picture of an eigenfunction on the surface of the pig,obtained using the CPM:


ContentsMotivation



Future Work

Finalize computations and bifurcation diagram for sphere

Study more interesting manifolds, like Anne’s Pig above, usingCPM.

Compute bifurcation diagrams for surfaces with mixedco-dimension.

Continue to generalize the well known variational theory tosolving on manifolds.


ContentsMotivation



Kaust Beacon

Here is one interesting manifold I am interested in solving our PDEon in the near future:


ContentsMotivation



Acknowledgments

I would like to thank the following people:

Dr. John Neuberger

Dr. Jim Swift

Tyler Diggans

Ian Douglas


ContentsMotivation



Questions?Thank you for listening!


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Solving Semilinear Elliptic PDEs on Manifolds

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