solving semilinear elliptic pdes on manifolds
TRANSCRIPT
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
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
1 Motivation
2 Overview of GNGA
3 Preliminary Experiments
4 Overview of the Closest Point Method (CPM)
5 Future Work
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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 .
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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 )
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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 .
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Illustration of tGNGA
S1 2 3 4
‖a‖
1
2
3
4
0
pinitial
pguess
pfinal
δ
1
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
The ring: x2 + y 2 = 1
∆Su + f (u) = 0 on the unit ring:
7.5cm
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Psi matrix for the ring
The Psi matrix for the ring is given by:
Ψ =
...
......
1√2π
cos(mθi )√π
sin(mθi )√π
......
...
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Eigenfunctions on the ring
Here is a picture of the seventh eigenfunction on the ring:
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Constant branch on the ring
Here is a picture of the constant bifurcation branch for the ring:
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
First four nontrivial branches on the ring
Here is a picture of the first four non-trivial bifurcation branchesfor the ring:
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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!
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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)
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
An eigenfunction on the Pig using CPM
Here is a picture of an eigenfunction on the surface of the pig,obtained using the CPM:
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
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.
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Kaust Beacon
Here is one interesting manifold I am interested in solving our PDEon in the near future:
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Acknowledgments
I would like to thank the following people:
Dr. John Neuberger
Dr. Jim Swift
Tyler Diggans
Ian Douglas
Springer Solving Semilinear Elliptic PDEs on Manifolds
ContentsMotivation
Overview of GNGAPreliminary Experiments
Overview of the Closest Point Method (CPM)Future Work
Questions?Thank you for listening!
Springer Solving Semilinear Elliptic PDEs on Manifolds