Optimal transport methods for mesh generation

Chris Budd (Bath), JF Williams (SFU)

Have a PDE with a rapidly evolving solution u(x,t)

How can we generate a mesh which effectively follows the solution structure?

• h-refinement

• p-refinement

• r-refinement

Also need some estimate of the solution/error structure which may be a-priori or a-posteriori

Will describe an efficient n-dimensional r-refinement strategy using a-priori estimates

r-refinement

Strategy for generating a mesh by mapping a uniform mesh from a computational domain into a physical domain

Need a strategy for computing the mesh mapping function F which is both simple and fast

F

CP

),( C ),( YXP

Equidistribution: 2D

Introduce a positive unit measure M(X,Y,t) in the physical domain which controls the mesh density

A : set in computational domain

F(A) : image set

)(

),,(AFA

dXdYtYXMdd

Equidistribute image with respect to the measure

JF: I wrote this talk using the assumption of unit measure for M as it simplifies the presentation. Of course we don’t need this in general

1),(

),(),,(

YX

tYXM

Differentiate:

Basic, nonlinear, equidistribution mesh equation

Equidistribution in one-dimension

This is a very well defined process

Let computational and physical domains both be the unit interval [0,1]. Mesh function X(xi)

X(0) = 0, X(1) = 11),( ddX

tXM

Basic mesh equation: unique solution if M>0

Hard, and unecessary, to solve exactly!

Various approaches to the solution …

0),(

ddX

tXMdt

d

0)( Xt MVM

1. Geometric conservation

t

XV

Solve for V

Some CJB notes on possible issues with GCL

• Have to start with an equidistributed mesh

• Have to know M_t

• Have to calculate V then calculate X

• Potential problems with mesh crossing

• Generalises to higher dimensions

0,0).( VMVM XXt

X

tXMX t ),(

X

tXMX t ),(

2. Relaxation

or

Very effective provided that the time-scales for the mesh evolution are smaller than those for the evolution of the underlying PDE: MOVCOL Code [Huang, Russell]

Evolve towards an equidistributed mesh

Mesh PDE [Russell]

(MMPDE5)

(MMPDE6)

Implementation:

)(tX i

),( txu

)),(()( ttXutU ii

]1,0[

Underlying PDE solution:

Moving mesh:

Approximate solution:

Discretise Underlying PDE (in Lagrangian form) and

Mesh PDE in the computational variable

Solve the resulting ODEs

Back to two-dimensions

Problem: in two-dimensions equidistribution does NOT uniquely define a mesh!

All have the same area

Need additional conditions to define the mesh

Also want to avoid mesh tangling and long thin regions

F

),( C ),( YXP

Optimally transported meshes

Argue: A good mesh is one which is as close as possible to a uniform mesh

Monge-Kantorovich optimal transport problem

ddYXYXIC

2),(),(),(

1),(

),(),,(

YX

tYXM

Minimise

Subject to

Also used in image registration,meteorology

Intuitively a good approach

A = 1

A = 2

A = 2

Small I

Larger I

Optimal transport helps to prevent small angles

Brenier’s polar factorisation theorem

Key result which makes everything work!!!!!

Theorem: [Brenier]

(a)There exists a unique optimally transported mesh

(b) For such a mesh the map F is the gradient of a convex function

),( P

P : Scalar mesh potential

Map F is a Legendre Transformation

Some corollaries of the Polar Factorisation Theorem

),(),( PPPYX

YX

Gradient map

Irrotational mesh

Convexity of P prevents mesh tangling

22

1 22 M

P

TT

bA

P 2

Some simple examples

)()( BAP

Uniform enlargement scale factor 1/M

Linear map. A is symmetric positive definite

Tensor product mesh

Monge-Ampere equation: fully nonlinear elliptic PDE

2det)(

),(

),(PPP

PP

PPPH

YX

1)(),( PHtPM

It follows immediately that

Hence the mesh equidistribustion equation becomes

(MA)

Basic idea: Solve (MA) for P with appropriate (Neumann) boundary conditions

Good news: Equation has a unique solution

Bad news: Equation is very hard to solve

Good news: We don’t need to solve it exactly!

Use relaxation as in the MMPDE equations

Relaxation uses a combination of a rescaled version of MMPDE5 and MMPDE6 in 2D

2/1)()( QHQMQI t

Spatial smoothing

(Invert operator using a spectral method)

Averaged monitor

Ensures right-hand-side scales like Q in 2D to give global existence

Parabolic Monge-Ampere equation

(PMA)

Discretise and solve PMA in parallel with the Lagrangian form of the PDE (possibly using a temporal rescaling)Useful properties of PMA

Lemma 1: [Budd,Williams 2006]

(a) If M(X,t) = M(X) then PMA admits the solution

(b) This solution is locally stable.

tPtQ )(),( PQX )(

Proof: Follows from the convexity of P which ensures that PMA behaves locally like the heat equation

Lemma 2: [Budd,Williams 2006]

If M(X,t) is slowly varying then the grid given by PMA is epsilon close to that given by solving the Monge Ampere equation.

Lemma 3: [Budd,Williams 2006]

The mapping is 1-1 and convex for all times

Additional issues for CJB and JFW

• If M is slowly varying then Q stays epsilon close to the solution of the Monge-Ampere equation

• Need rigorous proof that Q remains convex and of global convergence and a maximum principle

• When we solve in a rectangular domain then there is a mild loss of regularity in the corners.

• There is also an boundary orthogonality of the grid in the physical domain. This is both good and bad

Lemma 4: [Budd,Williams 2005]

If there is a natural length scale L(t) then for careful choices of M the PMA inherits this scaling and admits solutions of the form

)()(),( PtLtQ )()( YtLX

3uuu Xt

Extremely useful property when working with PDEs which have natural scaling laws eg.

2/12/1 )log()()( tTtTtL

2

1

2

1),(

4

4

dXdYu

utXM

Examples of applications

1. Prescribe M(X,t) and solve PMA

2. Solve in parallel with the PDE3uuu Xt

Mesh:

Solution:

XY

10 10^5

Solution in the computational domain

10^5

Conclusions

• Optimal transport is a natural way to determine meshes in dimensions greater than one

• It can be implemented using a relaxation process by using the PMA algorithm

• Method works well for a variety of problems, and there are rigorous estimates about its behaviour

• Still lots of work to be done eg. Finding efficient ways to couple PMA to the underlying PDE