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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