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Optimal transport methods for mesh generation, with applications
to meteorology
Chris Budd, Emily Walsh JF Williams (SFU)
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Have a PDE with a rapidly evolving solution u(x,t)
eg. A stormHow can we generate a mesh which effectively follows the solution structure on many scales?
• h-refinement
• p-refinement
• r-refinementAlso need some estimate of the solution/error structure which may be a-priori or a-posteriori
Will describe the Parabolic Monge-Ampere method: an efficient n-dimensional r-refinement strategy using a-priori estimates, based on optimal transport ideas
C.J. Budd and J.F. Williams, Journal of Physics A, 39, (2006), 5425--5463.C.J. Budd and J.F. Williams, SIAM J. Sci. Comp (2009) C.J. Budd, W-Z Huang and R.D. Russell, ACTA Numerica (2009)
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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
CΩPΩ
),( ηξCΩ
€
ΩP (x, y)
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Equidistribution: 2D
Introduce a positive unit measure M(x,y,t) in the physical domain which controls the mesh densityA : set in computational domain
F(A) : image set
€
dA
∫ ξ dη = M(x, y, t) dx dy
F (A )
∫
Equidistribute image with respect to the measure
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€
M(x,y, t)∂(x,y)
∂(ξ ,η )=1
Differentiate to give:
Basic, nonlinear, equidistribution mesh equation
Choose M to concentrate points where needed without depleting points elsewhere: error/physics/scaling
Note: All meshes equidistribute some function M
[Radon-Nicodym]
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Choice of the monitor function M(X)
• Physical reasoning
eg. Vorticity, arc-length, curvature
• A-priori mathematical arguments
eg. Scaling, symmetry, simple error estimates
• A-posteriori error estimates
eg. Residuals, super-convergence
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Mesh construction
Problem: in two/three -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
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F
),( ηξCΩ
€
ΩP (x,y)
Optimally transported meshes
Argue: A good mesh for solving a pde is often one which is as close as possible to a uniform mesh
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Monge-Kantorovich optimal transport problem
€
I(x,y) =ΩC
∫ (x,y) − (ξ ,η )2dξ dη
€
M(x,y, t)∂(x,y)
∂(ξ ,η )=1
Minimise
Subject to
Also used in image registration,meteorology …..
Optimal transport helps to prevent small angles, reduce mesh skewness and prevent mesh tangling.
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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
),( ηξP
P : Scalar mesh potential
Map F is a Legendre Transformation
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Some corollaries of the Polar Factorisation Theorem
€
(x,y) =∇ξ P = (Pξ ,Pη )
€
xη = yξ
Gradient map
Irrotational mesh
Same construction works in all dimensions
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Monge-Ampere equation: fully nonlinear elliptic PDE
€
∂(x,y)
∂(ξ ,η )= H(P) = det
Pξξ Pξη
Pξη Pηη
⎛
⎝ ⎜
⎞
⎠ ⎟= Pξξ Pηη − P 2
ξη
1)(),( =∇ PHtPM
It follows immediately that
Hence the mesh equidistribustion equation becomes
(MA)
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Basic idea: Solve (MA) for P with appropriate (Neumann or Periodic) 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, and can instead use parabolic relaxation Q P
Alternatively: Use Newton [Chacon et. al.]
Use a variational approach [van Lent]
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Relaxation uses a combination of a rescaled version of MMPDE5 and MMPDE6 in 2D
€
ε I −αΔξ( ) Qt = M (∇Q)H(Q)( )1/ 2
Spatial smoothing [Hou]
(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)
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Basic approach
• Carefully discretise PDE & PMA in computational domain
QuickTime™ and a decompressor
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Solve the coupled mesh and PDE system either
(i) As one large system (stiff!)
Velocity based Lagrangian approach. Works well for paraboloic blow-up type problems (JFW)
or
(ii) By alternating between PDE and mesh
Interpolation based - rezoning - approach
Need to be careful with conservation laws here!!! But this method works very well for advective meteorological problems. (EJW)
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Because PMA is based on a geometric approach, it has a set of useful regularity properties
1. System invariant under translations, rotations, periodicity
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2. Convergence 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/convergent and the mesh evolves to an equidistributed state
tPtQ Λ+= )(),( ξξ
€
x(ξ ) =∇ξ Q =∇ξ P
Proof: Follows from the convexity of P which ensures that PMA behaves locally like the heat equation
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Lemma 2: [B,W 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: [B,W 2006]The mapping is 1-1 and convex for all times:
No mesh tangling or points crossing the boundary
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Lemma 4: [B,W 2005] Multi-scale property
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)()(),( ξξ PtLtQ =
€
x = L(t)Y (ξ )
Extremely useful properties when working with PDEs which have natural scaling laws
4. For appropriate choices of M the coupled system is scale-invariant
Lemma 5: [B,W 2009] Multi-scale regularity
The resulting scaled meshes have the same local regularity as a uniform mesh
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€
ut = uxx + uyy + u3, u → ∞ t → T
2/12/1 )log()()( tTtTtL −−=
€
M(x,y, t) =1
2
u(x, y)4
u4∫ dx dy+
1
2
Example 1: Parabolic blow-up
M is locally scale-invariant, concentrates points in the peak and keeps 50% of the points away from the peak
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Solve in parallel with the PDE
€
ut = uxx + uyy + u3
Mesh:
Solution:
xy
10 10^5
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Solution in the computational domain
ξη
10^5
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Example 2: Tropical storm formation (Eady problem)
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Solve using rezoning and interpolation
• Update solution every 10 mins
• Update mesh every hour
• Advection and pressure correction on adaptive mesh
• Discontinuity singularity after 6.3 days
€
R=f 2 + f vx f vz
gθ0−1θx gθ0θ z
⎛
⎝ ⎜
⎞
⎠ ⎟
Monitor function: Maximum eigen-value of R:
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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
• Looking good on meteorological problems
• Still lots of work to be done eg. Finding efficient ways to couple PMA to the underlying PDE