# Optimal transport methods for mesh generation Chris Budd (Bath), JF Williams (SFU)

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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

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 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 dont need this in general

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) = 1 Basic mesh equation: unique solution if M>0

Hard, and unecessary, to solve exactly! Various approaches to the solution … 1. Geometric conservation 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

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

Monge-Kantorovich optimal transport problem Minimise Subject to Also used in image registration,meteorology

Intuitively a good approach A = 1 A = 2 Small I Larger I Optimal transport helps to prevent small angles

Breniers 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 : Scalar mesh potential Map F is a Legendre Transformation

Some corollaries of the Polar Factorisation Theorem Gradient map Irrotational mesh Convexity of P prevents mesh tangling

Some simple examples Uniform enlargement scale factor 1/M Linear map. A is symmetric positive definite Tensor product mesh

Monge-Ampere equation: fully nonlinear elliptic PDE 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 dont 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 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. 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 Extremely useful property when working with PDEs which have natural scaling laws eg.

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

2. Solve in parallel with the PDE Mesh: Solution: X Y 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

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