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

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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

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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

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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

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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

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Differentiate: Basic, nonlinear, equidistribution mesh equation

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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

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

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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

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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)

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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

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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

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

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

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

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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

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

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

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Monge-Ampere equation: fully nonlinear elliptic PDE It follows immediately that Hence the mesh equidistribustion equation becomes (MA)

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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

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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)

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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

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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

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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

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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.

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

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2. Solve in parallel with the PDE Mesh: Solution: X Y 10 10^5

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Solution in the computational domain 10^5

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

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