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Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the flow between two topologically-equivalent curves or surfaces which minimises some metric defined on the vector fields, emph{i.e.} the flow closest to the identity in some sense. In this paper, we describe a new particle-mesh discretisation for the evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms are very natural for this problem because Lagrangian particles (particles moving with the flow) can represent the movement of the shape whereas the vector field is Eulerian and hence best represented on a static mesh. We explain the derivation of the method, and prove conservation properties: the discrete method has a set of conserved momenta corresponding to the particle-relabelling symmetry which converge to conserved quantities in the continuous problem. We also introduce a new discretisation for the geometric current matching condition of (Vaillant and Glaunes, 2005). We illustrate the method and the derived properties with numerical examples.
The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angu
A Lagrangian-type numerical scheme called the comoving mesh method or CMM is developed for numerically solving certain classes of moving boundary problems which include, for example, the classical Hele-Shaw flow problem and the well-known mean curvat
The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions.
A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh par
We investigate several important issues regarding the Random Batch Method (RBM) for second order interacting particle systems. We first show the uniform-in-time strong convergence for second order systems under suitable contraction conditions. Second