Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Finite element methods for Maxwell's equations.

We present and analyze a perfectly matched, absorbing layer model for the velocity-stress formulation of elastodynamics. The principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components: a component orthogonal to the boundary and a component parallel to it. A system of equations governing these new unknowns then is constructed. A damping term finally is introduced for the component orthogonal to the boundary. This layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium. In practice, both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers. As we will show with several experiments, this model gives very satisfactory results; namely, the reflection coefficient, even in the case of heterogeneous, anisotropic media, is about 1% for a layer thickness of five space discretization steps.

Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.

An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation

In 1994 Berenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculations. In this paper we show how to derive and implement the Berenger layer in curvilinear coordinates (in two space dimensions). We prove that an infinite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated Berenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain.

/pdf/the-perfectly-matched-layer-in-curvilinear-coordinates-7dzv9lftok.pdf

The Perfectly Matched Layer in Curvilinear Coordinates

Optimizing the Perfectly Matched Layer

https://hal.inria.fr/inria-00073216/document

Francis Collino

Papers

The Perfectly Matched Layer in Curvilinear Coordinates

Optimizing the Perfectly Matched Layer

Domain Decomposition Method for Harmonic Wave Propagation : A General Presentation

Perfectly Matched Absorbing Layers for the Paraxial Equations

Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations