A numerical method for the solution of the elliptic MongeAmpère Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary conditions is combined with a finite difference scheme for the Monge-Ampère equation. Newton’s method is implemented leading to a fast solver, comparable to solving...

Numerical methods for the solution of partial di↵erential equations constitute an important class of techniques in scientific computing. Often, the discretization is based on approximating the partial derivatives by finite di↵erences on a regular Cartesian grid. The resulting computations are structured in the sense of updating a large, multidimensional array by a stencil operation. A stencil d...

Stencil computations are a class of algorithms which perform nearest-neighbor computation, often on a multi-dimensional grid. This type of calculation forms the basis for computer simulations across almost every field of science. The increasing computational speed of graphics processing units (GPUs) make their use for stencil computations an interesting goal. However, achieving highly efficient...

We present an object-oriented framework for constructing parallel implementations of stencil algorithms. This framework simplifres the development process by encapsulating the common aspects of stencil algorithms in a base stencil class so that application-specifre derived classes can be easily defined via inheritance and overloading. In addition, the stencil base class contains mechanisms for ...

We introduce an integral representation of the Monge–Ampère equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs for equation with either Dirichlet or optimal transport boundary conditions. use higher-order quadrature schemes allows substantial reduction in component error ...

We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High accuracy (up the sixth-order presently) is achieved, thanks polynomial reconstructions while stability provided with MOOD method which controls cell degree eliminating non-physical oscillations in vicinity of d...

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational me...

In this paper, we present a discontinuous Galerkin finite clement method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact ste...

The implementation of stencil computations on modern, massively parallel systems with GPUs and other accelerators currently relies on manually-tuned coding using low-level approaches like OpenCL and CUDA. This makes development of stencil applications a complex, time-consuming, and error-prone task. We describe how stencil computations can be programmed in our SkelCL approach that combines high...