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Gerard Awanou

Publications and source records attributed to Gerard Awanou.

At least 19 recordsLinked to original sources
arXiv2025

Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Ampère equation, we give the first proof of uniqueness. The boundary condition is enforced through the use of the notion of asymptotic cone while the differential operator is discretized based on a discrete analogue of the subdifferential. We establish the convergence of a subsequence of a damped Newton's method for the nonlinear system resulting from the discretization, thereby proving the existence of a solution. Using related arguments we then prove that such a solution is necessarily unique. Convergence of the discretization as well as numerical experiments are given.

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arXiv2024

The second boundary value problem for a discrete Monge-Ampere equation

In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

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arXiv2022

On uniqueness of weak solutions to the second boundary value problem for generated prescribed Jacobian equations

We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions.

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arXiv2021

Hybridization and postprocessing in finite element exterior calculus

We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential $k$-forms in $\mathbb{R}^n$. In the cases $k = 0$ and $k = n$, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for $0 < k < n$, we obtain new hybrid finite element methods, including methods for the vector Poisson equation in $n = 2$ and $n = 3$ dimensions. We also generalize Stenberg postprocessing from $k = n$ to arbitrary $k$, proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.

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arXiv2021

On weak convergence of Monge-Ampere measures for discrete convex mesh functions

To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of mesh functions implies the uniform convergence on compact subsets of their convex envelopes and hence the weak convergence of the associated Monge-Ampere measures. We also give conditions for mesh functions to have a subsequence which converges uniformly to a convex function. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampere equation and was used for a recently proposed discretization of the latter. For mesh functions which are uniformly bounded and satisfy a convexity condition at the discrete level, we show that there is a subsequence which converges uniformly on compact subsets to a convex function. The convex envelopes of the mesh functions of the subsequence also converge uniformly on compact subsets. If in addition they agree with a continuous convex function on the boundary, the limit function is shown to satisfy the boundary condition strongly.

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arXiv2019

Discrete Aleksandrov solutions of the Monge-Ampere equation

We prove the convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition, can be used to get a good initial guess for geometric methods solving optimal transport between two measures.

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arXiv2019

Uniform limit of discrete convex functions

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform limit on compact subsets of discrete convex mesh functions which are uniformly bounded is a continuous convex function. Furthermore, if the discrete convex mesh functions interpolate boundary data of a continuous convex function and their Monge-Ampere masses are uniformly bounded, the limit function satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. The result is applied to the convergence of some numerical methods for the Monge-Ampere equation.

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arXiv2018

A two-grid method for the $C^0$ interior penalty discretization of the Monge-Amp\`{e}re equation

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Amp\`ere equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.

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arXiv2015

A variational method for computing numerical solutions of the Monge-Ampere equation

We present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear constraints. When the equation is discretized with a certain monotone scheme, we prove that the unique minimizer of the discrete problem solves the finite difference equation. For the numerical results we use both the standard finite difference discretization and the monotone scheme. Results with standard tests confirm that the numerical approximations converge to the Aleksandrov solution.

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arXiv2014

Iterative methods for k-Hessian equations

On a domain of the n-dimensional Euclidean space, and for an integer k=1,...,n, the k-Hessian equations are fully nonlinear elliptic equations for k >1 and consist of the Poisson equation for k=1 and the Monge-Ampere equation for k=n. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate. In addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Ampere equation is established and new Gauss-Seidel type iterative methods for 2-Hessian equations are introduced.

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arXiv2014

Convergence of a hybrid scheme for the elliptic Monge-Ampere equation

We prove the convergence of a hybrid discretization to the viscosity solution of the elliptic Monge-Ampere equation. The hybrid discretization uses a standard finite difference discretization in parts of the computational domain where the solution is expected to be smooth and a monotone scheme elsewhere. A motivation for the hybrid discretization is the lack of an appropriate Newton solver for the standard finite difference discretization on the whole domain.

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arXiv2013

On standard finite difference discretizations of the elliptic Monge-Ampere equation

Given an orthogonal lattice with mesh length h on a bounded convex domain, we propose to approximate the Aleksandrov solution of the Monge-Ampere equation by regularizing the data and discretizing the equation in a subdomain using the standard finite difference method. The Dirichlet data is used to approximate the solution in the remaining part of the domain. We prove the uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain. The result explains the behavior of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions. We propose an algorithm which for smooth solutions appears faster than the popular Newton's method with a high accuracy for non smooth solutions. The convergence of the algorithm is independent of how close to the numerical solution the initial guess is, upon rescaling the equation and given a user's measure of the closeness of an initial guess.

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arXiv2013

Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions

We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0 or C1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Amp\`ere equation.

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arXiv2013

Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: classical solutions

We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation. Error estimates are given for Lagrange elements of degree d >= 2 in dimensions 2 and 3. No jump term is used in the variational formulation. We propose to solve the discrete nonlinear system of equations by a time marching method and numerical evidence is given which indicates that one approximates weak solutions in two dimensions.

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arXiv2013

Mixed finite elements for elasticity on quadrilateral meshes

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer $r \geq 2$ and with rate of convergence in the $L^2$ norm of order $r$ for all the variables. The methods use Raviart-Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.

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