Handling convexity-like constraints in variational problems

We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.Comment: 23 page

Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure

Lorenz gauged vector potential formulations for the time-harmonic eddy-current problem with L∞-regularity of material properties

In this paper we consider some Lorenz gauged vector potential formulations of the eddy-current problem for the time-harmonic Maxwell equations with material properties having only L1-regularity. We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.Comment: 40 pages using amsart style file, and 15 figure

Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering