Abstract. In this paper, we state a convergence result for an L 1 -based finite element approximation technique in one dimension. The proof of this result is constructive and provides the basis for an algorithm for computing L 1 -based almost minimizers with optimal complexity. Several numerical results are presented to illustrate the performance of the method.Key words. finite elements, best L 1 -approximation, viscosity solution, transport, ill-posed problem, HJ equation, eikonal equation AMS subject classifications. 65N35, 65N22, 65F05, 35J05 IntroductionThis paper is concerned with the approximation of first-order PDEs using finite element-based best L 1 -approximations. This type of approximation technique has been introduced by Lavery [13,14] and further explored in Guermond [8]. Numerical tests reported in these references suggest that L 1 -based minimization techniques can compute the viscosity solutions of some first-order PDEs. This fact has been proved in one space dimension for linear first-order PDEs equipped with ill-posed boundary conditions in Lavery [14] and Guermond and Popov [11]. The proofs in the two above references are quite technical and rely essentially on explicit computations of the minimizers. The technicalities therein are such that it is difficult to really understand from these two proofs why the L 1 -minimizer performs so well. The first objective of the present work is to revisit [11] and to give a very simple proof of the above statement. The key argument is to show that the L 1 -minimizer selects the up-wind solution. To the best of our knowledge, the present paper is the first showing that L 1 -minimization techniques automatically introduce up-winding on linear transport problems. Based on the constructive argument from the new proof, the second objective of the paper is to propose a fast algorithm for computing L 1 -minimizers. This algorithm involves O(N ) operations where N is the number of degrees of freedom.The paper is organized as follows. In Section 2 we revisit the one-dimensional ill-posed model problem considered in Guermond and Popov [11], and we give an elementary proof of the fact that L 1 -minimizers converge to the unique viscosity solution of this problem. The key to this result is that local L 1 -minimization selects the up-wind information as proved in Lemma 2.1. Based on the local minimization argument unveiled in Lemma 2.1, we construct in Section 3 a fast algorithm for solving the L 1 -minimization problem associated with the one-dimensional ill-posed model problem. The algorithm is tested on an ill-posed problem and on a transport problem with discontinuous velocity. In Section 4 we generalize the algorithm to nonlinear one-dimensional first-order PDEs. We essentially focus our attention on stationary

The stochastic integrate and fire neuron is one of the most commonly used stochastic models in neuroscience. Although some cases are analytically tractable, a full analysis typically calls for numerical simulations. We present a fast and accurate finite volume method to approximate the solution of the associated Fokker-Planck equation. The discretization of the boundary conditions offers a particular challenge, as standard operator splitting approaches cannot be applied without modification. We demonstrate the method using stationary and time dependent inputs, and compare them with Monte Carlo simulations. Such simulations are relatively easy to implement, but can suffer from convergence difficulties and long run times. In comparison, our method offers improved accuracy, and decreases computation times by several orders of magnitude. The method can easily be extended to two and three dimensional Fokker-Planck equations.

We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker-Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein-Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.

We derive a reaction-diffusion system modeling the spatial propagation of a disease with kinetics occurring on distinct spatial domains. This corresponds to the actual invasion of a disease from a species living in a given spatial domain toward a second species living in a different spatial domain. We study the global existence of solutions and discuss the long time behavior of solutions. Then we consider a special case, based on a model of brain worm infection from white-tailed deer to moose populations, for which we discuss the invasion success/failure process and disprove a conjecture stated in an earlier work.

This paper is devoted to the mathematical analysis of the miscible displacement of a set of radionuclides in a flow occurring in a heterogeneous porous medium. The flow is governed by Darcy's law, and the motion of the chemical species is given by a nonclassical advection-diffusion-reaction equations system. The novelty of the model lies in the adsorption phenomenon that leads to a time derivative of a nonlinear term in these equations. A semi-discretization method is used to establish the existence of weak solutions to this system. Uniform L ∞ -estimates on the solutions are specified.