Showing papers in "Esaim: Proceedings in 2021"
11 citations
TL;DR: The mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column, are presented and the characteristic equations of Riemann invariants are used to obtain the discretized transmission conditions.
Abstract: In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.
6 citations
TL;DR: In this paper, the authors analyzed the relevance of the use of the shallow water model and the Boussinesq model to simulate tsunamis generated by a landslide and determined if the two models are able to reproduce waves generated by the landslide.
Abstract: In this paper, we analyze the relevance of the use of the shallow water model and the Boussinesq model to simulate tsunamis generated by a landslide. In a first part, we determine if the two models are able to reproduce waves generated by a landslide. Each model has drawbacks but it seems that it is possible to use them together to improve the simulations. In a second part we try to recover the landslide displacement from the generated wave. This problem is formulated as a minimization problem and we limit the number of parameters to determine assuming that the bottom can be well described by an empirical law.
4 citations
TL;DR: This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, such that, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε.
Abstract: This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution x A. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {xe} (e ≥ 0), in a way such that: firstly, x0 = x A (compatibility), and secondly, for e sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(e 2), whereas the secondary cost functions are linearly decreasing functions of e. The theoretical results are recalled and the method is applied numerically to a SuperSonic Business Jet (SSBJ) sizing problem to optimize the flight performance.
3 citations
TL;DR: The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one and a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems.
Abstract: The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.
3 citations
TL;DR: An account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm.
Abstract: This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O 3 schemes.
2 citations
TL;DR: In this paper, a simplified shear-thinning fluid is obtained for a simplified rheology consisting of a piecewise linear stress tensor, resulting in a two-viscosity model.
Abstract: A lubrication equation is obtained for a simplified shear-thinning fluid. The simplified rheology consists of a piecewise linear stress tensor, resulting in a two-viscosity model. This can be interpreted as a modified Bingham fluid, which can be recovered in a specific limit. The lubrication equation is obtained in two steps. First two scalings are performed on the incompressible Navier-Stokes equations, namely the long-wave scaling and the slow motion scaling. Second, the resulting equations are averaged along the vertical direction. Numerical illustrations are provided, bringing to light the different possible behaviours.
2 citations
TL;DR: In this article, a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force are investigated, which rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance.
Abstract: We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.
2 citations
TL;DR: In this article, a non-parametric Bayesian approach is proposed to model the induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model.
Abstract: In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model. The considered models deliver probabilistic predictions of response distributions at candidate points, allowing for instance to perform (approximate) posterior simulations of probability density functions, to jointly predict multiple moments and other functionals of target distributions, as well as to quantify the impact of collecting new samples on the state of knowledge of the distribution field of interest. In particular, we introduce adaptive sampling strategies leveraging the potential of the considered random distribution field models to guide system evaluations in a goal-oriented way, with a view towards parsimoniously addressing calibration and related problems from non-linear (stochastic) inversion and global optimisation.
1 citations
TL;DR: This paper uses a new proximal splitting method for the minimization of a criterion using I2 norm both for the constraint and the loss function and provides a new efficient tailored primal Douglas-Rachford splitting algorithm which is very effective on high dimensional dataset.
Abstract: This paper deals with supervised classification and feature selection with application in the context of high dimensional features. A classical approach leads to an optimization problem minimizing the within sum of squares in the clusters (I 2 norm) with an I 1 penalty in order to promote sparsity. It has been known for decades that I 1 norm is more robust than I 2 norm to outliers. In this paper, we deal with this issue using a new proximal splitting method for the minimization of a criterion using I 2 norm both for the constraint and the loss function. Since the I 1 criterion is only convex and not gradient Lipschitz, we advocate the use of a Douglas-Rachford minimization solution. We take advantage of the particular form of the cost and, using a change of variable, we provide a new efficient tailored primal Douglas-Rachford splitting algorithm which is very effective on high dimensional dataset. We also provide an efficient classifier in the projected space based on medoid modeling. Experiments on two biological datasets and a computer vision dataset show that our method significantly improves the results compared to those obtained using a quadratic loss function.
1 citations
TL;DR: The optimization procedure for computing the discrete boxconstrained minimax classifier is presented, and a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain is considered.
Abstract: In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.
TL;DR: This work investigates an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers.
Abstract: Multigrid methods for two-body contact problems are mostly
based on special mortar discretizations, nonlinear Gauss-Seidel
solvers, and solution-adapted coarse grid spaces. Their high
computational efficiency comes at the cost of a complex implementation
and a nonsymmetric master-slave discretization of the nonpenetration
condition. Here we investigate an alternative symmetric and
overconstrained segment-to-segment contact formulation that
allows for a simple implementation based on standard multigrid and
a symmetric treatment of contact boundaries, but leads to nonunique
multipliers. For the solution of the arising quadratic programs,
we propose augmented Lagrangian multigrid with overlapping block
Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems.
TL;DR: In this article, the authors studied two-player one-dimensional discrete Hotelling pure location games with demand f (d ) = w d with 0 ≤ 1 and showed that this game admits a best response potential.
Abstract: We study two-player one-dimensional discrete Hotelling pure location games assuming that demand f (d ) as a function of distance d is constant or strictly decreasing. We show that this game admits a best-response potential. This result holds in particular for f (d ) = w d with 0 ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness of best-response correspondences will be made.
TL;DR: In this article, a functional formulation of the classical homicidal chauffeur Nash game is presented and a numerical framework for its solution is discussed, which combines a Hamiltonian based scheme with proximal penalty to determine the time horizon where the game takes place with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time.
Abstract: A functional formulation of the classical homicidal chauffeur Nash game is presented and a numerical framework for its solution is discussed. This methodology combines a Hamiltonian based scheme with proximal penalty to determine the time horizon where the game takes place with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time.
TL;DR: This paper uses a model in another domain based on a data-model coupling approach to simulate and predict SST, and describes the original and modified model.
Abstract: The Sea Surface Temperature (SST) plays a significant role in analyzing and assessing the dynamics of weather and also biological systems. It has various applications such as weather forecasting or planning of coastal activities. On the one hand, standard physical methods for forecasting SST use coupled ocean- atmosphere prediction systems, based on the Navier-Stokes equations. These models rely on multiple physical hypotheses and do not optimally exploit the information available in the data. On the other hand, despite the availability of large amounts of data, direct applications of machine learning methods do not always lead to competitive state of the art results. Another approach is to combine these two methods: this is data-model coupling. The aim of this paper is to use a model in another domain. This model is based on a data-model coupling approach to simulate and predict SST. We first introduce the original model. Then, the modified model is described, to finish with some numerical results.
TL;DR: A residual–based a posteriori error estimate is studied for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term.
Abstract: We study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems.
TL;DR: In this paper, a generalization of the indirect pseudo-spectral method, presented in [17], for the numerical solution of budget-constrained infinite horizon optimal control problems is presented.
Abstract: In this paper a generalization of the indirect pseudo-spectral method, presented in [17], for the numerical solution of budget-constrained infinite horizon optimal control problems is presented. Consideration of the problem statement in the framework of weighted functional spaces allows to arrive at a good approximation for the initial value of the adjoint variable, which is inevitable for obtaining good numerical solutions. The presented method is illustrated by applying it to the budget-constrained linear-quadratic regulator model. The quality of approximate solutions is demonstrated by an example.
TL;DR: In this paper, the sensitivity of the intermediate point c to the Legendre-Fenchel transformation for convex functions was studied and the expression of its gradient ∇c(d,d) was derived.
Abstract: We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ \\frac{f(a)-f(b)}{b-a}={f}^{\\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.
TL;DR: In this paper, the Serre-Green-Naghdi system under a non-hydrostatic formulation is studied and two numerical schemes are designed, based on a finite volume.
Abstract: We study the Serre–Green-Naghdi system under a non-hydrostatic formulation, modellingincompressible free surface flows in shallow water regimes. This system, unlike the well-known (non-linear) Saint-Venant equations, takes into account the effects of the non-hydrostatic pressure term aswell as dispersive phenomena. Two numerical schemes are designed, based on a finite volume – finitedifference type splitting scheme and iterative correction algorithms. The methods are compared bymeans of simulations concerning the propagation of solitary wave solutions. The model is also assessedwith experimental data concerning the Favre secondary wave experiments .
TL;DR: This paper uses exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set and uses the shift of the feasible region along the bisector of the positive orthant for this transformation.
Abstract: This paper presents a new method for global optimization. We use exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set. Quadratic regularization often allows you to convert a multimodal problem into a unimodal problem. For this, we use the shift of the feasible region along the bisector of the positive orthant. We use only local search (primal-dual interior point method) and a dichotomy method for search of a global extremum in the multimodal problems. The comparative numerical experiments have shown that this method is very efficient and promising.
TL;DR: A procedure for combining high order finite volumes and tree-based nonuniform grids and efficient algorithms for third order multidimensional volume interpolation and communication between levels of refinement is considered.
Abstract: We consider a procedure for combining high order finite volumes and tree-based non-uniform grids. Especially, we focus on efficient algorithms for third order multidimensional volume interpolation and communication between levels of refinement. In the end, numerical results are reviewed to validate our approach.
TL;DR: In this paper, the authors present and compare two methods: direct optimization and first-order Pontryagin type of conditions to compute the optimal sampling times for Dirac pulses, where a further refined numerical discretization is applied on the dynamics.
Abstract: Recent force-fatigue mathematical models in biomechanics [7] allow to predict the
muscular force response to functional electrical stimulation (FES) and leads to
the optimal control problem of maximizing the force. The stimulations are Dirac pulses
and the control parameters are the pulses amplitudes and times of application,
the number of pulses is physically limited and the model leads to a sampled data control problem.
The aim of this article is to present and compare two methods. The first method is a
direct optimization scheme where a further refined numerical discretization is applied on
the dynamics. The second method is an indirect scheme: first-order Pontryagin type
necessary conditions are derived and used to compute the optimal sampling times.
TL;DR: In this article, the authors discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations and explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme.
Abstract: In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.
TL;DR: In this article, the authors compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system, a staggered pseudo-Lagrangian scheme in which the mesh "follows" the fluid, and the Eulerian projection on a fixed mesh of the preceding scheme.
Abstract: In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: • a staggered scheme based on the Rusanov one for the inviscid (Euler) system, • a staggered pseudo-Lagrangian scheme in which the mesh "follows" the fluid, • the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.
TL;DR: Analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games, are discussed and are meant to serve as a starting point for the developement of efficient numerical solution methods formulti-leader-follower games.
Abstract: In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results.In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.