Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations

Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements.It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010), Rossmanith and Seal (2011),for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensionalPDEs. For second-order PDEs the idea of the schemeis a blending between weak Taylor approximations and projection on a DG basis.New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients.In particular we obtain high-order schemes, unconditionally stable and convergent,in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients.In the case of non-constant coefficients, we construct, in some particular cases,"almost" unconditionally stable second-order schemes and give precise convergence results.The schemes are tested on several academic examples

High-order filtered schemes for time-dependent second order HJB equations

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward difference formulae (BDF) for constructing the high order schemes.Comment: 27 pages, 16 figures, 4 table

On the Original Proof by Reductio ad Absurdum of the Hohenberg-Kohn Theorem for Many-Electron Coulomb Systems

It is shown that, for isolated many-electron Coulomb systems with Coulombic external potentials, the usual reductio ad absurdum proof of the Hohenberg-Kohn theorem is unsatisfactory since the to-be-refuted assumption made about the one-electron densities and the assumption about the external potentials are not compatible with the Kato cusp condition. The theorem is, however, provable by more sophisticated means and it is shown here that the Kato cusp condition actually leads to a satisfactory proof.Comment: 14 pages. Int. J. Quantum Chem. to appea

An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations

We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The proposed schemes are not monotone but still satisfy some $\epsilon$-monotone property. Convergence results and precise error estimates are given, of the order of $\sqrt{\Delta x}$ where $\Delta x$ is the mesh size. The framework allows to construct finite difference discretizations that are easy to implement, high--order in the domains where the solution is smooth, and provably convergent, together with error estimates. Numerical tests on several examples are given to validate the approach, also showing how the filtered technique can be applied to stabilize an otherwise unstable high--order scheme.Comment: 20 pages (including references), 26 figure

Safe Sequential Path Planning Under Disturbances and Imperfect Information

Multi-UAV systems are safety-critical, and guarantees must be made to ensure no unsafe configurations occur. Hamilton-Jacobi (HJ) reachability is ideal for analyzing such safety-critical systems; however, its direct application is limited to small-scale systems of no more than two vehicles due to an exponentially-scaling computational complexity. Previously, the sequential path planning (SPP) method, which assigns strict priorities to vehicles, was proposed; SPP allows multi-vehicle path planning to be done with a linearly-scaling computational complexity. However, the previous formulation assumed that there are no disturbances, and that every vehicle has perfect knowledge of higher-priority vehicles' positions. In this paper, we make SPP more practical by providing three different methods to account for disturbances in dynamics and imperfect knowledge of higher-priority vehicles' states. Each method has different assumptions about information sharing. We demonstrate our proposed methods in simulations.Comment: American Control Conference, 201

Unveiling and Revealing the Mirror: Mobile Reverberations in Patrick Bokanowski's Animated Films

[EN] Abstract The principal focus of this article is to provide an analysis of some of the most significant works by the independent filmmaker Patrick Bokanowski and the methodology he uses to create those animated films. It addresses Bokanowski¿s technique of filming on various kinds of reflective surfaces as a strategy for transforming the image through optical deformation and thereby expressing his subjective vision in films and animation. He explores the practical application of creating reverberation as visual echoes, as well as the use of unusual reflections that appear in mirrors and other more unstable reflective surfaces, such as liquid mercury, and the refraction occurring in sculpted lenses, glass or water. The aim of said strategy is the pure expression of subjective experience, with the director using a camera as if it were an entirely free-moving paintbrush that inspires creativity and breaks loose from automatic camera movements. Accordingly, this study examines the dynamic, expressive potential of the language of catoptrics, taking as paradigmatic examplestwo of Bokanowski's short films: La Plage (1992) and Au bord du lac (1994). Our analyses of these films are aimed at demonstrating how the artist manages to bring about qualitative, transformative changes of forms and space by reflecting images on mirrors, ripples in mercury and movements of water, and by combining these elements. The system developed by Bokanowski successfully transports us to an ever-changing, poetic universe and breaks new ground in the field of animation.The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article. This work was supported by the Spanish Ministry of Economy, Industry and Competitiveness, and was co-financed by the European Social Fund in the call for applications for Predoctoral Scholarships for Doctoral Training: [BES 2014-069330], within the framework of the R&D&I project: Alternative Strategies for the Creation of Animated Images Based on Its Essence: The Dynamic Expression [HAR2013-41708-P].Tamara Rama-Sotos; Lloret Ferrándiz, MC. (2019). Unveiling and Revealing the Mirror: Mobile Reverberations in Patrick Bokanowski's Animated Films. Animation. 14(2):1-19. https://doi.org/10.1177/1746847719856997S119142Bokanowski P (1992) Réflexions Optiques. Paris: In Bref, n.11, 34–38.Bokanowski P (2017a) Incandescent Material, Notes on the Cinematic Image. Published in booklet included in: Un Rêve Solaire DVD 2018. Paris: Re:Voir, 3–7.Buchan, S. (2013). Pervasive Animation. doi:10.4324/978020315257

Minimal Time Problems with Moving Targets and Obstacles

International audienceWe consider minimal time problems governed by nonlinear systems under general time dependant state constraints and in the two-player games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controlability assumption is made. In addition to these difficulties, we are interested here to the case when the target, the state constraints and the dynamics are allowed to be time-dependent. We introduce a particular "reachability" control problem, which has a supremum cost function but is free of state constraints. This auxiliary control problem allows to characterize easily the backward reachable sets, and then, the minimal time function, without assuming any controllability assumption. These techniques are linked to the well known level-set approachs. Partial results of the study have been published recently by the authors in SICON. Here, we generalize the method to more complex problems of moving target and obstacle problems. Our results can be used to deal with motion planning problems with obstacle avoidance

Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data

International audienceOn étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le caractère anti-diffusif du schéma est illustré par quelques exemples numériques

Some convergence results for Howard's algorithm

International audienceThis paper deals with convergence results of Howard's algorithm for the resolution of \min_{a\in \cA} (B^a x - b^a)=0 where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and \cA is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem

An efficient data structure to solve front propagation problems

International audienceIn this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an adapted anti-diffusive Ultra-Bee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d=2,3,4