183 research outputs found
Particle dynamics inside shocks in Hamilton-Jacobi equations
Characteristics of a Hamilton-Jacobi equation can be seen as action
minimizing trajectories of fluid particles. For nonsmooth "viscosity"
solutions, which give rise to discontinuous velocity fields, this description
is usually pursued only up to the moment when trajectories hit a shock and
cease to minimize the Lagrangian action. In this paper we show that for any
convex Hamiltonian there exists a uniquely defined canonical global nonsmooth
coalescing flow that extends particle trajectories and determines dynamics
inside the shocks. We also provide a variational description of the
corresponding effective velocity field inside shocks, and discuss relation to
the "dissipative anomaly" in the limit of vanishing viscosity.Comment: 15 pages, no figures; to appear in Philos. Trans. R. Soc. series
A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications
International audienceWe state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution a entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (Comparison Theorem). We next study a BGK-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the Comparison Theorem
Non-uniqueness of weak solutions for the fractal Burgers equation
The notion of Kruzhkov entropy solution was extended by the first author in
2007 to conservation laws with a fractional laplacian diffusion term; this
notion led to well-posedness for the Cauchy problem in the
-framework. In the present paper, we further motivate the
introduction of entropy solutions, showing that in the case of fractional
diffusion of order strictly less than one, uniqueness of a weak solution may
fail.Comment: 23 page
MeSLAM: Memory Efficient SLAM based on Neural Fields
Existing Simultaneous Localization and Mapping (SLAM) approaches are limited
in their scalability due to growing map size in long-term robot operation.
Moreover, processing such maps for localization and planning tasks leads to the
increased computational resources required onboard. To address the problem of
memory consumption in long-term operation, we develop a novel real-time SLAM
algorithm, MeSLAM, that is based on neural field implicit map representation.
It combines the proposed global mapping strategy, including neural networks
distribution and region tracking, with an external odometry system. As a
result, the algorithm is able to efficiently train multiple networks
representing different map regions and track poses accurately in large-scale
environments. Experimental results show that the accuracy of the proposed
approach is comparable to the state-of-the-art methods (on average, 6.6 cm on
TUM RGB-D sequences) and outperforms the baseline, iMAP. Moreover, the
proposed SLAM approach provides the most compact-sized maps without details
distortion (1.9 MB to store 57 m) among the state-of-the-art SLAM
approaches.Comment: Accepted paper at IEEE Systems, Man, and Cybernetics 2022 (IEEE SMC
2022), IEEE copyrigh
Smoothing properties for the higher order nonlinear Schr\"{o}dinger equation with constant coefficients
We study local and global existence and smoothing properties for the initial
value problem associated to a higher order nonlinear Schr\"odinger equation
with constant coefficients which appears as a model for propagation of pulse in
optical fiber
Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium
International audienceNeglecting capillary pressure effects in two-phase flow models for porous media may lead to non-physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non-zero capillarity. In this paper, we propose and compare several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi-dimensional case. It has been shown in [Andreianov&Canc'es, Comput. Geosci., 2013, to appear] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one-dimensional case which are used to motivate the construction of multi- dimensional finite volume schemes. We describe and compare a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Canc'es & Hilhorst, HAL preprint no.00675681, 2012) and two IMplicit Pressure - Explicit Saturation (IMPES) schemes with different level of coupling
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Design of assessment funds as a condition for FSES implementation
The implementation of new educational standards is focused on the transition from assessing students' cognitive indicators to assessing the level of the competence development, which is manifested in the need to develop a new control and evaluation system - a fund of assessment tools for the discipline (FAT). This circumstance has led to the formulation of the research problem: what is the mechanism for FAT design in the context of the implementation of federal higher education standards? The goal of the study is to substantiate methodological aspects of FAT design as a condition for the implementation of federal educational standards of higher education. During the study, the following methods were used: analysis, synthesis, analogy method, design method. The result of the presented research is the development of methodological aspects of the system of evaluation of educational results in the context of the implementation of the FSES HE, which provides for taking into account all the indicators obtained by a graduate in the process of passing the professional educational route (practical, cognitive, value-orientation and communication). Key conclusions: examples of materials of current control to assess the formation of competencies in the direction of training 43.03.02 Tourism have been proposed; criteria (scales) for assessing competence and levels of its formation in accordance with the form of control have been developed
Regularity of a kind of marginal functions in Hilbert spaces
We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient
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