734 research outputs found

    Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations.

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    We develop a very simple algorithm that permits to construct compact, high order schemes for steady first order Hamilton Jacobi equations. The algorithm relies on the blending of a first order scheme and a compact high order one. The blending is conducted in such a way that the scheme is formally high order accurate. A convergence proof is given. We provide several numerical illustrations that demonstrate the effective accuracy of the scheme. The numerical examples use triangular unstructured meshes, but our method may be applied to other kind of meshes. Several implementation remarks are also given

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

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    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 ϵ\epsilon-monotone property. Convergence results and precise error estimates are given, of the order of Δx\sqrt{\Delta x} where Δx\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

    Some non monotone schemes for Hamilton-Jacobi-Bellman equations

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    We extend the theory of Barles Jakobsen to develop numerical schemes for Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test some numerical schemes.Comment: 24 page

    Adaptive filtered schemes for first order Hamilton-Jacobi equations and applications

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    The accurate numerical solution of Hamilton-Jacobi equations is a challenging topic of growing importance in many fields of application but due to the lack of regularity of viscosity solutions the construction of high-order methods can be rather difficult. We consider a class of “filtered” schemes for first order time-dependent Hamilton-Jacobi equations. These schemes, already proposed in the literature, are based on a mixture of a high-order (possibly unstable) scheme and a monotone scheme, according to a filter function F and a coupling parameter epsilon. This construction allows to have a scheme which is high-order accurate where the solution is smooth and is monotone otherwise. This feature is crucial to prove that the scheme converges to the unique viscosity solutions. In this thesis we present an improvement of the classical filtered scheme, introducing an adaptive and automatic choice of the parameter epsilon at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold epsilon. Our smoothness indicator is based on some ideas developed for the construction of the WENO schemes, but other indicators with similar properties can be used. We present a convergence result and error estimates for the new scheme, the proofs are based on the properties of the scheme and of the indicators. All the constructions are extended to the multidimensional case, with main focus on the definition of new 2D-smoothness indicators, devised for functions with discontinuous gradient. A large number of numerical example are presented and critically discussed, confirming the reliability of the proposed smoothness indicators and the efficiency of the adaptive filtered scheme in many situations, improving previous results in the literature. Finally, we applied the constructed scheme to the problem of image segmentation via the level-set method, proposing also a simple and efficient modification of the classical model in order to improve the stability of the results. A series of numerical tests on synthetic and real images are presented and deeply commented
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