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

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

Some non monotone schemes for Hamilton-Jacobi-Bellman equations

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

A High-Order Scheme for Image Segmentation via a modified Level-Set method

In this paper we propose a high-order accurate scheme for image segmentation based on the level-set method. In this approach, the curve evolution is described as the 0-level set of a representation function but we modify the velocity that drives the curve to the boundary of the object in order to obtain a new velocity with additional properties that are extremely useful to develop a more stable high-order approximation with a small additional cost. The approximation scheme proposed here is the first 2D version of an adaptive "filtered" scheme recently introduced and analyzed by the authors in 1D. This approach is interesting since the implementation of the filtered scheme is rather efficient and easy. The scheme combines two building blocks (a monotone scheme and a high-order scheme) via a filter function and smoothness indicators that allow to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and real images confirm the accuracy of the proposed method and the advantages given by the new velocity.Comment: Accepted version for publication in SIAM Journal on Imaging Sciences, 86 figure

High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes

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 segmentation of astronomical images via level-set methods

Astronomical images are of crucial importance for astronomers since they contain a lot of information about celestial bodies that can not be directly accessible. Most of the information available for the analysis of these objects starts with sky explorations via telescopes and satellites. Unfortunately, the quality of astronomical images is usually very low with respect to other real images and this is due to technical and physical features related to their acquisition process. This increases the percentage of noise and makes more difficult to use directly standard segmentation methods on the original image. In this work we will describe how to process astronomical images in two steps: in the first step we improve the image quality by a rescaling of light intensity whereas in the second step we apply level-set methods to identify the objects. Several experiments will show the effectiveness of this procedure and the results obtained via various discretization techniques for level-set equations.Comment: 24 pages, 59 figures, paper submitte