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.