Georgios T. Kossioris

Finite volume schemes for Hamilton-Jacobi equations

Summary. We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton–Jacobi equation.

Convergence and error estimates of relaxation schemes for multidimensional conservation laws

M. A. Katsoulakis, G. Kossioris and Ch. Makridakis Abstract. We study discrete and semidiscrete relaxation schemes for multidimensional scalar conservation laws. We show convergence of the relaxation schemes to the entropy solution of the conservation law and derive error estimates that exhibit the precise interaction between the relaxation time and the space/time discretization parameters of the schemes.

Geometrical methods for level set based abdominal aortic aneurysm thrombus and outer wall 2D image segmentation

Abdominal aortic aneurysm (AAA) is a localized dilatation of the aortic wall. Accurate measurements of its geometric characteristics are critical for a reliable estimate of AAA rupture risk. However, current imaging modalities do not provide sufficient contrast to distinguish thrombus from surrounding tissue thus making the task of segmentation quite challenging. The main objective of this paper is to address this problem and accurately extract the thrombus and outer wall boundaries from cross sections of a 3D AAA image data set (CTA). This is achieved by new geometrical methods applied to the boundary curves obtained by a Level Set Method (LSM). Such methods address the problem of leakage of a moving front into sectors of similar intensity and that of the presence of calcifications. The versatility of the methods is tested by creating artificial images which simulate the real cases. Segmentation quality is quantified by comparing the results with a manual segmentation of the slices of ten patient data sets. Sensitivity to the parameter settings and reproducibility are analyzed. This is the first work to our knowledge that utilizes the level set framework to extract both the thrombus and external AAA wall boundaries.

On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Abstract In the present article, we study the system of eikonal and transport equations arising in geometrical optics. The mathematical analysis is performed by using the suitable notion of solution, i.e., the viscosity solution for the Hamilton–Jacobi equation and the measure solution for the transport equation defined via the generalized Filippov characteristics. We study the stability as well as the geometry of the solution to the system.

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in one space variable

In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on the convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which is the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves.

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations

In this work we study the generation of singularities (shock waves) of the solution of the Cauchy problem for HamiltonJacobi equations in several space variables, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We first examine the way the characteristics cross by identifying the set of critical points of the characteristic manifold with the caustic set of the related lagrangian mapping. We construct the viscosity solution by selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics. We finally discuss how the shocks propagate and undergo catastrophe in the case of two space variables. W This work was part of the authors PhD thesis in the Division of Applied Mathematics, Brown University. <> Partially supported by NSF Grants DMS-8801208 and DMS-8657464 (PYI) University Libraries Carnegie Mellon University Pittsburgh, PA 15213-3890

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions

In this work we study the generation of singularities (shock waves) of the solution of the Cauchy problem for HamiltonJacobi equations in several space variables, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We first examine the way the characteristics cross by identifying the set of critical points of the characteristic manifold with the caustic set of the related lagrangian mapping. We construct the viscosity solution by selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics. We finally discuss how the shocks propagate and undergo catastrophe in the case of two space variables. W This work was part of the authors PhD thesis in the Division of Applied Mathematics, Brown University. Partially supported by NSF Grants DMS-8801208 and DMS-8657464 (PYI) University Libraries Carnegie Mellon University Pittsburgh, PA 15213-3890

The value function of the shallow lake problem as a viscosity solution of a HJB equation

The economic analysis of a shallow lake ecological system requires the study of a nonstandard optimal control problem due to the conflicting services it provides and the nonlinearity of the governing dynamics. We first investigate the geometry of the optimal control-optimal path pair, by standard control analysis, for a given range of the discount factor. We then consider the welfare function (value function) as a viscosity solution of a reduced Hamilton-Jacobi-Bellman equation and we prove various regularity properties which are related to the dynamics of the problem. Finally, we approximate the welfare function by monotone convergent numerical schemes and present the numerical results.

Detection of Lumen, Thrombus and Outer Wall Boundaries of an Abdominal Aortic Aneurysm From 2D Medical Images Using Level Set Methods

Abdominal aortic aneurysm (AAA) is a localized dilatation of the aortic wall. Accurate geometric characterization is critical for a reliable patient specific estimate of AAA rupture risk. However, current imaging modalities do not provide sufficient contrast between thrombus, arterial wall and surrounding tissue thus making the task of segmenting these structures very challenging.Copyright

A Hamilton–Jacobi approach to the control of the trapping time of a soliton by an external potential

The control of the trapping time of a localized solution of the nonlinear Schrodinger equation (NLS) with the use of an external parabolic potential is studied. We reduce the dynamics of the position of the soliton center to those of a controlled linear oscillator and then study the viscosity solution of the associated Hamilton-Jacobi equation. A numerical scheme is proposed for the treatment of the problem.