Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations

A level set based Eulerian method for paraxial multivalued traveltimes

“…In fact, the success of the level-set method in the early stage was more or less attributed to the concept of viscosity solution and related high-order numerical methods [19,29], so that topology change and merging can be taken care of automatically when they occur. On the other hand, it also implies that in the current framework, i.e., Lagrangian formulation (3) and Eulerian formulation (1), it is hard to capture self-intersecting wavefronts unless special care is taken to keep track of some extra parameters, such as amplitude [1] or slowness [41,42], etc. These extra parameters are essentially used to parameterize the self-intersection, i.e., multivaluedness, and this viewpoint naturally leads one to consider phase-space formulations for computing multivalued solutions.…”

“…In this regard, a new field, so-called Eulerian geometrical optics, emerged [2] as many researchers have devoted a lot of efforts to developing efficient Eulerian methods for computing multivalued solutions since the early 90s. As a result, there are many different approaches in the literature: explicit caustic construction method [1,3], slowness matching method [41,42], segment projection method [13], dynamic surface extension method [37,38], kinetic method for multibranch entropy solutions [4,16], higher co-dimension level-set evolution method [6,7,20,26,31], Liouville equations for escape parameters [14], to name just a few.…”

GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type

“…A link can be made between Lagrangian and Eulerian viscosity solution using the theory of optimal control [5]. The viscosity solution can be characterized as the minimum of the associated Lagrangian phases:…”

“…These two approaches, Lagrangian and Eulerian, are mathematically equivalent only when the Lagrangian solution is well defined and single-valued (rays do not cross). The Eulerian method otherwise produces a weak "viscosity" solution which can be identified as the minimum value of the multi-valued Lagrangian solution (the problem of computing the Lagrangian solution using a Eulerian representation when it is multi-valued has been the subject of on-going work in the last ten years [1,4,5,7,10,11,14,16,17] ...).…”

Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation

“…It was used to compute the multivalued phase or velocity beyond caustics. The computations of multivalued solution in geometrical optics, or more generally for nonlinear PDEs, have been an active area of research in recent years, see [2,3,5,4,8,15,10,11,13,12,18,19,16,24,37,41,6,36,23]. However, all these works were developed without the interface.…”