Section: New Results

Other topics

Fast non-overlapping Schwarz domain decomposition methods for the neutron diffusion equation

Participant : Patrick Ciarlet.

A collaboration with Erell Jamelot (CEA Saclay/DEN). Investigating numerically the steady state of a nuclear core reactor can be very expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, which is then implemented on a parallel computer.

We model the problem using a mixed approach, which involves a scalar flux and a vector current. The equivalent variational formulation is then discretized with the help of Raviart-Thomas-Nédélec finite elements.

The domain decomposition method is based on the Schwarz iterative algorithm with Robin interface conditions to handle communications. This method is analyzed from the continuous to the discrete point of views: well-posedness, convergence of the finite element method, optimality of the parameter appearing in the Robin interface condition and algorithms. Numerical experiments carried out on realistic 3D configurations using the APOLLO3©code (of CEA/DEN) show the parallel efficiency of the algorithm.

Equivalent local boundary conditions for the Monge Kantorovitch Mass Transport problem

Participant : Jean-David Benamou.

This work is done in collaboration with Adam Oberman Britanny Froese from Simon Fraser University, Vancouver. In the last 20 years, the Monge Kantorovich Optimal Transport problem (OTP) and its relationship with Partial Differential Equations (PDE) experienced a spectacular research revival (the Fields medal was awarded to Cedric Villani in 2010 partly for his contributions to OTP). Applications appeared in fields as diverse as meteorology, medical image processing, astronomy and economy. This new area offers numerical challenges which go beyond current knowledge in the field. Novel computational tools are needed. The OTP can also be reformulated as a Monge-Ampere (MA) PDE with non standard/non local boundary conditions (BC). We would like to use the new and efficient wide-stencil Finite Difference MA solver developed by Oberman and Froese but we do not know how to deal with the OTP equivalent BCs. We pursue the investigation of the two following innovative strategies : 1. Iteratively construct Neumann BCs such that the solution of the MA equation satisfies the OTP BC in the limit. 2. Merge the data in all of space and design simplified asymptotic local BCs at infinity which can be used to formulate local transparent BC on a truncated domain.

Topological Effects in Quantum Mechanics and High-Velocity Estimates

Participant : Ricardo Weder.

This work is done in collaboration with Miguel Ballesteros. High-velocity -or high-energy- estimates for scattering solutions to the Schrödinger equation are important for many reasons. For example, in topological effects in quantum mechanics, where the space accessible to the particles has a non-trivial topology, like, for example, in the celebrated magnetic Aharonov-Bohm effect, where an electron is constrained to be on the exterior of a torus that contains a magnetic flux inside. Here, the solution acquires a phase if the electron travels inside the hole of the magnet and, on the contrary, it acquires no phase if the particle travels outside the hole. We obtain precise high-velocity estimates for the scattering solutions, that prove that quantum mechanics actually predicts the existence of the magnetic Aharonov-Bohm effect, under the conditions of the celebrated Tonomura et al. experiments. Moreover, in the case of the electric Aharonov-Bohm effect, we provide precise conditions for the validity of the Aharonov-Bohm Ansatz and we give a rigorous proof that quantum mechanics predicts the existence of this effect.

Entanglement Creation in Low-Energy Scattering

Participant : Ricardo Weder.

We study the entanglement creation in the low-energy scattering of two particles of mass, $m_1,m_2$ in three dimensions. We consider a general class of interaction potentials that are not required to be spherically symmetric. The incoming asymptotic state, before the collision, is a product of two normalized Gaussian states with the same variance, $\sigma$, and opposite mean momentum. After the scattering the particles are in the outgoing asymptotic state that is not a product state. We take as a measure of the entanglement created by the collision the purity of one of the particles in the outgoing asymptotic state. In the incoming asymptotic state the purity is one. We provide a rigorous explicit computation, with error bound, of the leading order of the purity at low-energy. The leading order depends strongly in the difference of the masses. The entanglement takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses. It is quite remarkable that the anisotropy of the potential plays no role, on spite of the fact that entanglement is a second order effect.

Open Scattering Channels in Manifolds with Horns

Participant : Ricardo Weder.

This work is done in collaboration with Olaf Post (Humboldt University, Berlin) and Rainer Hempel (Mount Allison University, Sackville New Brunswick). In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension $n$. The smallness condition for the perturbation is expressed in purely geometric terms using the harmonic radius; therefore, the size of the perturbation can be controlled in terms of local bounds on the radius of injectivity and the Ricci-curvature, and no global assumption is needed. As an application of these ideas we obtain a stability result for the scattering matrix. As a consequence we find that a scattering channel which interacts with other channels preserves this property under small perturbations.