Advection modes by optimal mass transfer

Abstract

Kármán wake and hurricane Dean advection are discussed.

Model Order Reduction for Problems with Large Convection Effects

“…Well chosen here means that the calibrated equation has to be well posed, and the dynamics of the shape component should be much simpler than those of the original solution. The second paper [13] differs on two points. There is a unique reference mode u 0 that allows to characterize the mapping, obtained from an optimal transport problem.…”

“…The two closest methods to ours are first the one developed in [2,19], second the one presented in [13]. In the former method, there is also the search for a phase component: F(t; µ) and a shape component: v := u(·,t; µ) • F(t; µ) that they name the "calibrated solution".…”

“…The method presented in [13] is similar to our work in many aspects, in particular in looking for a change of variable for better representing the solution manifold. Their approach relies on the existence of a main mode u 0 that, by convection, represents most of the solution.…”

“…We start here by presenting the general framework and notations. We introduce the notion of "preconditioning" of a solution manifold based on our knowledge of the process (differing here somehow from the optimal transport problem approach in [13]). We then apply our method to the specific problem of the one dimensional unsteady viscous Burger equation and we then present some numerical simulations confirming the feasibility of the method.…”

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

“…Lagrangian approaches propose to exploit a linear method Z N : R N → Z N ⊂ X to approximate the mapped solution z µ := z µ • Φ µ where Φ µ : Ω → Ω is a suitably-chosen bijection from Ω into itself: the mapping Φ µ should be chosen to make the mapped solution manifold M more amenable for linear approximations. Examples of Lagrangian approaches have been proposed in 1 [24,34,37,54]: in [37,54] the construction of the map is performed separately from the construction of the solution coefficients, while in [34] the authors propose to build the mapping Φ µ and the solution coefficients α µ simultaneously. Note that Lagrangian approaches are significantly less general than Eulerian approaches -any Lagrangian method is equivalent to an Eulerian method with…”