A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

The class of continuous piecewise linear (PL) functions represents a useful family of approximants because invertibility can be readily imposed, and if a PL function is invertible, then it can be inverted in closed form. Many applications, arising, for example, in control systems and robotics, involve the simultaneous construction of a forward and inverse system model from data. Most approximation techniques require that separate forward and inverse models be trained, whereas an invertible continuous PL affords, simultaneously, the forward and inverse system model in a single representation. The minvar algorithm computes a continuous PL approximation to data. Local convergence of minvar is proven for the case when the data generating function is itself a PL function and available directly rather than through data

Algorithmes d'adaptation de maillages anisotropes et application à l'aérodynamique

The goal of this thesis is to study an anisotropic adaptive algorithm for transonic compressible viscous flow around an airwing. A convection-diffusion model problem is considered, an anisotropic a posteriori error estimator for the H1 semi-norm of the error is derived. The equivalence between the error and the estimator is proved, which provides the efficiency and the reliability of this estimator. A goal oriented anisotropic a posteriori error estimator is introduced. The equivalence with the error is not proved but lower and upper bounds are obtained. Based on this error estimator, an anisotropic mesh algorithm is proposed and applied to the transonic compressible flow around an airwing. The mesh is structured close to the boundary layer and is kept as is, while the mesh outside the boundary layer is adapted according to the anisotropic error estimator. This anisotropic adaptive algorithm allows shocks to be captured accurately, while keeping the number of vertices as low as possible