Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations

Abstract. In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computa-tional cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its con-vergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution

A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants

Having in mind the modelling of marble degradation under chemical pollutants, e.g. the sulfation process, we consider governing nonlinear diffusion equations and their numerical approximation. The space domain of a computation is the pristine marble object. In order to accurately discretize it while maintaining the simplicity of finite difference discretizations, the domain is described using a level-set technique. A uniform Cartesian grid is laid over a box containing the domain, but the solution is defined and updated only in the grid nodes that lie inside the domain, the level-set being employed to select them and to impose accurately the boundary conditions. We use a Crank-Nicolson scheme in time, while for the space variables the discretization is performed by a standard Finite-Difference scheme for grid points inside the domain and by a ghost-cell technique on the ghost points (by using boundary conditions). The solution of the large nonlinear system is obtained by a Newton-Raphson procedure and a tailored multigrid technique is developed for the inner linear solvers. The numerical results, which are very satisfactory in terms of reconstruction quality and of computational efficiency, are presented and discussed at the end of the paper

Diffusion equations and inverse problems regularization.

The present thesis can be split into two dfferent parts: The first part mainly deals with the porous and fast diffusion equations. Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators. The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration