Well-posedness classes for sparse regularization

Well-posedness Classes for Sparse Regularization∗

Abstract. Because of their sparsity enhancing properties, `1 penalty terms have recently received much attention in the field of inverse problems. Also, it has been shown that certain properties of the linear operator A to be inverted imply that `1-regularization is equivalent to `0-regularization, which tries to minimise the number of non-zero coefficients. In the context of compressed sensing...

Well - Posedness and Convergenceof Some Regularization

In this paper we carry out a theoretical analysis of two regularization methods for nonlinear ill-posed problems. The rst is a penalty method called Tikhonov regularization, in which one solves an unconstrained optimization problem while the second is based on a constrained optimization problem. For each method we examine the well-posedness of the respective optimization problem. We then show s...

Well-posedness for Hall-magnetohydrodynamics

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions. Acknowledgements: The first and third authors wish to thank the hospitality of the Institut de Mathématiques de Toulouse, France, where this research has been car...

Well-posedness for Hyperbolic Problems

Well-posedness for General 2

We consider the Cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension u t + F (u)] x = 0; u(0; x) = u(x); (1) which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic elds. If r i (u); i = 1; 2; denotes the i-th right eigenvector of DF (u) and i (u) the corresponding eigenvalue, then the set f...