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No abstract uploaded!

We investigate a problem of approximation of a large class of nonlinear expressions$f$($x, u$, ∇$u$), including polyconvex functions. Here$u$: Ω→$R$^{$m$}, Ω⊂$R$^{$n$}, is a mapping from the Sobolev space$W$^{$1,p$}. In particular, when$p=n$, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in addition$n=m$, satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation of$f$($x, u$, ∇$u$) are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions.

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We propose a computational framework named iterative local adaptive majorize-minimization (I-LAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. I-LAMM is a two-stage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasi-likelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family