Ming-Jun Lai

Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed

In this paper, we first study

\ell_q

\ell_q

Minimization

minimization and its associated iterative reweighted algorithm for recovering sparse vectors. Unlike most existing work, we focus on unconstrained

On the approximation power of bivariate splines

\ell_q

Augmented

minimization, for which we show a few advantages on noisy measurements and/or approximately sparse vectors. Inspired by the results in [Daubechies et al., Comm. Pure Appl. Math., 63 (2010), pp. 1--38] for constrained

\ell_1

\ell_q

and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm

minimization, we start with a preliminary yet novel analysis for unconstrained

Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets

\ell_q

Macro-elements and stable local bases for splines on Powell-Sabin triangulations

minimization, which includes convergence, error bound, and local convergence behavior. Then, the algorithm and analysis are extended to the recovery of low-rank matrices. The algorithms for both vector and matrix recovery have been compared to some state-of-the-art algorithms and show superior performance on recovering sparse vectors and low-rank matrices.

Scattered data interpolation and approximation using bivariate C 1 piecewise cubic polynomials

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces Sdr(Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain.