LSMR: An iterative algorithm for sparse least-squares problems

An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problem \min \norm{Ax-b}_2, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation A\T Ax = A\T b, so that the quantities \norm{A\T r_k} are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). In practice we observe that \norm{r_k} also decreases monotonically. Compared to LSQR, for which only \norm{r_k} is monotonic, it is safer to terminate LSMR early. Improvements for the new iterative method in the presence of extra available memory are also explored.Comment: 21 page

Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$

Average norms of polynomials

In this paper we study the average \NL_{2\alpha}-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average \NL_{2\alpha}-norm over all the polynomials of degree exactly $n$ with coefficients in $T$, where $T$ is a finite set of complex numbers, $\alpha$ is a positive integer, and $n\geq0$. In particular, we give a complete answer for the cases of Littlewood polynomials and polynomials of a given height. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.Comment: 13 pages, key words: Littlewood polynomials, Polynomials of height $h