1,188,276 research outputs found
LSMR: An iterative algorithm for sparse least-squares problems
An iterative method LSMR is presented for solving linear systems and
least-squares problem \min \norm{Ax-b}_2, with 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 is the residual for the current iterate
). 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 if either the scaled -norm of the velocity
with , , or the -norm of the
vorticity with , , or the
-norm of the gradient of the vorticity with , , , is sufficiently small near
Average norms of polynomials
In this paper we study the average \NL_{2\alpha}-norm over -polynomials,
where 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 with coefficients in , where is a finite set of complex
numbers, is a positive integer, and . 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
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