This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.

The doubling algorithms are very efficient iterative methods for computing the unique minimal nonnegative solution to an M-matrix algebraic Riccati equation (MARE). They are globally and quadratically convergent, except for MARE in the critical case where convergence is linear with the linear rate 1/2. However, the initialization phase and the doubling iteration kernel of any doubling algorithm involve inverting nonsingular M-matrices. In particular, for MARE in the critical case, the M-matrices in the doubling iteration kernel, although nonsingular, move towards singular M-matrices at convergence. These inversions are causes of concerns on entrywise relative accuracy of the eventually computed minimal nonnegative solution. Fortunately, a nonsingular M-matrix can be inverted by the GTH-like algorithm of Alfa, Xue, and Ye [Math. Comp., 71:217--236, 2002] to almost full entrywise relative accuracy, provided a triplet representation of the matrix is known. Recently, Nguyen and Poloni [Numer. Math., 130(4):763--792, 2015] discovered a way to construct triplet representations in a cancellation-free manner for all involved M-matrices in the doubling iteration kernel, for a special class of MAREs arising from Markov-modulated fluid queues. In this paper, we extend Nguyen's and Poloni's work to all MAREs by also devising a way to construct the triplet representations cancellation-free. Our construction, however, is not a straightforward extension of theirs. It is made possible by an introduction of novel recursively computable auxiliary nonnegative vectors. As the second contribution, we propose an entrywise relative residual for an approximate solution. The residual has an appealing feature of being able to reveal the entrywise relative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residual which reflects relative accuracies of large entries well but not so much those of very tiny entries. Numerical examples are presented to demonstrate and confirm our claims.

We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new examples of projective manifolds for which such global Torelli theorem holds. As applications we prove that, in these cases, a canonical completion of the Torelli space is a bounded pseudoconvex domain in complex Euclidean space, and show that generic Torelli implies global Torelli on moduli space with certain level structure.

We establish by exact, nonperturbative methods a universality for the correlation functions in Kraichnan's``rapid-change''model of a passively advected scalar field. We show that the solutions for separated points in the convective range of scales are unique and independent of the particular mechanism of the scalar dissipation. Any non-universal dependences therefore must arise from the large length-scale features. The main step in the proof is to show that solutions of the model equations are unique even in the idealized case of zero diffusivity, under a very modest regularity requirement (square-integrability). Within this regularity class the only zero-modes of the global many-body operators are shown to be trivial ones (ie constants). In a bounded domain of size L , with physical boundary conditions, the``ground-state energy''is strictly positive and scales as L with an exponent L .

We study the Ginzburg dg algebra Γ_T associated with the quiver with potential arising from a triangulation T of a decorated marked surface S_△⁠, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories D_{fd}(Γ_T)⁠, denoted by D_{fd}(S_△)⁠. As an application, we show that the spherical twist group ST(S_△) associated with D_{fd}(S_△) acts faithfully on its space of stability conditions.