Extrapolated cyclic subgradient projection methods for the convex feasibility problems and their numerical behaviour

ABSTRACT

We present two versions of the extrapolated cyclic subgradient projections method for solving the convex feasibility problem. Moreover, we present the results of numerical tests, where we compare the methods with the classical cyclic subgradient projections method.     

On a Class of Projections on *-Rings

ABSTRACT

In this article we describe the structure of projections acting on semiprime *-rings and satisfying a certain functional identity. The main result is applied to bicircular projections on C *-algebras.     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

The Eclipse of Confidentiality

Abstract

Change's periodic projections of leading economic indicators for academic institutions continue with these new 1980 findings. (See February 1979 Change for its 1978–79 analysis.) These projections are prepared for Change by John Minter Associates, a financial analysis organization specializing in the academic field. Change will publish its full 1981 projections in a fall 1980 issue.     

On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach

Using elementary ODE tools we show that for a class of dissipative PDEs, including Navier-Stokes equations on the plane with periodic boundary conditions, Galerkin projections are C¹-convergent, i.e. the partial derivatives with respect to initial conditions for Galerkin projections converge giving rise to smooth dependence on initial conditions for full PDE.     

Finding Optimal Shadows of Polytopes

This paper deals with the problem of projecting polytopes in finite-dimensional Euclidean spaces on subspaces of given dimension so as to maximize or minimize the volume of the projection. As to the computational complexity of the underlying decision problems we show that maximizing the volume of the orthogonal projection on hyperplanes is already NP-hard for simplices. For minimization, the problem is easy for simplices but NP-hard for bipyramids over parallelotopes. Similar results are given for projections on lower-dimensional subspaces. Several other related NP-hardness results are also proved including one for inradius computation of zonotopes and another for a location problem. On the positive side, we present various polynomial-time approximation algorithms. In particular, we give a randomized algorithm for maximizing orthogonal projections of CH-polytopes in R ⁿ on hyperplanes with an error bound of essentially . Received February 17, 1999.     

Smooth Feshbach map and operator-theoretic renormalization group methods

A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map. It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.     

Notes on pre-Frölicher spaces

In this work we introduce a class of Sikorski differential spaces (M, D) called pre-Fr¨olicher spaces, on which the process of yielding a Fr¨olicher structure on the underlying set M is D preserving, their category we denote by preFrl. We investigate some algebraic properties on these spaces whose subsequent geometric properties are mostly similar to those of smooth manifolds, except for the invariance of dimension, and also that preFrl naturally induces a Cartesian closed subcategory of the category Frl in which there is no discrete object. Using this Cartesian property, it is shown that the Gelfand representation is a smooth map, that the tangent as well as cotangent bundles are made smooth spaces in an unusual but more natural way via smooth curves.     

Fredholm Properties of the Difference of Orthogonal Projections in a Hilbert Space

Buckholtz (Proc. Amer. Math. Soc. 128 (2000), 1415–1418) gave necessary and sufficient conditions for the invertibility of the difference of two orthogonal projections in a Hilbert space. We generalize this result by investigating when the difference of such projections is a Fredholm operator, and give an explicit formula for its Fredholm inverse.     

An Analogue of the Kerzman-Stein Formula for the Bergman and Szegö Projections

We show that the difference between the Bergman and Szegö projections on a smooth, bounded planar domain gains a derivative in the L ^p -Sobolev and Lipschitz spaces.     

Homotopy method for solving mathematical programs with bounded box-constrained variational inequalities

ABSTRACT

On the basis of Robinson's normal equation and smoothing projection operator, a homotopy method for solving mathematical programs with box-constrained variational inequalities (MPBVI) is presented. In which, the Chen–Harker–Kanzow–Smale smooth function is used to transform MPBVI into a smooth optimization problem. Under some mild assumptions, the existence and global convergence of a smooth path from almost any initial point to the GKKT point of the approximate problems is proven. And, the convergence of the GKKT point to a strong C-stationary point of the original problems is proved. Finally, some numerical results are given to show the effectiveness and feasibility of the homotopy method.     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

Interactive High-Dimensional Data Visualization

Abstract

We propose a rudimentary taxonomy of interactive data visualization based on a triad of data analytic tasks: finding Gestalt, posing queries, and making comparisons. These tasks are supported by three classes of interactive view manipulations: focusing, linking, and arranging views. This discussion extends earlier work on the principles of focusing and linking and sets them on a firmer base. Next, we give a high-level introduction to a particular system for multivariate data visualization—XGobi. This introduction is not comprehensive but emphasizes XGobi tools that are examples of focusing, linking, and arranging views; namely, high-dimensional projections, linked scatterplot brushing, and matrices of conditional plots. Finally, in a series of case studies in data visualization, we show the powers and limitations of particular focusing, linking, and arranging tools. The discussion is dominated by high-dimensional projections that form an extremely well-developed part of XGobi. Of particular interest are the illustration of asymptotic normality of high-dimensional projections (a theorem of Diaconis and Freedman), the use of high-dimensional cubes for visualizing factorial experiments, and a method for interactively generating matrices of conditional plots with high-dimensional projections. Although there is a unifying theme to this article, each section—in particular the case studies—can be read separately.     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

The Dimension of the Hilbert Scheme of Special Threefolds #

The Hilbert scheme of 3-folds in ? ⁿ , n ≥  6 , that are scrolls over ? ² or over a smooth quadric surface Q  ? ? ³ or that are quadric or cubic fibrations over ? ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.     

Stochastic sensitivity of attractors for a piecewise smooth neuron model

ABSTRACT

We consider a one-dimensional model of neural activity, given by a piecewise smooth discontinuous map. Fold bifurcations as well as border collision bifurcations are described in detail. Using the method of stochastic sensitivity functions, noise-induced phenomena, such as transitions within attractor and between attractors, and spike generation, are described. Statistical characteristics of interspike intervals depending on noise intensity are studied.     

Hausdorff Dimension and Non-degenerate Families of Projections

We study parameterized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from the n-dimensional Euclidean space to the m-dimensional one.     

Constructing convex solutions via Perron’s method

Abstract In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05     

Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR ⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL ^p estimates of multiparameter maximal functions over compact hypersurfaces.     

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R ⁿ obtained by eliminating from R ⁿ the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles. Dedicated to N. N. Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 7–48.