An alternating projections method for certain linear problems in a Hilbert space

1995 ◽  
Vol 15 (2) ◽  
pp. 291-305 ◽  
Author(s):  
WILLIAM K. GLUNT
Keyword(s):  
Hilbert Space ◽  
Alternating Projections ◽  
Linear Problems ◽  
Alternating Projections Method

scholarly journals A variational approach to the alternating projections method

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Convex Sets ◽  
Nonempty Intersection ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Limit Sets ◽  
Von Neumann ◽  
Convex Feasibility ◽  
Starting Point ◽  
Method Of Alternating Projections ◽  
Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

scholarly journals Global convergence of the alternating projection method for the Max-Cut relaxation problem

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 737-746
Author(s):  
Suliman Al-Homidana
Keyword(s):  
Convex Sets ◽  
Alternating Projections ◽  
Correlation Matrices ◽  
Von Neumann ◽  
Relaxation Problem ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Convergent Method ◽  
Alternating Projection Method ◽  
Alternating Projections Method

The Max-Cut problem is an NP-hard problem [15]. Extensions of von Neumann?s alternating projections method permit the computation of proximity projections onto convex sets. The present paper exploits this fact by constructing a globally convergent method for the Max-Cut relaxation problem. The feasible set of this relaxed Max-Cut problem is the set of correlation matrices.

Complementary bivariational principles for linear problems involving non-self-adjoint operators

1980 ◽  
Vol 86 (1-2) ◽  
pp. 115-128 ◽  
Author(s):  
R. J. Cole
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Linear Operator ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fredholm Type ◽  
Linear Problems ◽  
Adjoint Operators

SynopsisBivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

Sign in / Sign up

Export Citation Format

Share Document