relaxation methods for problems with strictly convex separable costs and linear constraints we consider the minimization problem with strictly convex possibly nondifferentiable separable cost and linear constraints the dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on gauss seidel relaxation we show that these methods yield the optimal primal solution and under additional assumptions an optimal dual solution to do this it is necessary to extend the classical gauss seidel convergence results because the dual cost may not be strictly convex and may have unbounded level sets

asymptotic behaviour of non linear parabolic equations with monotone principal part abstract we consider non linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non lipschitz perturbations the case of globally lipschitz perturbations of a maximal monotone operator has been addressed in boll un mat ital b 8 2 2000 693 706 in the case of perturbations which are not globally lipschitz the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors we overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed the results developed here are applied to the case when the subdifferential operator is the p laplacian to obtain existence of attractors and the existence of periodic solutions

a simultaneous projections method for linear inequalities abstract an iterative method for solving general systems of linear inequalities is considered the method a relaxed generalization of cimmino s scheme for solving linear systems was first suggested by censor and elfving each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities the algorithm is particularly suitable for implementation on computers with parallel processors we prove convergence from any starting point for both consistent and nonconsistent systems to a feasible point in the first case and to a weighted least squares type solutions in the second

a method for approximating the solution set of a system of convex inequalities by polytopes abstract in this note a method for computing approximating by polytopes of the solution set q of a system of convex inequalities is presented it is shown that such approximations can be determined by an algorithm which converges in finitely many steps when the solution set of the given system of inequalities is bounded in this case the algorithm generates inner and outer approximations having the hausdorff distance to each other and to the set q no greater than an a priori fixed and having their extreme points in q and in the relative exterior of q respectively

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

optimization of log x entropy over linear equality constraints in this paper we develop a special purpose iterative algorithm of the row action type for solving the problem of maximizing the log x entropy functional over linear equality constraints the algorithm employs projections onto hyperplanes which we call log x entropy projections a complete proof of convergence is given

a computational solution of the inverse problem in radiation therapy treatment planning radiation therapy concerns the delivery of the proper dose of radiation to a tumor volume without causing irreparable damage to healthy tissue and critical organs the forward problem refers to calculating the dose distribution that would be delivered to a measured patient cross section when the radiation field generated by the beam sources is specified the inverse problem refers to calculating the radiation field that will provide a specified dose distribution in the patient the forward and inverse problems of radiation therapy treatment planning are first formulated in their continuous versions and the point is made that in this field of application the inverse problem calls for the inversion of an operator for which no analytic closed form mathematical representation exists to attack the inverse problem under such circumstances a discretized model is set up in which both patient section and radiation field are finely discretized this leads to a linear feasibility problem which is solved by a relaxation method the paper includes full details of this new approach with discussion of technical aspects such as dose apportionment software development and experimental results consequences and limitations as well as a precise comparison with other methodologies are also discussed

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

extensions of hildreth s row action method for quadratic programming an extended version of hildreth s iterative quadratic programming algorithm is presented geometrically interpreted and proved to produce a sequence of iterates that i converges to the solution and ii has an important intermediate optimality property this extended hildreth algorithm is cast into a new form which more pronouncedly brings out its primal dual nature the application of the algorithm may be governed by an index sequence which is more general than a cyclic sequence namely by an almost cyclic control and a sequence of relaxation parameters is incorporated without ruining convergence the algorithm is a row action method which is particularly suitable for handling large or huge and sparse systems

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

continuity properties on p for p laplacian parabolic problems abstract in this work we obtain some continuity properties on the parameter p at p 2 for the takeuchi yamada problem which is a degenerate p laplacian version of the chafee infante problem we prove the continuity of the flows and the equilibrium sets and the upper semicontinuity of the global attractors

an automatic relaxation method for solving interval linear inequalities abstract the problem of solving iteratively a large and possibly sparse system of interval linear inequalities ax is considered an algorithm of the row action type is proposed which in each iterative step effectively takes account of a pair of inequalities describing a single interval inequality the algorithm realizes in an automatic manner a relaxation principle proposed by goffin but also allows further external relaxation parameters

new methods for linear inequalities abstract the iterative method of cimmino for solving linear equations is generalized to linear inequalities we also present a richardson type iterative method for solving the inequality problem which includes the generalized cimmino scheme convergence proofs are provided

strong underrelaxation in kaczmarz s method for inconsistent systems we investigate the behavior of kaczmarz s method with relaxation for inconsistent systems we show that when the relaxation parameter goes to zero the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system this point minimizes the sum of the squares of the euclidean distances to the hyperplanes of the system if the starting point is chosen properly then the limits approach the minimum norm weighted least squares solution the proof is given for a block kaczmarz method

classifying semisimple orbits of theta groups i consider the problem of classifying the semisimple orbits of a theta group for this purpose once a preliminary presentation of the theoretical subjects where my problem arises from i first give an algorithm to compute a cartan subspace subsequently i describe how to compute the little weyl group

iterative algorithms for large partitioned linear systems with applications to image reconstruction abstract we present a unifying framework for a wide class of iterative methods in numerical linear algebra in particular the class of algorithms contains kaczmarz s and richardson s methods for the regularized weighted least squares problem with weighted norm the convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for kaczmarz s and richardson s methods the algorithms in the class may be characterized as being group iterative and incorporate relaxation matrices as opposed to a single relaxation parameter we show that some well known iterative methods of image reconstruction fall into the class of algorithms under consideration and are thus covered by the convergence theory we also describe a novel application to truly three dimensional image reconstruction

a simultaneous iterative method for computing projections on polyhedra a simultaneous version of hildreth s iterative algorithm for norm minimization over linear inequalities is presented proofs are given showing that the algorithm converges from any starting point to its projection on the linear constraints set in the feasible case and to the nearest least squares solution in the general case

convergence results for an accelerated nonlinear cimmino algorithm we present an accelerated version of cimmino s algorithm for solving the convex feasibility problem in finite dimension the algorithm is similar to that given by censor and elfving for linear inequalities we show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case applications to the linear problem are suggested

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

lap number properties for p laplacian problems investigated by lyapunov methods abstract in this work we consider a one dimensional quasilinear parabolic equation and we prove that the lap number of any solution cannot increase through orbits as the time passes if the initial data is a continuous function we deal with the lap number functional as a lyapunov function and apply lap number properties to reach an understanding on the asymptotic behavior of a particular problem

volume approximation of convex bodies by inscribed polytopes a convex body c in d dimensional euclidean space e is a compact convex subset of le a with non empty interior we say that c is of differentiability class k if its boundary considered as a manifold is of class k the symmetric difference metric 6 s also called nikodym metric on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference given a convex body c let c n d l d 2 denote the family of convex polytopes p having at most n vertices and being inscribed into c that is all vertices are on the boundary bdc of c it is well known that for each n there is a polytope p such that s c p s c inf s c p p

cyclic subgradient projections a cyclically controlled method of subgradient projections csp for the convex feasibility problem of solving convex inequalities is presented the features of this method make it an efficient tool in handling huge and sparse problems a particular application to an image reconstruction problem of emission computerized tomography is mentioned