image recovery by convex combinations of projections the problem of image recovery in a hilbert space setting by using convex projections by projection we always mean minimum distance projection onto a closed convex set may be stated as follows the original unknown image f is known a priori to belong to the intersection co of r well defined dosed convex sets c 1 c r in a hilbert space h hence f e co n 1 ci given only the projection operators pi onto the individual sets ci i 1 r recover f by an iterative scheme practically the image f is a function of two real variables and the hilbert space h is the real or complex space l2 q where q c r2 since the sets ci are only assumed to be convex the projections pi are in general nonlinear iterative methods of finding a common point of sets by means of contraction like operators can be found in 1 2 5j these methods have been applied to image processing first by youla and webb 11 a procedure with essentially the same ideas has been used in 7j for the extrapolation of bandlimited functions in the methods of image recovery used until now 9 12 7j from the projections pi i 1 r r operators ti are constructed and from these the composition operator t tr tr 1 t2 t1 starting from an arbitrary element x of h the sequence tnx 0 is shown to converge at least weakly to an element of co two cases are important the case where each ti equals pi hence t prpr 1 p2p1 and the case where each ti is given by ti l oi pi l with 1 the identity operator on h and ai a relaxation parameter with 0 ai 2 when oi 1 for all i this reduces to the foregoing case so at each iteration step the computation has to be done sequentially when a parallel computer is available the computing process could be speeded up a lot if at each iteration step the contribution of the different 413

metric subregularity and the proximal point method we examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators we begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator this result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods

a generalization of polyak s convergence result for subgradient optimization this paper generalizes a practical convergence result first presented by polyak this new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach

block iterative surrogate projection methods for convex feasibility problems abstract a unified framework is presented for studying the convergence of projection methods for finding a common point of finitely many closed convex sets in r n every iteration approximates each set by a half space given either by an approximate projection of the current iterate or by an aggregate inequality derived from the convex inequalities describing this set the next iterate is found by projecting the current one on a surrogate half space formed by taking a convex combination of the half space inequalities convergence to a solution is established under weak conditions that allow various acceleration techniques and choices of aggregating weights the resulting methods are block iterative and hence lend themselves to parallel implementation we show that the idea of forming cut maps via surrogate inequalities encompasses many classical as well as recently proposed methods for set intersection problems and convex feasibility problems with nondifferentiable inequalities and linear equations and inequalities

block transitive automorphism groups of designs in this note we consider subgroups of the automorphism group of a 2 v k 1 design the notation is standard see for example 2 the number of blocks through a point is r v l k 1 and the number of blocks b is vr k recall that a flag is an incident point block pair there are two main results in this note the first is straightforward but essential to the proof of the second

parallel projected aggregation methods for solving the convex feasibility problem convergence conditions are established for new sequential and parallel projected aggregation methods pams that find a feasible point of a large system of convex inequalities and linear equations

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

convergence studies on block iterative algorithms for image reconstruction the sequential block iterative scheme based on landweber s method is a general iterative one for image reconstruction in this paper we give the finite termination convergence for this scheme which provides an approach to choose relaxation coefficients furthermore sufficient and necessary convergence conditions are also established for the sequential block iterative scheme case

the rate of convergence for the method of alternating projections ii abstract the purpose of the paper is threefold 1 to develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember 2 to exhibit a counterexample to a conjecture of kayalar and weinert 3 to show that in the case of at least three subspaces any error bound which only depends on the angles between the various subspaces involved can never be sharp

iterative algorithm for solving a class of convex feasibility problem abstract in this paper we consider a class of convex feasibility problem which arises in quantum computation based on the matrix equation theory the feasible sets are characterized by exploiting the special structure of the linear constraints and its analytic expression is given by making use of the nice structure properties and the kkt condition we derive the projection formulas of a matrix onto the feasible sets the relaxed alternating projection method is designed to solve the convex feasibility problem numerical experiments show that the new method is feasible and effective

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

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

surrogate projection methods for finding fixed points of firmly nonexpansive mappings we present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a hilbert space at every iteration an approximation to each mapping generates a halfspace containing its set of fixed points the next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities this acceleration technique extends one for convex feasibility problems cfps since projection operators onto closed convex sets are firmly nonexpansive the resulting methods are block iterative and hence lend themselves to parallel implementation we extend to accelerated methods some recent results of bauschke and borwein siam rev 38 1996 pp 367 426 on the convergence of projection methods

random products of contractions in metric and banach spaces abstract suppose x d is a metric space and t0 tn is a family of quasinonexpansive self mappings on x we give conditions sufficient to guarantee that every possible iteration of mappings drawn from t0 tn converges as a consequence if c0 cn are closed convex subsets of a hilbert space with nonempty intersection one of which is compact and the proximity mappings are iterated in any order provided only that each is used infinitely often then the resulting sequence converges strongly to a point of the common intersection

stadium norm and douglas rachford splitting a new approach to road design optimization the basic optimization problem of road design is quite challenging due to an objective function that is the sum of nonsmooth functions and the presence of set constraints in this paper we model and solve this problem by employing the douglas rachford splitting algorithm this requires a careful study of new proximity operators related to minimizing area and to the stadium norm we compare our algorithm to a state of the art projection algorithm our numerical results illustrate the potential of this algorithm to significantly reduce cost in road design

online sparse system identification and signal reconstruction using projections onto weighted ell_ 1 balls this paper presents a novel projection based adaptive algorithm for sparse signal and system identification the sequentially observed data are used to generate an equivalent sequence of closed convex sets namely hyperslabs each hyperslab is the geometric equivalent of a cost criterion that quantifies data mismatch sparsity is imposed by the introduction of appropriately designed weighted l1 balls and the related projection operator is also derived the algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted l1 balls the resulting scheme exhibits linear dependence with respect to the unknown system s order on the number of multiplications additions and an o llog2l dependence on sorting operations where l is the length of the system signal to be estimated numerical results are also given to validate the performance of the proposed method against the least absolute shrinkage and selection operator lasso algorithm and two very recently developed adaptive sparse schemes that fuse arguments from the lms rls adaptation mechanisms with those imposed by the lasso rational

upper estimates for the number of periodic solutions to multi dimensional systems abstract the maximal number of zeros of multi dimensional real analytic maps with small parameter is studied by means of the multi dimensional generalization of rouche s theorem the obtained result is applied to study the maximal number of periodic solutions to multi dimensional differential systems an application to a class of three dimensional autonomous systems is given

bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

on the convergence of the products of firmly nonexpansive mappings consider a finite collection of firmly nonexpansive self mappings on a hilbert space whose fixed point sets intersect it is shown that in the finite dimensional case any iteration of mappings drawn from this collection converges this resolves for the finite dimensional case at least a popular conjecture concerning the convergence of the successive projection method in the infinite dimensional case it is shown that if the mappings are drawn according to a certain order called the quasi cyclic order then the iteration converges weakly in a sense the quasi cyclic order may be viewed as an extension of the well known cyclic order in which the lengths of the cycles are permitted to grow without bound

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

the efficiency of subgradient projection methods for convex optimization part i general level methods we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we present several variants and show that they enjoy almost optimal efficiency estimates in another paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations they can also exploit accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a hilbert space the projection onto convex sets pocs algorithm in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule has been the most prevalent tool to solve such problems nonetheless pocs has several shortcomings it converges slowly it is ill suited for implementation on parallel processors and it requires the computation of exact projections at each iteration we propose a general parallel projection method emopsp that overcomes these shortcomings at each iteration of emopsp a convex combination of subgradient projections onto some of the sets is formed and the update is obtained via relaxation the relaxation parameter may vary over an iteration dependent extrapolated range that extends beyond the interval 0 2 used in conventional projection methods emopsp not only generalizes existing projection based schemes but it also converges very efficiently thanks to its extrapolated relaxations theoretical convergence results are presented as well as numerical simulations

network flows that solve linear equations we study distributed network flows as solvers in continuous time for the linear algebraic equation mathbf z mathbf h mathbf y each node i has access to a row mathbf h i mathrm t of the matrix mathbf h and the corresponding entry z i in the vector mathbf z the first consensus projection flow under investigation consists of two terms one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the mathbf h i and z i the second projection consensus flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback without dwell time assumption on switching graphs we prove that all node states converge to a common solution of the linear algebraic equation if there is any the convergence is global for the consensus projection flow while local for the projection consensus flow in the sense that the initial values must lie on the affine subspaces if the linear equation has no exact solutions we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the consensus projection flow for a fixed bidirectional graph semi global convergence to approximate least squares solutions is also demonstrated for switching balanced directed graphs under suitable conditions it is also shown that the projection consensus flow drives the average of the node states to the least squares solution with a complete graph numerical examples are provided as illustrations of the established results

averaging methods of arbitrary order periodic solutions and integrability in this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems this result extends and improves results on averaging theory of periodic analytic differential systems and it unifies many different kinds of averaging methods applying our theory to autonomous analytic differential systems we obtain some conditions on the existence of limit cycles and integrability for polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues we prove that there always exists a positive number n such that if its first n averaging functions vanish then all averaging functions vanish and consequently there exists a neighborhood of the origin filled with periodic orbits consequently if all averaging functions vanish the origin is a center for n 2 furthermore in a punctured neighborhood of the origin the system is c infty completely integrable for n 2 provided that each periodic orbit has a trivial holonomy finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

a proximal projection method for finding zeros of set valued operators in this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set valued operators in a reflexive banach space this algorithm which we call the proximal projection method is essentially a fixed point procedure and our convergence results are based on new generalizations of the browder s demiclosedness principle we show how the proximal projection method can be applied for solving ill posed variational inequalities and convex optimization problems with data given or computable by approximations only the convergence properties of the proximal projection method we establish also allow us to prove that the proximal point method with bregman distances whose convergence was known to occur for maximal monotone operators still converges when the operator involved in it is monotone with sequentially weakly closed graph

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

a norm convergence result on random products of relaxed projections in hilbert space suppose x is a hilbert space and c 1 ldots c n are closed convex intersecting subsets with projections p 1 ldots p n suppose further r is a mapping from bbb n onto 1 ldots n that assumes every value infinitely often we prove a more general version of the following result begin quote if the n tuple c 1 ldots c n is innately boundedly regular then the sequence x n defined by x 0 in x mbox arbitrary x n 1 p r n x n mbox for all n geq 0 converges in norm to some point in bigcap i 1 n c i end quote examples without the usual assumptions on compactness are given methods of this type have been used in areas like computerized tomography and signal processing

bifurcation theory for finitely smooth planar autonomous differential systems abstract in this paper we establish bifurcation theory of limit cycles for planar c k smooth autonomous differential systems with k n the key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of hopf bifurcation at a fine focus or a center and of poincare bifurcation in a period annulus we especially study the smoothness of the first order melnikov function in degenerate hopf bifurcation at an elementary center as we know the smoothness problem was solved for analytic and c differential systems but it was not tackled for finitely smooth differential systems here we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case

convergence of sequential and asynchronous nonlinear paracontractions we establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces applications to the solution of linear systems of equations with convex constraints are outlined a first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented

steered sequential projections for the inconsistent convex feasibility problem we study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step sizes this algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the bregman distances to the members of the family of convex sets the resulting algorithm is a new steered sequential bregman projection method

perturbed projections and subgradient projections for the multiple sets split feasibility problem we study the multiple sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space by casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case we further derive and analyze a perturbed projection method for the multiple sets split feasibility problem and additionally furnish alternative proofs to two known results

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

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

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 general iterative method for a finite family of nonexpansive mappings abstract we study the approximation of common fixed points of a finite family of nonexpansive mappings and suggest a modification of the iterative algorithm without the assumption of any type of commutativity also we show that the convergence of the proposed algorithm can be proved under some types of control conditions

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

variational analysis applied to the problem of optical phase retrieval we apply nonsmooth analysis to a well known optical inverse problem phase retrieval the phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years the state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem despite widespread use of these algorithms current mathematical theory cannot explain their success at the heart of projection algorithms is a nonconvex nonsmooth optimization problem we obtain some insight into these algorithms by applying techniques from nonsmooth analysis in particular we show that the weak closure of the set of directions toward the projection generate the subdifferential of the corresponding squared set distance function following a pattern of proof described in f h clarke yu s ledyaev r j stern and p r wolenski nonsmooth analysis and control theory springer verlag new york 1998 this result is generalized to provide conditions under which the subdifferential of an integral function equals the integral of the subdifferential

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 parallel splitting method for coupled monotone inclusions a parallel splitting method is proposed for solving systems of coupled monotone inclusions in hilbert spaces and its convergence is established under the assumption that solutions exist unlike existing alternating algorithms which are limited to two variables and linear coupling our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes the breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions variational problems best approximation and network flows

the efficiency of subgradient projection methods for convex optimization part ii implementations and extensions in the first part of this paper we studied subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we presented several variants and showed that they enjoy almost optimal efficiency estimates in this part of the paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations we discuss accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

bifurcation of periodic orbits by perturbing high dimensional piecewise smooth integrable systems abstract this paper deals with the problem of periodic orbit bifurcations for high dimensional piecewise smooth systems under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane a formula for the first order melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits we especially can use the function to study the number of periodic orbits both in degenerate hopf bifurcations and in degenerate homoclinic bifurcations finally we present two examples to illustrate an application of the theoretical results

a family of projective splitting methods for the sum of two maximal monotone operators a splitting method for two monotone operators a and b is an algorithm that attempts to converge to a zero of the sum a b by solving a sequence of subproblems each of which involves only the operator a or only the operator b prior algorithms of this type can all in essence be categorized into three main classes the douglas peaceman rachford class the forward backward class and the little used double backward class through a certain extended solution set in a product space we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in hilbert space our algorithms are essentially standard projection methods using splitting decomposition to construct separators we prove convergence through fejer monotonicity techniques but showing fejer convergence of a different sequence to a different set than in earlier splitting methods our projective algorithms converge under more general conditions than prior splitting methods allowing the proximal parameter to vary from iteration to iteration and even from operator to operator while retaining convergence for essentially arbitrary pairs of operators the new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases

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

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

optical wavefront reconstruction theory and numerical methods optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in nasa s hubble space telescope hst nasa s success with reconstruction algorithms on the hst has led to an effort to develop software that can aid and in some cases replace complicated expensive and error prone hardware among the many applications is hst s replacement the next generation space telescope ngst indent this work details the theory of optical wavefront reconstruction reviews some numerical methods for this problem and presents a novel numerical technique that we call extended least squares we compare the performance of these numerical methods for potential inclusion in prototype ngst optical wavefront reconstruction software we begin with a tutorial on rayleigh sommerfeld diffraction theory