coulomb plus power law potentials in quantum mechanics we study the discrete spectrum of the hamiltonian h v r for the coulomb plus power law potential v r 1 r sgn q rq where 0 q 2 and q 0 we show by envelope theory that the discrete eigenvalues enl of h may be approximated by the semiclassical expression enl q minr 0 1 r2 1 r sgn q r q values of and are prescribed which yield upper and lower bounds accurate upper bounds are also obtained by use of a trial function of the form r rl 1 e xr d we give detailed results for v r 1 r rq q 0 5 1 2 for n 1 l 0 1 2 along with comparison eigenvalues found by direct numerical methods

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

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

massively parallel row action algorithms for some nonlinear transportation problems row action iterative algorithms are developed for two classes of nonlinear optimization problems with transportation constraints entropy problems where the objective function is a sum of x ln x a 1 terms and quadratic problems where the objective function is a sum of frac 1 2 wx 2 cx terms the algorithms are specialized to take advantage of both the structure of the transportation constraints and the special forms of the objective functions both generalized and pure networks are dealt with in this paper the algorithms are well suited for parallel computing implementations are developed on a massively parallel connection machine cm 2 with up to 32k processing elements the algorithms solve test problems with 4000 nodes and 4 million arcs in less than one minute of computer time on a maximally configured machine with 64k processing elements they achieve a peak computing rate of 3 gflops

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

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

weighted p laplacian problems on a half line abstract we study the weighted half line eigenvalue problem y x p 1 sgn y x p 1 r x q x y x p 1 sgn y x 0 x for 1 p with initial condition y 0 sin y 0 cos 0 using a modified prufer angle x the eigenvalues k k 0 with k as k are characterized by k x k 1 p and x k 1 p if k k 1 as x we allow the weight r to be locally integrable and definite semidefinite or indefinite in the first two cases the sequence of eigenvalues accumulates at one of and in the third the sequence accumulates at both in all cases solutions y are nonoscillatory on 0 for all

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

upper and lower bounds for an eigenvalue associated with a positive eigenvector when an eigenvector of a semibounded operator is positive we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue this theorem is a substantial generalization of barta type inequalities and can be applied to non necessarily purely quadratic hamiltonians an application for a magnetic hamiltonian is given and the case of a discrete schrodinger operator is also discussed it is shown how this approach leads to some explicit bounds on the ground state energy of a system made of an arbitrary number of attractive coulombian particles

a computational methodology for shape optimization of structures in frictionless contact this paper presents a computational methodology for shape optimization of structures in frictionless contact which provides a basis for developing user friendly and efficient shape optimization software for evaluation it has been implemented as a subsystem of a general finite element software the overall design and main principles of operation of this software are outlined the parts connected to shape optimization are described in more detail the key building blocks are analytic sensitivity analysis an adaptive finite element method an accurate contact solver and a sequential convex programing optimization algorithm results for three model application examples are presented in which the contact pressure and the effective stress are optimized

shape optimization with p adaptivity the p method of analysis when used with shape optimization has certain advantages over the h method the method of h adaptivity results in fine meshes with small elements in areas of stress concentration the presence of these small elements prevents the shape from changing significantly during optimization due to mesh distortion the p adaptive method uses relatively large sized elements and adaptivity is achieved by increasing the polynomial order of the shape functions rather than by introducing small elements as in the h method thus the p adaptive method allows for greater shape changes without sacrificing the accuracy of the analysis a methodology is developed in this paper to integrate p adaptive analysis with shape optimization for general velocity fields the key contributions lie in an easily implementable shape update strategy and an adaptive scheme to locally increase the polynomial order based on error indicators numerical results show that significant shape change is achievable prior to serious mesh distortion in addition need for an h p adaptive strategy rather than a pure p adaptive strategy is also highlighted

shape sensitivity analysis in linear elastic fracture mechanics shape sensitivity analysis of an elastic solid in equilibrium with a known load system applied over its boundary is presented in this work the domain and boundary integral expressions of the first and second order shape derivatives of the total potential energy are established by using an arbitrary change of the domain characterized by a velocity field defined over the initial body configuration in these expressions we recognize free divergence tensors that are denoted in this paper as energy shape change tensors next shape sensitivity analysis is applied to cracked bodies for that purpose a suitable velocity distribution field is adopted to simulate the crack advance of a unit length in a two dimensional body finally the corresponding domain and the equivalent path independent integral expressions of the first and second order potential energy release rate of fracture mechanics are also derived

hypersingular residuals a new approach for error estimation in the boundary element method summary this paper presents a new approach for a posteriori pointwise error estimation in the boundary element method the estimator relies upon evaluation of the residual of hypersingular integral equations and is therefore intrinsic to the boundary integral equation approach a methodology is developed for approximating the error on the boundary as well as in the interior of the domain extensive computational experiments have been performed for the two dimensional laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve moreover a reasonable estimate of the magnitude of the actual error is also predicted

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

state space newton s method for topology optimization abstract we introduce a new algorithm for solving certain classes of topology optimization problems which enjoys fast local convergence normally achieved by the full space methods while working in a smaller reduced space the computational complexity of newton s direction finding subproblem in the algorithm is comparable with that of finding the steepest descent direction in the traditional first order nested reduced space algorithms for topology optimization that is the space reduction is computationally inexpensive and more importantly it does not ruin the sparsity of the full space system of optimality conditions the fast local convergence of the algorithm allows one to efficiently solve a sequence of optimization problems for varying parameters numerical continuation this can be utilized for eliminating the errors introduced by the approximate enforcement of the boundary conditions or 0 1 type constraints on the design field through penalties in many topology optimization approaches we test the algorithm on the benchmark problems of dissipated power minimization for stokes flows and in all cases the algorithm outperforms the traditional first order reduced space nested approaches by a factor varying from two to almost twenty in terms of the number of iterations while attaining an almost unprecedented accuracy in solving the discretized topology optimization problem finally we present a few extensions to the algorithm one involving computations on adaptively refined meshes and another related to solving topology optimization problems for non newtonian fluids

a general methodology for structural shape optimization problems using automatic adaptive remeshing this methodology is based on the use of the adaptive mesh refinement techniques in the context of 2d shape optimization problems a technique for the parametrization of the optimization problem using b splines to define the boundary is first presented then mesh generation using the advancing frontal method the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems in particular the analytical sensitivity analysis of the different items ruling the problem b splines finite element mesh structural behaviour and error estimator is studied in detail

on efficient solutions to the continuous sensitivity equation using automatic differentiation shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation pde these shape sensitivities are described by a continuous sensitivity equation cse automatic differentiation ad can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation the approximate solution of the pde uses a spatial discretization mesh that often depends on the shape parameters therefore the straightforward application of ad introduces derivatives of the mesh there are two drawbacks to this approach first extra computational effort especially memory is used in these calculations due to mesh sensitivities second this mesh sensitivity information needs to be computed in order to obtain accurate results in this work we provide a methodology that avoids mesh sensitivities and their drawbacks by defining a modified pde on a fixed domain i e independent of the shape parameter such that ad provides the desired approximation of the cse using two examples we demonstrate significant improvement in the computational effort both in terms of floating point operations and memory requirements we explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code these changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation

generalized comparison theorems in quantum mechanics this paper is concerned with the discrete spectra of schrodinger operators h v where v r is an attractive potential in n spatial dimensions two principal results are reported for the bottom of the spectrum of h in each angular momentum subspace l i an optimized lower bound when the potential is a sum of terms v r v 1 r v 2 r and the bottoms of the spectra of v 1 r and v 2 r in l are known and ii a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials v 1 r and v 2 r intersect in a controlled way pure power law potentials are studied and an application of the results to the coulomb plus linear potential v r a r br is presented in detail for this problem an earlier formula for energy bounds is sharpened and generalized to n dimensions

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

ground state energy bounds for potentials x ν a theory is developed from which both upper and lower analytic bounds on schrodinger eigenvalues can be obtained we propose a recursion algorithm with which ground energies for certain potentials can be rigorously bounded to arbitrary precision these analytic and numerical methods together with existing techniques are applied to the ground state problem for power potentials x 0

optimal bounds for ratios of eigenvalues of one dimensional schrodinger operators with dirichlet boundary conditions and positive potentials consider the schrodinger equation u v x u u on the intervali wherev x 0 forx i and where dirichlet boundary conditions are imposed at the endpoints ofi we prove the optimal bound frac lambda n lambda 1 leqq n 2 for n 2 3 4 on the ratio of thenth eigenvalue to the first eigenvalue for this problem this leads to a complete treatment of bounds on ratios of eigenvalues for such problems extensions of these results to singular problems are also presented a modified prufer transformation and comparison techniques are the key elements of the proof

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

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

bound state energy eigenvalues for a general class of one dimensional problems on the whole axis via the prufer transformation an extension of the prufer phase function method for the bound state energy calculation is presented it is applicable to one dimensional problems described by the schrodingere quation on the whole axis with a general class of potentials theorems are given which are a generalization of the analogous ones concerning the half axis 0 problems that have been presented in previous papers the method is suitable especially for numerical calculations of the bound state energy eignevalues

a geometrical theory of energy trajectories in quantum mechanics suppose f r is an attractive central potential of the form f r ki 1 g i f i r where f i is a set of basis potentials powers log hulthen sech2 and g i is a set of smooth increasing transformations which for a given f are either all convex or all concave formulas are derived for bounds on the energy trajectories enl fnl v of the hamiltonian h vf r where v is a coupling constant the transform f f is carried out in two steps f f f where f s is called the kinetic potential of f and is defined by f s inf f subject to d l2 r3 where d is the domain of h 1 and s a table is presented of the basis kinetic potentials f i s the general trajectory bounds f v are then shown to be given by a legendre transformation of the form s f s v f v where f s ki 1g i f i s and f v mins 0 s v f s with the aid of this potential construction set a kind of schrodinger lego ground state trajectory bounds are derived for a vari

parallel optimization for traffic assignment most large scale optimization problems exhibit structures that allow the possibility of attack via algorithms that exhibit a high level of parallelism the emphasis of this paper is the development of parallel optimization algorithms for a class of convex block structured problems computational experience is cited for some large scale problems arising from traffic assignment applications the algorithms considered here have the property that they allow such problems to be decomposed into a set of smaller optimization problems at each major iteration these smaller problems correspond to linear single commodity networks in the traffic assignment case and they may be solved in parallel results are given for the distributed solution of such problems on the crystal multicomputer

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

on the fine grain decomposition of multicommodity transportation problems a simple algorithm for nonlinear optimization problems with multicommodity transportation constraints is developed the algorithm is of the row action type and when properly applied decomposes the underlying graph alternatingly by nodes and arcs hence a fine grain decomposition scheme is developed that is suitable for massively parallel computer architectures of the simd i e single instruction stream multiple data stream class data structures are developed for the implementation of both dense and sparse problems and details of an implementation on a connection machine cm 2 with 32k processing elements are given the dense implementation achieves computing rate of up to three gflops several aspects of the algorithm are investigated empirically computational results are reported for the solution of quadratic programs with approximately 10 million columns and 100 thousand rows

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

a differential method for bounding the ground state energy for a wide class of hamiltonians a novel method for obtaining lower and upper bounds for the lowest energy is presented unlike perturbative or variational techniques this method does not involve the computation of any integral a normalization factor or a matrix element it just requires the determination of the absolute minimum and maximum in the whole configuration space of the local energy associated with a normalizable trial function the calculation of the norm is not needed after a general introduction the method is applied to three non integrable systems the asymmetric annular billiard the many body spinless coulombian problem the hydrogen atom in a constant and uniform magnetic field being more sensitive than the variational methods to any local perturbation of the trial function this method can be used to systematically improve the energy bounds with a local skilled analysis an algorithm relying on this method can therefore be constructed and an explicit example for a one dimensional problem is given

fast eigenvalue algorithm for central potentials an algorithm is described that seeks out schrodinger eigenvalues enl for a given central potential v r the algorithm has in principle arbitrary precision a particular implementation has absolute error mod e enl mod decreasing as t 3 where t is machine computing time the method is tested on the central gaussian potential v r a exp lambda r2 of recent interest

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

high order sensitivity analysis in shape optimization problems abstract a formulation for high order directional differentiation of functions defined through integration is presented the formulation has been developed initially for sensitivity analysis in optimal shape design problems by the finite elements method however its application range is much wider explicit analytical expressions are given for sequential directional derivatives special emphasis has been made in holding recurrence and simplicity of intermediate operations the proposed scheme does not depend strongly on any particular form of the state equations and its numerical implementation in finite elements method programs can be considered easy and efficient as an example up to second order sensitivity techniques are applied to the optimization of a concrete roof structural analysis has been performed in linear elastic conditions with quadratic three dimensional elements for the discretization of the roof

interval constrained matrix balancing abstract we define two interval constrained matrix balancing problems the models encompass several known formulations of problems that appear in regional input output analysis estimation of traffic over transportation and telecommunications networks and estimation of social accounting matrices we develop primal dual row action algorithms for the solution of these models and establish their convergence both algorithms generalize existing scaling algorithms for equality constrained matrix balancing problems one of the algorithms is a generalization of the well known procedure for matrix balancing called ras that can handle the range constraints and is thus called range ras the structure of the problem makes the algorithm suitable for implementation on massively parallel computers details of our implementation on a connection machine cm 2 are given numerical results for the solution of problems of size up to 500 500 are given and the performance of the algorithm is compared with that of ras