on the global convergence of trust region algorithms for unconstrained minimization many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large 2 we consider how large these approximations have to be if they prevent convergence when the objective function is bounded below and continuously differentiable thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number

numerical validation of solutions of complementarity the nonlinear case this paper proposes a validation method for solutions of nonlinear complementarity problems the validation procedure performs a computational test if the result of the test is positive then it is guaranteed that a given multi dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem

curvature characteristics and congruence of the thumb carpometacarpal joint differences between female and male joints abstract three dimensional geometric models of the articular surfaces of the thumb carpometacarpal cmc joint were constructed using precise data obtained from stereophotogrammetry spg it was demonstrated that by using a least squares surface fitting technique the spg data on the surface can accurately be described by a single parametric biquintic spline function from this mathematical description curvature maps of the surfaces were calculated for 13 cmc joints eight females average 64 yr old five males average 70 yr old the surface geometry of each joint was analyzed comparisons were made between trapezial and metacarpal surfaces of the joint and differences determined between males and females with regard to joint surface areas the female trapezium is significantly smaller than that of the metacarpal the shape of the female trapezial surface is also fundamentally different from of males no gender related difference exists regarding the shape of the metacarpal surface congruence of the two opposing articular surfaces was defined by their relative principal curvatures from these definitions congruence in the radioulnar and dorsovolar anatomic directions as well as the global congruence of the joint were calculated most cmc joints were found to be more congruent along the radioulnar direction than the dorsovolar direction and globally female joints were found to be less congruent than male joints the concept of joint congruence has played a central role in a number of hypotheses relating to the etiology of cmc joint osteoarthritis oa although conflicting hypotheses do exist the precise quantitative findings of this study may lead to an improved understanding of cmc joint oa and perhaps explain its prevalence in the female population over 55

multilevel algorithms for constrained compact fixed point problems in this paper the authors extend the multilevel algorithm of atkinson and brakhage for compact fixed point problems and the projected newton method of bertsekas to create a fast multilevel algorithm for parabolic boundary control problems having bound constraints on the control results are extended from finite dimension on constraint identification this approach permits both adaptive integration in time and inexact evaluation of the cost functional

pseudo transient continuation for nonsmooth nonlinear equations pseudo transient continuation is a newton like iterative method for computing steady state solutions of differential equations in cases where the initial data are far from a steady state the iteration mimics a temporal integration scheme with the time step being increased as steady state is approached the iteration is an inexact newton iteration in the terminal phase in this paper we show how steady state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo transient continuation an example of such a case is a discretized pde with a lipschitz continuous but nondifferentiable constitutive relation as part of the nonlinearity in this case we can approximate a generalized derivative with a difference quotient the existing theory for pseudo transient continuation requires lipschitz continuity of the jacobian newton like methods for nonsmooth equations have been globalized by trust region methods smooth approximations and splitting methods in the past but these approaches are not designed to find steady state solutions of time dependent problems the method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo transient continuation

global convergence of trust region algorithms for convex constrained minimization without derivatives in this work we propose a trust region algorithm for the problem of minimizing a function within a convex closed domain we assume that the objective function is differentiable but no derivatives are available the algorithm has a very simple structure and allows a great deal of freedom in the choice of the models under reasonable assumptions for derivative free schemes we prove global convergence for the algorithm that is to say that all accumulation points of the sequence generated by the algorithm are stationary

a decomposition method for large scale box constrained optimization a decomposition method for solving large scale box constrained optimization is proposed the algorithm is motivated by the successful use of the decomposition method presented by joachims for training support vector machines in particular a new technique based on the new definition kkt violating index is introduced for working set identification finally the numerical experiments and implementation details show that this method is practical for large scale problems

primal dual interior point methods for pde constrained optimization this paper provides a detailed analysis of a primal dual interior point method for pde constrained optimization considered are optimal control problems with control constraints in l p it is shown that the developed primal dual interior point method converges globally and locally superlinearly not only the easier l setting is analyzed but also a more involved l q analysis q is presented in l the set of feasible controls contains interior points and the frechet differentiability of the perturbed optimality system can be shown in the l q setting which is highly relevant for pde constrained optimization these nice properties are no longer available nevertheless a convergence analysis is developed using refined techniques in parti cular two norm techniques and a smoothing step are required the l q analysis with smoothing step yields global linear and local superlinear convergence whereas the l analysis without smoothing step yields only global linear convergence

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

adjoint sensitivity analysis for differential algebraic equations algorithms and software an efficient numerical method for sensitivity computation of large scale differential algebraic systems is developed based on the adjoint method issues that are critical for the implementation are addressed complexity analysis and numerical results demonstrate that the adjoint sensitivity method is advantageous over the forward sensitivity method for applications with a large number of sensitivity parameters and few objective functions

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

on the lagrange newton sqp method for the optimal control of semilinear parabolic equations a class of lagrange newton sqp methods is investigated for optimal control problems governed by semilinear parabolic initial boundary value problems distributed and boundary controls are given restricted by pointwise upper and lower bounds the convergence of the method is discussed in appropriate banach spaces based on a weak second order sufficient optimality condition for the reference solution local quadratic convergence is proved the proof is based on the theory of newton methods for generalized equations in banach spaces

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

a trust region algorithm for parametric curve and surface fitting abstract let a family of curves or surfaces be given in parametric form via the model equation x s where x rn r m and s s rd d such that the manifold m s s s is a best fit toscattered data zi i 1y rn in the sense that for some s i i 1 n the sum of the squared least distances i 1 n z i s i 2 2 from the data points to the manifold m is minimal among all such manifolds for robustness our algorithm uses a globally convergent trust region approach in which at each iteration an approximation to the objective function is minimized in a given neighborhood of the current iterate the set s may be all of rd or a closed convex subset in particular it may be chosen so that our theory is applicable to splines

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

an adaptive moving mesh method for two dimensional thin film flow equations with surface tension abstract in this paper we extend our previous work alharbi and naire 2017 on a one dimensional r adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations mmpdes to two dimensions as a test problem we consider the gravity driven thin film flow down an inclined and pre wetted plane including surface tension and a moving contact line this technique accurately captures and resolves the moving contact line and associated fingering instability moreover the computational effort is hugely reduced in comparison to a fixed uniform mesh

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

the accuracy of joint surface models constructed from data obtained with an electromagnetic tracking device electromagnetic tracking devices are widely used in biomechanics in this article a method is evaluated to construct models of articular surfaces using an electromagnetic tracking device first the accuracy of the space tracker was examined and optimised then from several joint surfaces random points were measured and eighth degree polynomials were fitted to these measurements to check if the fit converged well plots of cross sections of the model with corresponding data points were examined the accuracy of the models was determined by comparing them with computed tomography data and by reproducibility tests all the fits converged well to the data the root mean square rms error of the models varied from 0 07 to 0 18mm and was proportional to the size and complexity of the surface this was mainly due to systematic errors made by the space tracker which were also proportional to the size and complexity of the surface

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

tensor product adaptive grids based on coordinate transformations in this paper we discuss a two dimensional adaptive grid method that is based on a tensor product approach adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations pdes and ordinary differential equations a traditional and widely used form of adaptivity is the concept of equidistribution which is well defined and well understood in one space dimension the extension of the equidistribution principle to two or three space dimensions however is far from trivial and has been the subject of investigation of many researchers during the last decade besides the nonsingularity of the transformation that defines the nonuniform adaptive grid the smoothness of the grid or transformation plays an important role as well we will analyse these properties and illustrate their importance with numerical experiments for a set of time dependent pde models with steep moving pulses fronts and boundary layers

time domain green s function for an infinite sequentially excited periodic planar array of dipoles the present paper is a continuation of previous explorations by the authors aimed at gaining a basic understanding of the time domain td behavior of large periodic phased i e sequentially turned on array antennas and related configurations our systematic investigation of the relevant canonical td dipole excited green s functions has so far included those for infinite and truncated sequentially pulsed line periodic arrays parameterized in terms of radiating propagating and nonradiating evanescent conical td floquet waves fw and truncation induced td fw modulated tip diffractions the present contribution extends these investigations to an infinite periodic sequentially pulsed planar array which generates pulsed plane propagating and evanescent fw starting from the familiar frequency domain fd transformation of the linearly phased element by element summation synthesis into summations of propagating and evanescent fws we access the time domain by fourier inversion the inversion integrals are manipulated in a unified fashion into exact closed forms which are parameterized by the single nondimensional quantity spl eta c v sup p sub u1 where v sup p sub u1 and c are the excitation phase speed along a preferred phasing direction u sub 1 in the array plane and the ambient wave speed respectively the present study deals with the practically relevant rapidly phased propagating case spl eta 1 regime for a future manuscript numerical reference data generated via element by element summation over the fields radiated by the individual dipoles with ultrawide band limited excitation are compared with results obtained much more efficiently by inclusion of a few td fws physical interpretation of the formal td fw solutions is obtained by recourse to asymptotics instantaneous frequencies and wavenumbers and related constructs of special interest is the demonstration that the td fws emerge along equal delay ellipses from the array plane this furnishes a novel and physically appealing interpretation of the planar array td fw phenomenology

a note on exploiting structure when using slack variables we show how to exploit the structure inherent in the linear algebra for constrained nonlinear optimization problems when inequality constraints have been converted to equations by adding slack variables and the problem is solved using an augmented lagrangian method

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

a sequential quadratic programming based algorithm for optimization of gas networks abstract british gas uses a complex heavily looped network of pipes and controllable units compressors and regulators to transmit gas from coastal supply terminals to regional demand points computer algorithms are required for efficient management of the system this paper describes an algorithm for optimal control over periods of up to a day the problem is large scale and highly nonlinear in both objective function and constraints the method is based on sequential quadratic programming and takes account of the structure of the pipeflow equations by means of a reduced gradient technique which eliminates most of the variables from the quadratic subproblems the latter involve only simple bound constraints which are handled efficiently by a conjugate gradient active set algorithm trust region techniques permit use of the exact hessian preserving sparsity more general constraints are handled at an outer level by a truncated augmented lagrangian method results are included for some realistic problems the algorithm is generally applicable to problems with a control structure

finite difference methods and spatial a posteriori error estimates for solving parabolic equations in three space dimensions on grids with irregular nodes adaptive methods for solving systems of partial differential equations have become widespread much of the effort has focused on finite element methods in this paper modified finite difference approximations are obtained for grids with irregular nodes the modifications are required to ensure consistency and stability asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method these estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives of the solution at grid nodes the interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids computational results demonstrate the convergence of the finite difference method and a posteriori error estimates for cases not covered by the theory

consistent approximations and approximate functions and gradients in optimal control because of the unavoidable use of numerical integration methods such as runge kutta or finite elements the numerical solution of optimal control problems with either ode or pde dynamics is governed by a discretization parameter such as the integration mesh size usually when explicit integration techniques are used function and derivative values can be computed exactly for the discretized problems recently we have come across some examples where function and derivative values of the explicitly discretized problems had to be approximated by the outcome of n iterations of a solver consequently the discretization of these problems is controlled by two parameters the mesh size and the number of iterations of the solver referring to e polak optimization algorithms and consistent approximations springer verlag 1997 we find a theory for solving optimization problems that require discretization it deals with two situations in the first which is referred to as that of consistent approximations it is assumed that an infinite dimensional optimization problem can be suitably approximated by a family of progressively higher dimensional optimization problems in this case strategies in the form of algorithm models are presented for diagonalizing the solution process in the second situation it is assumed that numerical solution of the dynamic equations does not result in a family of finite dimensional consistent approximations e g when implicit integration methods are used for this case the theory provides models for the implementation of conceptual algorithms unfortunately neither of these situations envisions the possibility of two discretization parameters in this paper we present new algorithm models that can be used with two discretization parameters the first one controls the mesh size of an explicit integration scheme and the second one controls the precision with which functions and gradients associated with a fixed mesh size are computed the result can be seen as a framework of quasi consistent approximations we implemented these new algorithm models using an approximate steepest descent method for the solution of two problems a two point boundary value problem in which the discretized linear ode dynamics are solved approximately using the gauss seidel method and a distributed control problem in which the discretized dynamics are solved using a domain decomposition algorithm which can be implemented on parallelized computers our numerical results show that these new algorithms perform quite well and are fairly insensitive to the selection of user set parameters also they appear to be superior to some alternative ad hoc schemes

a fast multilevel algorithm for integral equations we show how the discretization of integral equations by composite gauss rules can be related to approximations of integral operators that converge in the operator norm rather than strongly converge from this norm convergent formulation a two level approximate inverse can be constructed whose evaluation requires no fine mesh evaluations of the integral operator the resulting multilevel algorithm therefore is roughly half as costly as the atkinson brakhage iteration the algorithm is applicable to both linear and nonlinear equations

superlinear convergence of affine scaling interior point newton methods for infinite dimensional nonlinear problems with pointwise bounds we develop and analyze a superlinearly convergent affine scaling interior point newton method for infinite dimensional problems with pointwise bounds in lp space the problem formulation is motivated by optimal control problems with lp controls and pointwise control constraints the finite dimensional convergence theory by coleman and li siam j optim 6 1996 pp 418 445 makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity since these features are not available in our infinite dimensional framework algorithmic changes are necessary to ensure fast local convergence the main building block is a newton like iteration for an affine scaling formulation of the kkt condition we demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with mbox q rate 1 under a slightly stronger condition if a smoothing step is available we discuss how the algorithm can be embedded in the class of globally convergent trust region interior point methods recently developed by m heinkenschloss and the authors numerical results for the control of a heating process confirm our theoretical findings

on the resolution of the generalized nonlinear complementarity problem minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem gncp defined on a polyhedral cone it is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems theoretical results that relate stationary points of the function that is minimized to the solutions of the gncp are presented perturbations of the gncp are also considered and results are obtained related to the resolution of gncps with very general assumptions on the data these theoretical results show that local methods for box constrained optimization applied to the associated problem are efficient tools for solving the gncp numerical experiments are presented that encourage the use of this approach

a new approach for surface fitting method of articular joint surfaces the application of joint contact mechanics requires a precise configuration of the joint surfaces b spline and nurbs have been widely used to model joint surfaces but because these formulations use a structured data set provided by a rectangular net first then a grid there is a limit to the accuracy of the models they can produce however new imaging systems such as 3d laser scanners can provide more realistic unstructured data sets what is needed is a method to manipulate the unstructured data we created a parametric polynomial function and applied it to unstructured data sets obtained by scanning joint surfaces we applied our polynomial model to unstructured data sets from an artificial joint and confirmed that our polynomial produced a smoother and more accurate model than the conventional b spline method next we applied it to a diarthrodial joint surface containing many ripples and found that our function s noise filtering characteristics smoothed out existing ripples since no formulation was found to be optimal for all applications we used two formulations to model surfaces with ripples first we used our polynomial to describe the global shape of the objective surface minute undulations were then specifically approximated with a fourier series function finally both approximated surfaces were superimposed to reproduce the original surface in a complete fashion

implicit interpolation error based error estimation for reaction diffusion equations in two space dimensions several authors have proposed an error estimation strategy for the finite element method applied to linear reaction diffusion equations in two space dimensions based on an odd even order dichotomy principle for odd order approximations the estimates are computed directly for even order approximations a second solution is computed although both estimators are asymptotically exact the latter are more robust than the former herein the even order method is extended to all orders greater than one thereby recovering robustness for odd orders proofs of asymptotic exactness are extended to nonlinear reaction diffusion equations computational results demonstrating their effectiveness are presented

a moving mesh study for diffusion induced effects in avascular tumour growth abstract studying the dynamical evolution of complex systems as biological ones from the continuum point of view requires monitoring several parameters involved whose modelling leads to system of non linear coupled partial differential equations the interaction of the urokinase plasminogen activator system with a model for cancer cell in the avascular phase is faced with the moving mesh partial differential equation numerical technique monitoring the dynamical evolution of the system as a function of the diffusion properties of cancer cells and of cell proliferation factor over a one dimensional biological domain the computations are consistent with previous results confirming that cancer proliferation in the very early stage of invasion occurs through highly irregular spatio temporal pattern which depends essentially on cancer motility characteristics but non obvious effects are observed which depend on the model proliferation parameters

new adaptive move limit management strategy for approximate optimization part 2 approximations play an important role in multidisciplinary design optimization by offering system behavior information at a relatively low cost most approximate optimization strategies are sequential in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence the move limits are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information to ensure convergence of the sequence of approximate optimizations to a karush kuhn tucker solution a move limit management strategy is required in a companion paper issues of move limit management are reviewed and a new adaptive strategy for move limit management is developed wujek b a and renaud j e new adaptive move limit management strategy for approximate optimization part 1 aiaa journal vol 36 no 10 1998 pp 1911 1921 with its basis in the provably convergent trust region methodology the trust region ratio approximation method tram strategy utilizes available gradient information and employs a backtracking process using various two point approximation techniques to provide a flexible move limit adjustment factor the tram strategy is successfully implemented in application to several multidisciplinary design optimization test problems in addition a comprehensive study comparing the performance of the tram strategy to existing move limit strategies is conducted results indicate that application of the tram strategy results in increased efficiency for approximate optimization processes these implementation studies highlight the ability of the tram strategy to control the amount of approximation error and efficiently manage the convergence to a karush kuhn tucker solution

numerical computation of the incomplete lipschitz hankel integral je o a z abstract two factorial neumann series expansions are derived for the incomplete lipschitz hankel integral je 0 a z these expansions are used together with the neumann series expansion given by agrest in an algorithm which efficiently computes je 0 a z to a user defined number of significant digits other expansions for je 0 a z which are found in the literature are also discussed but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm

a penalty free type nonmonotone trust region method for nonlinear constrained optimization in this paper we propose a penalty free type nonmonotone trust region method for solving general nonlinear programming problem the algorithmic framework yields global convergence without using any penalty function we analyze the global convergence of the main algorithm for the degenerate problems and also give the globally convergent results under the linear independence constraint qualification the preliminary numerical tests are reported

an aggregation based domain decomposition preconditioner for groundwater flow we consider theoretical and computational issues associated with an aggregation based domain decomposition preconditioner applied to a bi cgstab iterative solver used to solve both laplace s equation and an important nonlinear model from hydrology used to simulate unsaturated flow richards equation theoretical results for laplace s equation provide estimates of the condition number and the rate of convergence for a two level schwarz domain decomposition preconditioner computational results for laplace s equation and richards equation show excellent scalability although no theory is yet available to support the results for the difficult nonlinear problem

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

primal dual active set strategy for a general class of constrained optimal control problems an algorithm for efficient solution of optimal control problems with pointwise constraints is introduced it is based on an active set strategy using primal as well as dual variables under certain assumptions a decrease of an appropriately chosen merit function and convergence of the algorithm are proved numerical examples for a parabolic optimal control problem demonstrate the efficiency of the proposed method for control constrained problems

network oriented dyadic time domain green s function for a sequentially excited infinite planar array of dipoles in free space during the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas infinite periodic structures frequency selective surfaces and related applications with emphasis on gaining physical insight into the phenomenology of short pulse radiation the present contribution shows the current status of our network oriented dyadic time domain green s function td gf for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short pulsed radiation by infinite periodic arrays the dispersive effects of the td floquet waves fw and the consequences of a te and tm field decomposition are discussed in details in this network formulation the equivalent td transmission line tl voltages and currents excited by equivalent td current generators are determined an expression for the tm voltage tl gf is provided for the first time in terms of incomplete lipschitz hankel integrals and bessel functions together with some remarks for its evaluation as shown in previous publications for the scalar potentials real td physical observables are determined by pairing td floquet waves with positive and negative indices it is shown that the vectorial td radiated field is reconstructed at any time with the superposition of a few td fws

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

an incomplete assembly with thresholding algorithm for systems of reaction diffusion equations in three space dimensions iat for reaction diffusion systems solving systems of reaction diffusion equations in three space dimensions can be prohibitively expensive both in terms of storage and cpu time herein i present a new incomplete assembly procedure that is designed to reduce storage requirements incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values the algorithm is incorporated in a finite element method of lines code and tested on a set of reaction diffusion systems the effect of incomplete assembly on cpu time and storage and on the performance of the temporal integrator daspk algebraic solver gmres and preconditioner ilut is studied

three dimensional mathematical model analysis of the patellofemoral joint this paper is concerned with a mathematical model analysis of the patellofemoral joint in the human knee taking into account the articular surface geometry and mechanical properties of the ligament it was made by the application of a computer aided design theory previously studied and it was possible to express the articular surface geometries in a mathematical formulation and hence elucidate the joint movement mechanics this method was then applied to a three dimensional geometrical model of the patellofemoral joint for the modelling of tendofemoral contact at large angles of knee flexion the geodestic line theory was adopted applying the newton raphson method and the runge kutta gil method to the model variables such as patellar attitudes patellofemoral contact force and tensile force of the patellar ligament for various knee flexion angles were computed applying the hertzian elastic theory contact stress was also computed these results showed good agreement with the previously reported experimental results as an application for the model some parameter analyses were performed in terms of the contact stress variations and compared with those of the normal knee the simulation results indicated that both the q angle increase and decrease increased contact stress the patella alta showed undulating variations of stress while the patella infera showed little change of stress and the tibial tuberositas elevation showed 20 30 reduction of stress

an active set quasi newton method with projected search for bound constrained minimization we analyze an active set quasi newton method for large scale bound constrained problems our approach combines the accurate active set identification function and the projected search both of these strategies permit fast change in the working set the limited memory method is employed to update the inactive variables while the active variables are updated by simple rules a further division of the active set enables the global convergence of the new algorithm numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies

finite element model updating for structural applications abstract a novel method for performing model updating on finite element models is presented the approach is particularly tailored to modal analyses of buildings by which the lowest frequencies obtained by using sensors and system identification approaches need to be matched to the numerical ones predicted by the model this is done by optimizing some unknown material parameters such as mass density and young s modulus of the materials and or the boundary conditions which are often known only approximately in particular this is the case when considering historical buildings the straightforward application of a general purpose optimizer can be impractical given the large size of the model involved in the paper we show that by slightly modifying the projection scheme used to compute the eigenvalues at the lowest end of the spectrum one can obtain local parametric reduced order models that embedded in a trust region scheme form the basis for a reliable and efficient specialized algorithm we describe an optimization strategy based on this approach and we provide numerical experiments that confirm its effectiveness and accuracy

periodicity induced generalized fourier transform pair the field radiated by an infinite periodic structure can be expressed in terms of floquet waves fws both in the frequency domain fd and time domain td see felsen l b and capolino f ieee trans antennas propagat vol 48 p 921 31 2000 a new periodicity induced generalized fourier transform ft pair is derived relating fd fws to td fws and vice versa based on tabulated transforms and physical conditions at infinity the new fts are directly related to the simple canonical problem of a line array of sequentially excited dipoles that is a basic building block for more general phased periodic structures

fitting parametrized polynomials with scattered surface data currently used joint surface models require the measurements to be structured according to a grid with the currently available tracking devices a large quantity of unstructured surface points can be measured in a relatively short time in this paper a method is presented to fit polynomial functions to three dimensional unstructured data points to test the method spherical cylindrical parabolic hyperbolic exponential logarithmic and sellar surfaces with different undulations were used the resulting polynomials were compared with the original shapes the results show that even complex joint surfaces can be modelled with polynomial functions in addition the influence of noise and the number of data points was also analyzed from a surface diam 20 mm which is measured with a precision of 0 2 mm a model can be constructed with a precision of 0 02 mm

gip integrators for matrix riccati differential equations matrix riccati differential equations mrdes are initial value problems of the form x a 2 1 xa 1 1 a 2 2x xa 1 2x x 0 x 0 these equations arise frequently throughout applied mathematics science and engineering it can happen that even when the a i j are smooth functions of t or constant the solution x may have a singularity or even infinitely many singularities this paper shows several classes of numerical algorithms which we call gip integrators that can solve for x past its singularities furthermore none of the algorithms require knowledge of the placement or even existence of singularities in x also it is shown how embedded runge kutta methods can be used to construct gip integrators to not only approximate x past singularities but also provide for error estimation to allow efficient time stepping finally several examples are shown to validate the theory

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

a trust region method for parabolic boundary control problems in this paper we develop a trust region algorithm for constrained parabolic boundary control problems for the computation of a trust region step we propose an iterative scheme which is a projected form of the steihaug trust region conjugate gradient method to ensure the good local convergence properties in the terminal phase a smoothing step at each iteration is added this step and the projection require the modification of the standard trust region algorithm and its convergence proof the algorithm has sup norm convergence in the terminal phase and l2 convergence in the global phase the results are illustrated for a parabolic boundary control problem

software and algorithms for sensitivity analysis of large scale differential algebraic systems sensitivity analysis for dae systems is important in many engineering and scientific applications the information contained in the sensitivity trajectories is useful for parameter estimation optimization model reduction and experimental design in this paper we present algorithms and software for sensitivity analysis of large scale dae systems of index up to two the new software provides for consistent initialization of the solutions and the sensitivities interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations and is capable via mpi of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations

an accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space te case we develop a method for the analytical evaluation of the inverse laplace transform representations for transient transverse magnetic tm plane wave obliquely incident on a conductive half space we assume that the permittivity and conductivity of the dispersive half space are independent of frequency the time domain expressions for the reflected and transmitted waves are first represented as inverse laplace transforms the transient fields are then shown to consist of two canonical integrals the canonical integrals in turn are solved analytically thereby yielding close form solutions involving incomplete lipschitz hankel integrals ilhis the ilhis are computed numerically using efficient convergent and asymptotic series expansions thus enabling the efficient computation of the transient fields the exact closed form expressions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast fourier transform fft algorithms an asymptotic series representation for the ilhis is also employed to obtain a relatively simple late time approximation for the transient fields this approximate late time expression is shown to accurately model the fields over a large portion of its time history

an adaptive moving mesh method for thin film flow equations with surface tension we present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension we follow the r adaptive moving mesh technique which utilises a mesh density function and moving mesh partial differential equations mmpdes to adapt and move the mesh coupled to the pde s describing the thin film flow problem numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method this technique accurately resolves the multiple one dimensional structures observed in these test problems moreover it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh we are the first to implement r adaptive schemes to higher order parabolic pdes this method accurately resolves the solution and reduces the computational effort we introduce a new mesh density function that resolves multiple solution structures this mesh density function is also adapted to include multiple solution components

using krylov methods in the solution of large scale differential algebraic systems in this paper a new algorithm for the solution of large scale systems of differential algebraic equations is described it is based on the integration methods in the solver dassl but instead of a direct method for the associated linear systems which arise at each time step we apply the preconditioned gmres iteration in combination with an inexact newton method the algorithm along with those in dassl is implemented in a new solver called daspk we outline the algorithms and strategies used and discuss the use of the solver we develop and analyze some preconditioners for a certain class of dae stems and finally demonstrate the application of daspk on two example problems

complex time representation of the sommerfeld type integrals and its application to 1 d periodic structures the concept of complex time is employed to derive the time domain representation of the green s function of a periodic structure for the first time the spatial domain green s function of the periodic structure in the frequency domain is separated into singular and nonsingular terms the nonsingular terms are approximated by using exponential functions resulting in the well known complex time representation of the time domain green s function in order to find a closed form expression for the singular terms a novel time domain representation of the sommerfeld type integrals is derived the proposed procedure is applied to the computation of the time domain green s function of a one dimensional periodic structure numerical experiments demonstrate the high accuracy of this method

a sequential homotopy method for mathematical programming problems we propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract hilbert spaces under the guignard constraint qualification the method is equivalent to performing projected backward euler timestepping on a projected gradient antigradient flow of the augmented lagrangian the projected backward euler equations can be interpreted as the necessary optimality conditions of a primal dual proximal regularization of the original problem the regularized problems are always feasible satisfy a strong constraint qualification guaranteeing uniqueness of lagrange multipliers yield unique primal solutions provided that the stepsize is sufficiently small and can be solved by a continuation in the stepsize we show that equilibria of the projected gradient antigradient flow and critical points of the optimization problem are identical provide sufficient conditions for the existence of global flow solutions and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient antigradient flow practically eradicating convergence to saddle points and maxima the sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework we demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth newton approach for the regularized subproblems

analysis of the rayleigh pulse seismologists make frequent use of wavelets also referred to as signals signatures or pulses particularly in such fields as seismic filtering wavelet processing wave propagation modeling and trace inversion whenever possible the actual seismic wavelet of the real source should be considered hosken 1988 however frequently particularly in wave propagation modeling one must consider a synthetic source wavelet this should if possible be given by a simple mathematical formula and possess an easy description for its most important spectral properties e g amplitude and phase spectrum main frequency hilbert transform etc moreover the mathematical expression should be such that with a minimum number of parameters a large flexibility in the form of the wavelet can be obtained

convergence to second order stationary points in inequality constrained optimization we propose a new algorithm for the nonlinear inequality constrained minimization problem and prove that it generates a sequence converging to points satisfying the kkt second order necessary conditions for optimality the algorithm is a line search algorithm using directions of negative curvature and it can be viewed as a nontrivial extension of corresponding known techniques from unconstrained to constrained problems the main tools employed in the definition and in the analysis of the algorithm are a differentiable exact penalty function and results from the theory of lc1 functions

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

fast interior point solution of quadratic programming problems arising from pde constrained optimization interior point methods provide an attractive class of approaches for solving linear quadratic and nonlinear programming problems due to their excellent efficiency and wide applicability in this paper we consider pde constrained optimization problems with bound constraints on the state and control variables and their representation on the discrete level as quadratic programming problems to tackle complex problems and achieve high accuracy in the solution one is required to solve matrix systems of huge scale resulting from newton iteration and hence fast and robust methods for these systems are required we present preconditioned iterative techniques for solving a number of these problems using krylov subspace methods considering in what circumstances one may predict rapid convergence of the solvers in theory as well as the solutions observed from practical computations

preconditioned implicit solution of linear hyperbolic equations with adaptivity this paper describes a method for solving hyperbolic partial differential equations using an adaptive grid the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves the solution is advanced in time via a backward differentiation formula the discretisation used is second order accurate and stable on cartesian grids the resulting system of linear equations is solved by gmres at every time step with the convergence of the iteration being accelerated by a semi toeplitz preconditioner the efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

an adaptive finite element method for parabolic differential systems some algorithmic considerations in solving in three space dimensions an adaptive h refinement method is described for solving systems of parabolic partial differential equations in three space dimensions on hexahedral grids these grids typically have irregular hanging nodes solutions are calculated using galerkin s method with a piecewise trilinear basis in space and a backward difference formula bdf code in time new a posteriori error indicators based on interpolation error estimates for irregular grids are used to control refinement and coarsening a more efficient algorithm for assembling banded portions of the jacobian is introduced a simple strategy for dealing with storage limitations by limiting the level of refinement is developed computational results demonstrate the effectiveness of the adaptive method on linear and nonlinear problems

partitioned variable metric updates for large structured optimization problems this paper presents a minimization method based on the idea of partitioned updating of the hessian matrix in the case where the objective function can be decomposed in a sum of convex element functions this situation occurs in a large class of practical problems including nonlinear finite elements calculations some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented

a parallelizable preconditioner for the iterative solution of implicit runge kutta type methods the main diculty in the implementation of most standard implicit runge kutta irk methods applied to sti ordinary dierential equations odes is to eciently solve the nonlinear system of equations in this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit euler method the preconditioner is based on the w transformation of the rk coecient matrices discovered by hairer and wanner for sti odes the preconditioner is by construction asymptotically exact for methods with an invertible rk coecient matrix the methodology is particularly useful when applied to super partitioned additive runge kutta spark methods the nonlinear system can be solved by inexact simplied newton iterations at each simplied newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system c 1999 published by elsevier science b v all rights reserved

mesh independence of matrix free methods for path following in this paper we consider a matrix free path following algorithm for nonlinear parameter dependent compact fixed point problems we show that if these problems are discretized so that certain collective compactness and strong convergence properties hold then this algorithm can follow smooth folds and capture simple bifurcations in a mesh independent way

a direct active set algorithm for large sparse quadratic programs with simple bounds we show how a direct active set method for solving definite and indefinite quadratic programs with simple bounds can be efficiently implemented for large sparse problems all of the necessary factorizations can be carried out in a static data structure that is set up before the numeric computation begins the space required for these factorizations is no larger than that required for a single sparse cholesky factorization of a matrix with the same sparsity structure as the hessian of the quadratic we propose several improvements to this basic algorithm a new way to find a search direction in the indefinite case that allows us to free more than one variable at a time and a new heuristic method for finding a starting point these ideas are motivated by the two norm trust region problem additionally we also show how projection techniques can be used to add several constraints to the active set at each iteration our experimental results show that an algorithm with these improvements runs much faster than the basic algorithm for positive definite problems and finds local minima with lower function values for indefinite problems

frequency and time domain green s functions for a phased semi infinite periodic line array of dipoles we extend a previous prototype study of felsen and capolino see ibid vol 48 p 921 931 june 2000 of frequency domain fd and time domain td green s functions for an infinite periodic phased line array of dipoles to account for the effects of truncation as modeled by a semi infinite array these canonical problems are to be used eventually for the systematic analysis and synthesis of actual rectangular td plane phased arrays in the presentation we rely on the analytic results and phenomenologies pertaining to the infinite array which are reviewed major emphasis is then placed on the modifications introduced by the truncation finite poisson summation is used to convert the individual dipole radiations into collective truncated wavefields the fd and td floquet waves fw in the td exact closed form solutions are obtained and are examined asymptotically to extract fd and td periodicity matched conical truncated fw fields both propagating and nonpropagating corresponding tip diffracted periodicity matched spherical waves and uniform transition functions connecting both across the fd and td fw truncation boundaries these new effects can again be incorporated in a fw modulated geometrical theory of diffraction a numerical example of radiation from a finite phased td dipole array with band limited excitation demonstrates the accuracy and efficiency of the fw diffracted field asymptotic algorithm when compared with an element by element summation reference solution

global convergence of trust region interior point algorithms for infinite dimensional nonconvex minimization subject to pointwise bounds a class of interior point trust region algorithms for infinite dimensional nonlinear optimization subject to pointwise bounds in l p banach spaces 2 le p le infty is formulated and analyzed the problem formulation is motivated by optimal control problems with l p controls and pointwise control constraints the interior point trust region algorithms are generalizations of those recently introduced by coleman and li siam j optim 6 1996 pp 418 445 for finite dimensional problems many of the generalizations derived in this paper are also important in the finite dimensional context all first and second order global convergence results known for trust region methods in the finite dimensional setting are extended to the infinite dimensional framework of this paper

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables

robust solution of richards equation for nonuniform porous media capillary pressure saturation relative permeability relations described using the van genuchten 1980 and mualem 1976 models for nonuniform porous media lead to numerical convergence difficulties when used with richards equation for certain auxiliary conditions these difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure convergence difficulties are illustrated using standard numerical approaches to simulate such problems we investigate constitutive relations interblock permeability nonlinear algebraic system approximation methods and two time integration approaches an integral permeability approach approximated by hermite polynomials is recommended and shown to be robust and economical for a set of test problems which correspond to sand loam and clay loam media

a limited memory quasi newton trust region method for box constrained optimization by means of wolfe conditions strategy we propose a quasi newton trust region method to solve box constrained optimization problems this method is an adequate combination of the compact limited memory bfgs and the trust region direction while the generated point satisfies the wolfe conditions and therefore maintains a positive definite approximation to the hessian of the objective function the global convergence and the quadratic convergence of this method are established under suitable conditions finally we compare our algorithms iwtral and ibwtral with an active set trust region algorithm astral xu and burke 2007 on the cutest box constrained test problems presented by gould et al 2015 numerical results show that the presented method is competitive and totally interesting for solving box constrained optimization