the riemann boundary value problem on closed riemann surfaces and integrable systems the riemann problem has different forms in several areas of physics scattering theory integrable systems and so on it is necessary to build an analytic function or a matrix with prescribed poles and zeros on a riemann surface another similar problem is to construct a section of a complex vector bundle over a riemann surface for example anti self dual yang mills equation as was established by a grothendieck and h rohrl this problem is equivalent to the riemann boundary value problem let m be a compact riemann surface a contour on m g p an n n matrix on determine an analytic matrix f p in m which is multiple of a given divisor and satisfies the boundary condition on f p g p f p this problem is well understood on the complex plane it is related to singular integral operators wiener hopf equations and banach algebras in the case of a riemann surface there are additional with the riemann roch and abel theorems jacobi varieties and riemann theta functions these relations are described here the riemann problem is equivalent to the so called problem u au recently this elliptic equation was used to study the inverse scattering problem finally we consider some physical problems finite gap potentials the landau lifschitz equation the painleve problems to illustrate the applications of the riemann problem on a riemann surface acquaintance with riemann surface theory and algebraic topology is not assumed

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

crack tip enrichment in the xfem using a cutoff function we consider a variant of the extended finite element method xfem in which a cutoff function is used to localize the singular enrichment surface the goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical xfem with a fixed enrichment area we give a mathematical result of quasi optimal error estimate one of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments finally we present some numerical computations validating the theoretical result these computations are compared with those of the classical xfem and a non enriched method copyright 2008 john wiley sons ltd

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

on the generic properties of convex optimization problems in conic form we prove that strict complementarity primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties in this paper we say generic to mean that the set of data possessing the desired property or properties has strictly larger hausdorff dimension than the set of data that does not our proof is elementary and it employs an important result due to larman 7 on the boundary structure of convex bodies

on optimality conditions for structured families of nonlinear programming problems optimality conditions for families of nonlinear programming problems inrn are studied from a generic point of view the objective function and some of the constraints are assumed to depend on a parameter while others are held fixed techniques of differential topology are used to show that under suitable conditions certain strong second order conditions are necessary for optimality except possibly for parameter values lying in a negligible set

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

identifying structure of nonsmooth convex functions by the bundle technique we consider the problem of minimizing nonsmooth convex functions defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite hessian on the set of interest this is a particular case of functions with primal dual gradient structure a notion closely related to the so called cal vu space decomposition at a given point nonsmoothness is locally restricted to the directions of the subspace cal v while along the subspace cal u the behavior of the function is twice differentiable constructive identification of the two subspaces is important because it opens the way to devising fast algorithms for nonsmooth optimization by following iteratively the manifold of smoothness on which superlinear cal u newton steps can be computed in this work we show that for the class of functions in consideration the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points provided a minimizer satisfies the nondegeneracy and strong transversality conditions

a stable and optimally convergent generalized fem sgfem for linear elastic fracture mechanics abstract in this paper we investigate the accuracy and conditioning of the stable generalized fem sgfem and compare it with standard generalized fem gfem for a 2 d fracture mechanics problem the sgfem involves localized modifications of enrichments used in the gfem and the conditioning of the stiffness matrix in this method is of the same order as in the fem numerical experiments show that using the sgfem with only the modified heaviside functions which are used as enrichments in the gfem to approximate the solution of fracture problems in 2 d gives inaccurate results however the sgfem using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the sgfem are also presented this set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the gfem we show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2 d the selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity a simple and yet generic implementation strategy for the sgfem in an existing gfem xfem software is described the implementation can be used with 2 d and 3 d elements it leads to an efficient evaluation of sgfem enrichment functions

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

genericity results in linear conic programming a tour d horizon this paper is concerned with so called generic properties of general linear conic programs many results have been obtained on this subject during the last two decades for example it is known that uniqueness strict complementarity and nondegeneracy of optimal solutions hold for almost all problem instances strong duality holds generically in a stronger sense i e it holds for a generic subset of problem instances in this paper we survey known results and present new ones in particular we give an easy proof of the fact that slater s condition holds generically in linear conic programming we further discuss the problem of stability of uniqueness nondegeneracy and strict complementarity we also comment on the fact that in general a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

sufficient optimality conditions hold for almost all nonlinear semidefinite programs we derive a new genericity result for nonlinear semidefinite programming nlsdp namely almost all linear perturbations of a given nlsdp are shown to be nondegenerate here nondegeneracy for nlsdp refers to the transversality constraint qualification strict complementarity and second order sufficient condition due to the presence of the second order sufficient condition our result is a nontrivial extension of the corresponding results for linear semidefinite programs sdp from alizadeh et al math program 77 2 ser b 111 128 1997 the proof of the genericity result makes use of forsgren s derivation of optimality conditions for nlsdp in forsgren math program 88 1 ser a 105 128 2000 due to the latter approach the positive semidefiniteness of a symmetric matrix g x depending continuously on x is locally equivalent to the fact that a certain schur complement s x of g x is positive semidefinite this yields a reduced nlsdp by considering the new semidefinite constraint s x succeq 0 s x 0 instead of g x succeq 0 g x 0 while deriving optimality conditions for the reduced nlsdp the well known and often mentioned h term in the second order sufficient condition vanishes this allows us to access the proof of the genericity result for nlsdp

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

mesh based construction of flat top partition of unity functions a novel idea to construct flat top partition of unity functions in a closed form on a general structured or unstructured finite element mesh is presented an efficient and practical construction method of a flat top partition of unity function is important in the generalized finite element method gfem details on how to construct flat top partition of unity functions on a provided mesh are given the generalized finite element approximation with the use of the new flat top partition of unity function is presented with various numerical examples that demonstrate the effectiveness of proposed flat top partition of unity functions

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

the structure of completely positive matrices according to their cp rank and cp plus rank we study the topological properties of the cp rank operator mathrm cp a and the related cp plus rank operator mathrm cp a which is introduced in this paper in the set mathcal s n of symmetric n times n matrices for the set of completely positive matrices mathcal cp n we show that for any fixed p the set of matrices a satisfying mathrm cp a mathrm cp a p is open in mathcal s n setminus mathrm bd mathcal cp n we also prove that the set mathcal a n of matrices with mathrm cp a mathrm cp a is dense in mathcal s n by applying the theory of semi algebraic sets we are able to show that membership in mathcal a n is even a generic property we furthermore answer several questions on the existence of matrices satisfying mathrm cp a mathrm cp a or mathrm cp a neq mathrm cp a and establish genericity of having infinitely many minimal cp decompositions

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

linear dependence problems of partition of unity based generalized fems a known problem of partition of unity based generalized finite element methods referred to as gfem is the linear dependence problem which leads to singular global stiffness matrices thus far attempts to eliminate the linear dependence problem have been unsuccessful numerical experiments are carried out among several gfems to investigate the problem based on the numerical experiments simple but effective approaches to eliminating the linear dependence problem are suggested

a robust preconditioning technique for the extended finite element method the extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions but the resulting stiffness matrices can become ill conditioned in that case iterative solvers need a large number of iterations to obtain an acceptable solution in this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments a domain decomposition is employed and the algorithm is very well suited for parallel computations the method was tested in numerical experiments to show its effectiveness the experiments have been conducted for structures containing cracks and material interfaces we show that the corresponding enrichments can result in arbitrarily ill conditioned matrices the method proposed here however provides well conditioned matrices and can be applied to any sort of enrichment the complexity of this approach and its relation to the domain decomposition is discussed computation times have been measured for a structure containing multiple cracks for this structure the computation times could be decreased by a factor of 2

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

a note on alternating projections for ill posed semidefinite feasibility problems we observe that sturm s error bounds readily imply that for semidefinite feasibility problems the method of alternating projections converges at a rate of mathcal o big k frac 1 2 d 1 2 big o k 12d 1 2 where d is the singularity degree of the problem the minimal number of facial reduction iterations needed to induce slater s condition consequently for almost all such problems in the sense of lebesgue measure alternating projections converge at a worst case rate of mathcal o big frac 1 sqrt k big o 1k

generic minimizing behavior in semialgebraic optimization we present a theorem of sard type for semialgebraic set valued mappings whose graphs have dimension no larger than that of their range space the inverse of such a mapping admits a single valued analytic localization around any pair in the graph for a generic value parameter this simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems typical semialgebraic problems have finitely many critical points around each of which they admit a unique active manifold analogue of an active set in nonlinear optimization moreover such critical points satisfy strict complementarity and second order sufficient conditions for optimality are indeed necessary

generic optimality conditions for semialgebraic convex programs we consider linear optimization over a nonempty convex semialgebraic feasible region f semidefinite programming is an example if f is compact then for almost every linear objective there is a unique optimal solution lying on a unique active manifold around which f is partly smooth and the second order sufficient conditions hold perturbing the objective results in smooth variation of the optimal solution the active manifold consists locally of these perturbed optimal solutions it is independent of the representation of f and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes these results extend to unbounded sets f

the dimension of semialgebraic subdifferential graphs a b s t r a c t a corollary of a celebrated theorem of minty is that the subdifferential graph of a closed convex function on r n has uniform local dimension n in contrast there exist nonconvex closed functions whose subdifferentials have large graphs we consider how far minty s corollary extends to functions that are nonconvex but semi algebraic

a well conditioned and optimally convergent xfem for 3d linear elastic fracture summary a variation of the extended finite element method for three dimensional fracture mechanics is proposed it utilizes a novel form of enrichment and point wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two dimensional and three dimensional crack problems a bespoke benchmark problem is introduced to determine the method s accuracy in the general three dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40 for moderately refined meshes and topological enrichment moreover when a fixed enrichment volume is used the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method diminishing the number of iterations by almost one order of magnitude copyright 2015 john wiley sons ltd

prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes in the partition of unity pu based methods the global approximation is built by multiplying a partition of unity by local approximations within this framework high order approximations are achieved by directly adopting high order polynomials as local approximations and therefore nodes along sides or inside elements which are usually adopted in the conventional finite element methods are no more required however the pu based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom in this paper the origin of the rank deficiency in the pu based approximation space is first dissected at an element level and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency finally examples are investigated to validate the present approach the current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix it can be resolved in its own manner

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

a unified enrichment approach addressing blending and conditioning issues in enriched finite elements abstract we present a combination of techniques to improve the convergence and conditioning properties of partition of unity pu enriched finite element methods by applying these techniques to different types of enrichment functions namely polynomial discontinuous and singular higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

a frank wolfe type theorem for nondegenerate polynomial programs in this paper we study the existence of optimal solutions to a constrained polynomial optimization problem more precisely let f 0 f 0 and f 1 ldots f p mathbb r n rightarrow mathbb r f 1 f p r n r be convenient polynomial functions and let s x in mathbb r n f i x le 0 i 1 ldots p ne emptyset s x r n f i x 0 i 1 p under the assumption that the map f 0 f 1 ldots f p mathbb r n rightarrow mathbb r p 1 f 0 f 1 f p r n r p 1 is non degenerate at infinity we show that if f 0 f 0 is bounded from below on s s then f 0 f 0 attains its infimum on s s

3d corrected xfem approach and extension to finite deformation theory in this paper the modified or corrected extended finite element method originally presented in fries int j numer meth engng 2008 75 503 532 for the 2d case is extended to 3d including different remedies for the problem that the crack front enrichment functions are linearly dependent in the blending elements in the context of this extension we address a number of computational issues of the 3d xfem in particular possible quadrature rules for elements with discontinuities also the influence of finite deformation theory for crack simulations in comparison to linear elastic fracture mechanics is investigated a number of numerical examples demonstrate the behavior of the presented possibilities copyright 2010 john wiley sons ltd

semi algebraic functions have small subdifferentials we prove that the subdifferential of any semi algebraic extended real valued function on mathbf r n has n dimensional graph we discuss consequences for generic semi algebraic optimization problems

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

higher order stable generalized finite element method the generalized finite element method gfem is a galerkin method where the trial space is obtained by augmenting the trial space of the standard finite element method fem by non polynomial functions called enrichments that mimic the local behavior of the unknown solution of the underlying variational problem the gfem has excellent approximation properties but its conditioning could be much worse than that of the fem however if the enrichments satisfy certain properties then the conditioning of the gfem is not worse than that of the standard fem and the gfem is referred to as the stable gfem sgfem in this paper we address the higher order sgfem that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning yielding a robust implementation of the higher order sgfem

typical convex program is very well posed in this paper we consider the collection of convex programming problems with inequality and equality constraints in which every problem of the collection is obtained by linear perturbations of the cost function and right hand side perturbation of the constraints while the core cost function and the left hand side constraint functions are kept fixed the main result shows that the set of the problems which are not well posed is porous in a certain strong sense our results concern both the infinite and finite dimensional case in the last case the conclusions are significantly sharper

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm