generalized mixed variational principles and solutions of ill conditioned problems in computational mechanics part ii shear locking abstract although the finite element method fem has been extensively applied to various areas of engineering the ill conditioned problems occurring in many situations are still thorny to deal with this study attempts to provide a high performing and simple approach to the solutions of ill conditioned problems the theoretical foundation of it is the parametrized variational principles called the generalized mixed variational principles gmvps initiated by rong in 1981 gmvps can solve many kinds of ill conditioned problems in computational mechanics among them four cases are investigated in detail the volumetric locking the shear locking the inhomogeneousness and the membrane locking problems composing four parts of the study part i part iv respectively this paper is part ii wherein a gmvp specially suited to the reissner plate theory and timoshenko beam theory is constructed providing a mathematical foundation for establishing fem formulations which can automatically unlock the shear locking and produce no spurious zero energy modes

proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

an isogeometric reissner mindlin shell element based on bezier dual basis functions overcoming locking and improved coarse mesh accuracy abstract we develop a mixed geometrically nonlinear isogeometric reissner mindlin shell element for the analysis of thin walled structures that leverages bezier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses the accuracy of computed solutions over coarse meshes that have highly non interpolatory control meshes is achieved through the application of a continuous rotational approach the starting point of the formulation is the modified hellinger reissner variational principle with independent displacement membrane and shear strains as the unknown fields to overcome locking the strain variables are interpolated with lower order spline bases while the variations of the strain variables are interpolated with the corresponding bezier dual bases leveraging the orthogonality property of the bezier dual basis the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system the condensed approach preserves the accuracy of the non condensed mixed approach but with fewer degrees of freedom from a practical point of view since the bezier dual basis is completely specified through bezier extraction any spline space that admits bezier extraction can utilize the proposed approach directly

multiplicity results for a class of gradient systems depending on two parameters abstract in this paper we prove the existence of at least three solutions for a class of two point boundary value systems of the type u i u i f u i x u 1 u n g u i x u 1 u n u i a u i b 0 for 1 i n the approach is fully based on a recent three critical points theorem of ricceri b ricceri a three critical points theorem revisited nonlinear anal 70 9 2009 3084 3089

on stress interpolation for hybrid models a technique is proposed for the selection of stress interpolations for hybrid models the present paper applies this approach to plane problems the stiffness matrix is derived using the hellinger reissner variational principle this formulation uses infinitesimal equilibrium relationships and divides the assumed stress into its lower order and higher order parts the patch test can be passed and the resulting elements are generally invariant a plane four node quadrilateral element is described and compared with existing elements numerical studies show that the accuracy of the element is generally good

the development of hybrid axisymmetric elements based on the hellinger reissner variational principle we present a general procedure for the development of hybrid axisymmetric elements based on the hellinger reissner principle within the context of linear elasticity similar to planar elements the stress interpolation is obtained by an identification of the zero energy modes we illustrate our procedure by designing a lower order four node and a higher order nine node element both elements are of correct rank and moreover use the minimum number of stress parameters namely seven and 17 several examples are presented to show the excellent performance of both elements under various demanding situations such as when the material is almost incompressible when the thickness to radius ratio is very small etc when the variation of the field variables is along the radial direction alone when the mesh is uniform and the loading is of pressure type the developed elements are superconvergent i e they yield the exact nodal displacement values

blind motion deblurring using multiple images recovery of degraded images due to motion blurring is a challenging problem in digital imaging most existing techniques on blind deblurring are not capable of removing complex motion blurring from the blurred images of complex structures one promising approach is to recover the clear image using multiple images captured for the scene however in practice it is observed that such a multi frame approach can recover a high quality clear image of the scene only after multiple blurred image frames are accurately aligned during pre processing which is a very challenging task even with user interactions in this paper by exploring the sparsity of the motion blur kernel and the clear image under certain domains we propose an alternative iteration approach to simultaneously identify the blur kernels of given blurred images and restore a clear image our proposed approach is not only robust to image formation noises but is also robust to the alignment errors among multiple images a modified version of linearized bregman iteration is then developed to efficiently solve the resulting minimization problem experiments show that our proposed algorithm is capable of accurately estimating the blur kernels of complex camera motions with minimal requirements on the accuracy of image alignment as a result our method is capable of automatically recovering a high quality clear image from multiple blurred images

linearized bregman iterations for compressed sensing finding a solution of a linear equation au f with various minimization properties arises from many applications one such application is compressed sensing where an efficient and robust to noise algorithm to find a minimal l 1 norm solution is needed this means that the algorithm should be tailored for large scale and completely dense matrices a while au and a t u can be computed by fast transforms and the solution we seek is sparse recently a simple and fast algorithm based on linearized bregman iteration was proposed in 28 32 for this purpose this paper is to analyze the convergence of linearized bregman iterations and the minimization properties of their limit based on our analysis here we derive also a new algorithm that is proven to be convergent with a rate furthermore the new algorithm is simple and fast in approximating a minimal l 1 norm solution of au f as shown by numerical simulations hence it can be used as another choice of an efficient tool in compressed sensing

higher order shell elements based on a cosserat model and their use in the topology design of structures for the purpose of carrying out topology optimization of shell structures we show that it is advantageous to use shell elements based on a classical shell theory such as the cosserat theory rather than elements based on the degenerated solid approach since the shell thickness appears explicitly in the formulation thereby greatly simplifying the sensitivity analysis one of the well known shell elements based on the cosserat shell theory is the four node element presented by simo et al comput methods appl mech engrg 72 1989 267 73 1989 53 although one could use this element the use of lower order elements often results in instabilities such as the checkerboard instability in the resulting topologies this provides the motivation for developing higher order shell elements based on a classical shell theory in this work we present the formulation and implementation details for six node and seven node triangular and nine node quadrilateral shell elements which are based on the variational formulation of simo et al and show that good accuracy is obtained even with coarse meshes in fairly demanding problems we also present the topology optimization formulation and some examples of optimal topologies that are obtained using this formulation

a minimax inequality and its applications to ordinary differential equations the aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of b ricceri below equivalent formulations are shown and characterization is proved in particular for a special class of functionals as an application a multiplicity result for an ordinary dirichlet problem is emphasized

extensions of the mountain pass theorem on considere une fonctionnelle c 1 i x r ou x est un espace de banach reel et ou i satisfait la condition de compacite de palais smale on presente une serie de theoremes qui generalisent le theoreme du col

a new efficient approach to the formulation of mixed finite element models for structural analysis abstract a new finite element formulation is developed based on the hellinger reissner principle with independent strain by dividing the assumed strain into the lower order part and the higher order part the new formulation can be made much more efficient than the standard mixed formulation in addition the present new approach provides a rational way of introducing stabilization matrix to suppress undesirable kinematic modes author

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

split bregman iteration algorithm for total bounded variation regularization based image deblurring many existing algorithms taking the seminorm in bv for regularization have achieved great success in image processing however this paper considers the total bounded variation regularization based approach to perform image deblurring based on this novel model we introduce an extended split bregman iteration to obtain the optimum solution quickly we also provide the rigorous convergence analysis of the iterative algorithm here compared with the results of the rof method numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm

multiplicity theorems for the dirichlet problem involving the p laplacian abstract in this paper we establish some results on the existence of at least three weak solutions for dirichlet problems involving the p laplacian by a variational approach

multiple solutions for discrete boundary value problems a recent multiplicity theorem for the critical points of a functional defined on a finite dimensional hilbert space established by ricceri is extended an application to dirichlet boundary value problems for difference equations involving the discrete p laplacian operator is presented

a chaotic iterative algorithm based on linearized bregman iteration for image deblurring in this paper we propose a new chaotic iterative algorithm for image deblurring the algorithm is based on generalized inverse iteration and linearized bregman iteration for the basis pursuit problem min u r n u 1 au f numerical experiments show that the chaotic algorithm for image restoration is effective and efficient

a new efficient mixed formulation for thin shell finite element models abstract a nine node shell element is developed by a new and more efficient mixed formulation the new shell element formulation is based on the hellinger reissner principle with independent strain and the concept of degenerate solid shell the new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower part and a higher order part numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable in fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix

elimination of spurious pressure and kinematic modes in biquadratic nine node plane element in two dimensional penalty finite element analysis of incompressible materials q9 3p the 9 node quadrilateral with three assumed quasi pressure modes is the most popular element not bothered by spurious pressure in constructing the penalty matrix of q9 3p it is necessary to form a 3 18 matrix and a 3 3 symmetric matrix the inverse of the symmetric matrix is then post and pre multiplied by the 3 18 matrix and its transpose respectively by employing a rank subtraction technique a new and more efficient implementation scheme is devised for the penalty matrix same as the conventional q9 element when q9 3p is fully integrated it becomes expensive and too stiff on the other hand there are two spurious kinematic modes should the element be sub integrated in the proposed q9 3p element the two mechanisms will be annihilated by judiciously chosen higher order assumed stress modes in conjunction with a modified hellinger reissner functional it will be demonstrated that the element is of good accuracy and high efficiency

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

a finite element formulation with stabilization matrix for geometrically non linear shells an assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non linear problems of isotropic and laminated composite shells the present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement the assumed independent strain field is divided into a lower order part and a higher order part subsequently the lower order part is set equal to the displacement dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix the strains and the determinant of the jacobian matrix are assumed to vary linearly in the thickness direction this assumption allows analytical integration through thickness independent of the number of plies a nine node element with a judiciously chosen set of higher order assumed strain field is developed numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation

two higher order shell finite elements with stabilization matrix a 32 node three dimensional solid element and a 16 node degenerate solid element that are applicable to analysis of thin shell structural problems are presented in this paper both elements are formulated by using the hellinger reissner principle with independent strain the assumed independent strain field is divided into higherand lower order polynomial terms the element stiffness matrix associated with the higher order assumed strain plays the role of the stabilization matrix various types of the higher order assumed strain are tested for the present elements the resulting finite element models require much less time to compute the element stiffness matrix than the corresponding conventional mixed models or assumed displacement models with a full gaussian quadrature rule numerical results demonstrate that with a properly chosen set of higher order assumed strain the present elements produce reliable and very accurate solutions even for very thin plates and thin shells with deeply curved geometry

sparse canonical correlation analysis new formulation and algorithm in this paper we study canonical correlation analysis cca which is a powerful tool in multivariate data analysis for finding the correlation between two sets of multidimensional variables the main contributions of the paper are 1 to reveal the equivalent relationship between a recursive formula and a trace formula for the multiple cca problem 2 to obtain the explicit characterization for all solutions of the multiple cca problem even when the corresponding covariance matrices are singular 3 to develop a new sparse cca algorithm and 4 to establish the equivalent relationship between the uncorrelated linear discriminant analysis and the cca problem we test several simulated and real world datasets in gene classification and cross language document retrieval to demonstrate the effectiveness of the proposed algorithm the performance of the proposed method is competitive with the state of the art sparse cca algorithms

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

an adaptive algorithm for tv based model of three norms lq q 12 1 2 in image restoration abstract in this paper we present an adaptive method for the tv based model of three norms l q q 1 2 1 2 for the image restoration problem the algorithm with the l2 norm is used in the smooth regions where the value of u is small the algorithm with the l 1 2 norm is applied for the jumps where the value of u is large when the value of u is moderate the algorithm with the l1 norm is employed thus the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted the numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images specifically in contrast to the l1 norm as in the rudin osher fatemi model the l2 norm yields better results in the smooth and flat regions and the l 1 2 norm is more suitable in regions with strong discontinuities therefore our adaptive algorithm is efficient and robust even for images with large noises

an iterative algorithm for signal reconstruction from bispectrum the author presents a procedure to reconstruct a minimum or nonminimum phase discrete time signal from its bispectrum the algorithm is iterative and based on using the method of projections onto convex sets pocs in an inner product space in which the vector space consists of real sequences and vector addition is defined in terms of the convolution operation prior information such as an energy bound on the signal can be incorporated into this algorithm an algorithm that employs energy information was found to produce better results in simulation than one that ignores energy constraints

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

an explicit hybrid stabilized 9 node lagrangian shell element abstract this paper presents an explicit hybrid stabilized 9 node lagrangian shell element in the formulation the stress modes pertinent to the stabilization are contravariant in nature and their products with the displacement derived covariant strain are explicit linear functions of the nodal geometric parameters the element is particularly efficient because the leverage matrix or the stabilization vectors can be constructed without resorting to numerical integration or the gram schmidt orthogonalization accuracy of the element is promising and comparable to the state of the art stabilized and hybrid elements

an improved algorithm for basis pursuit problem and its applications abstract we propose an algorithm for solving the basis pursuit problem min u c n u 1 a u f our starting motivation is the algorithm for compressed sensing proposed by qiao li and wu which is based on linearized bregman iteration with generalized inverse qiao li and wu defined new algorithm for solving the basis pursuit problem in compressive sensing using a linearized bregman iteration and the iterative formula of linear convergence for computing the matrix generalized inverse in our proposed approach we combine a partial application of the newton s second order iterative scheme for computing the generalized inverse with the bregman iteration our scheme takes lesser computational time and gives more accurate results in most cases the effectiveness of the proposed scheme is illustrated in two applications signal recovery from noisy data and image deblurring

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

the augmented lagrangian method for equality and inequality constraints in hilbert spaces in this paper we consider an augmented lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a hilbert space an inequality constraint whose image space is finite dimensional and an affine inequality constraint whose image space is in an infinite dimensional hilbert space we obtain local convergence of this method without imposing strict complementarity conditions when the equality as well as the inequality constraint with finite dimensional image space are augmented to the author s knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

a boundary value problem for fourth order elastic beam equations multiplicity results for a fourth order nonlinear boundary value problem are presented the proof of the main result is based on the critical point theory

projected landweber iteration for matrix completion recovering an unknown low rank or approximately low rank matrix from a sampling set of its entries is known as the matrix completion problem in this paper a nonlinear constrained quadratic program problem concerning the matrix completion is obtained a new algorithm named the projected landweber iteration plw is proposed and the convergence is proved strictly numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of the unknown low rank matrix

multiscale methods for polyhedral regularizations in this paper we present the extension and generalization of the adaptive inverse scale space aiss method proposed for ell 1 regularization in math comp 82 2013 pp 269 299 to arbitrary polyhedral functions we will see that the representation of a convex polyhedral function as a finitely generated function allows us to conclude that the inverse scale space flow the time continuous formulation of the augmented lagrangian method can be solved exactly without discretization we present an algorithm for the computation of such a solution for arbitrary polyhedral functions analyze its convergence and interpret the well known forward scale space flow as the inverse scale space flow on the convex conjugate functional thus including this class of flows in our analysis a surprising result is the equivalence of the scale space or gradient flow with a standard variational problem finally we give examples of the applications for the aiss algorithm for polyhedral functions which illustrate th

existence of three solutions for a non homogeneous neumann problem through orlicz sobolev spaces abstract the aim of this paper is to establish a multiplicity result for an eigenvalue non homogeneous neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition precisely a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate orlicz sobolev space

a dual split bregman method for fast l minimization in this paper we propose a new algorithm for fast l1 minimization as frequently arising in compressed sensing our method is based on a split bregman algorithm applied to the dual of the problem of minimizing u 1 1 2 u 2 such that u solves the under determined linear system au f which was recently investigated in the context of linearized bregman methods furthermore we provide a convergence analysis for split bregman methods in general and show with our compressed sensing example that a split bregman approach to the primal energy can lead to a different type of convergence than split bregman applied to the dual thus making the analysis of different ways to minimize the same energy interesting for a wide variety of optimization problems

control of spurious modes in the nine node quadrilateral element the nine node quadrilateral is much more accurate than its eight node relative if elements are non rectangular however the nine node element may display spurious low energy modes under 2 2 gauss quadrature these modes can be controlled by using the 3 3 quadrature rule but at added cost this paper investigates two inexpensive control devices that can be used with 2 2 gauss quadrature

multiplicity of solutions of a two point boundary value problem for a fourth order equation we study the existence of multiple solutions for semi linear fourth order differential equation describing elastic deflections the proof of the main result is based on a three critical point theorem

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

regularization preconditioners for frame based image deblurring with reduced boundary artifacts thresholding iterative methods were recently successfully applied to image deblurring problems in this paper we investigate the modified linearized bregman algorithm mlba used in image deblurring problems with a proper treatment of the boundary artifacts we consider two standard approaches the imposition of boundary conditions and the use of the rectangular blurring matrix the fast convergence of the mlba depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block bccb matrix diagonalized by the discrete fourier transform we show that the standard approach based on the bccb preconditioner may provide low quality restored images and we propose different preconditioning strategies that improve the quality of the restoration and save some computational cost at the same time motivated by a recent nonstationary preconditioned iteration we propose a new algorithm that combines such a method with the mlba we p

mixed models and reduced selective integration displacement models for nonlinear analysis of curved beams simple mixed models are developed for the geometrically nonlinear analysis of deep arches a total lagrangian description of the arch deformation is used and the analytical formulation is based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included the fundamental unknowns consist of the six internal forces and generalized displacements of the arch and the element characteristic arrays are obtained by using hellinger reissner mixed variational principle the polynomial interpolation functions used in approximating the forces are one degree lower than those used for approximating the displacements and the forces are discontinuous at the interelement boundaries the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms is discussed the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and compare their performance with other mixed models reported in the literature