extended variational quasicontinuum methodology for lattice networks with damage and crack propagation abstract lattice networks with dissipative interactions are often employed to analyse materials with discrete micro or meso structures or for a description of heterogeneous materials which can be modelled discretely they are however computationally prohibitive for engineering scale applications the variational quasicontinuum qc method is a concurrent multiscale approach that reduces their computational cost by fully resolving the dissipative lattice network in small regions of interest while coarsening elsewhere when applied to damageable lattices moving crack tips can be captured by adaptive mesh refinement schemes whereas fully resolved trails in crack wakes can be removed by mesh coarsening in order to address crack propagation efficiently and accurately we develop in this contribution the necessary generalizations of the variational qc methodology first a suitable definition of crack paths in discrete systems is introduced which allows for their geometrical representation in terms of the signed distance function second special function enrichments based on the partition of unity concept are adopted in order to capture kinematics in the wakes of crack tips third a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed finally as our standpoint is variational we discuss implications of the mesh refinement and coarsening from an energy consistency point of view all theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces energy evolutions and crack paths are compared to those of the direct numerical simulations

minimum density hyperplanes associating distinct groups of objects clusters with contiguous regions of high probability density high density clusters is central to many statistical and machine learning approaches to the classification of unlabelled data we propose a novel hyperplane classifier for clustering and semi supervised classification which is motivated by this objective the proposed minimum density hyperplane minimises the integral of the empirical probability density function along it thereby avoiding intersection with high density clusters we show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent thus linking this approach to maximum margin clustering and semi supervised support vector classifiers we propose a projection pursuit formulation of the associated optimisation problem which allows us to nd minimum density hyperplanes efficiently in practice and evaluate its performance on a range of benchmark data sets the proposed approach is found to be very competitive with state of the art methods for clustering and semi supervised classification

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

spectral embedded clustering a framework for in sample and out of sample spectral clustering spectral clustering sc methods have been successfully applied to many real world applications the success of these sc methods is largely based on the manifold assumption namely that two nearby data points in the high density region of a low dimensional data manifold have the same cluster label however such an assumption might not always hold on high dimensional data when the data do not exhibit a clear low dimensional manifold structure e g high dimensional and sparse data the clustering performance of sc will be degraded and become even worse than k means clustering in this paper motivated by the observation that the true cluster assignment matrix for high dimensional data can be always embedded in a linear space spanned by the data we propose the spectral embedded clustering sec framework in which a linearity regularization is explicitly added into the objective function of sc methods more importantly the proposed sec framework can naturally deal with out of sample data we also present a new laplacian matrix constructed from a local regression of each pattern and incorporate it into our sec framework to capture both local and global discriminative information for clustering comprehensive experiments on eight real world high dimensional datasets demonstrate the effectiveness and advantages of our sec framework over existing sc methods and k means based clustering methods our sec framework significantly outperforms sc using the nystrom algorithm on unseen data

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

unsupervised learning by minimal entropy encoding following basic principles of information theoretic learning in this paper we propose a novel approach to data clustering referred to as minimal entropy encoding mee which is based on a set of functions features projecting each input onto a minimum entropy configuration code inspired by traditional parsimony principles we seek solutions in reproducing kernel hilbert spaces and then we prove that the encoding functions are expressed in terms of kernel expansion in order to avoid trivial solutions the developed features must be as different as possible by means of a soft constraint on the empirical estimation of the entropy associated with the encoding functions this leads to an unconstrained optimization problem that can be efficiently solved by conjugate gradient we also investigate an optimization strategy based on concave convex algorithms the relationships with maximum margin clustering are studied showing that mee overcomes some of its critical issues such as the lack of a multiclass extension and the need to face problems with a large number of constraints a massive evaluation on several benchmarks of the proposed approach shows improvements over state of the art techniques both in terms of accuracy and computational complexity

unsupervised learning of finite mixture models this paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data the adjective unsupervised is justified by two properties of the algorithm 1 it is capable of selecting the number of components and 2 unlike the standard expectation maximization em algorithm it does not require careful initialization the proposed method also avoids another drawback of em for mixture fitting the possibility of convergence toward a singular estimate at the boundary of the parameter space the novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models instead we seamlessly integrate estimation and model selection in a single algorithm our technique can be applied to any type of parametric mixture model for which it is possible to write an em algorithm in this paper we illustrate it with experiments involving gaussian mixtures these experiments testify for the good performance of our approach

quasicontinuum based multiscale approaches for plate like beam lattices experiencing in plane and out of plane deformation the quasicontinuum qc method is a multiscale approach that aims to reduce the computational cost of discrete lattice computations the method incorporates smallscale local lattice phenomena e g a single lattice defect in macroscale simulations since the method works directly and only on the beam lattice qc frameworks do not require the construction and calibration of an accompanying continuum model e g a cosserat micropolar description furthermore no coupling procedures are required between the regions of interest in which the beam lattice is fully resolved and coarse domains in which the lattice is effectively homogenized hence the method is relatively straightforward to implement and calibrate in this contribution four variants of the qc method are investigated for their use for planar beam lattices which can also experience out of plane deformation the different frameworks are compared to the direct lattice computations for three truly multiscale test cases in which a single lattice defect is present in an otherwise perfectly regular beam lattice

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

on the solution of hill s equation using milne s method milne s method is applied to the hill equation i e the schrodinger equation for a periodic one dimensional potential the method allows a compact discussion of the organization and parameter dependence of the stability bands as well as an efficient numerical computation of the band edges as an example a class of potentials which show a transition from two to one minimum per period is studied numerically furthermore periodic solutions of the milne equation are discussed and constructed

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

maximum margin clustering made practical motivated by the success of large margin methods in supervised learning maximum margin clustering mmc is a recent approach that aims at extending large margin methods to unsupervised learning however its optimization problem is nonconvex and existing mmc methods all rely on reformulating and relaxing the nonconvex optimization problem as semidefinite programs sdp though sdp is convex and standard solvers are available they are computationally very expensive and only small data sets can be handled to make mmc more practical we avoid sdp relaxations and propose in this paper an efficient approach that performs alternating optimization directly on the original nonconvex problem a key step to avoid premature convergence in the resultant iterative procedure is to change the loss function from the hinge loss to the laplacian square loss so that overconfident predictions are penalized experiments on a number of synthetic and real world data sets demonstrate that the proposed approach is more accurate much faster hundreds to tens of thousands of times faster and can handle data sets that are hundreds of times larger than the largest data set reported in the mmc literature

higher order narrow tube quantizations of quasienergies the narrow tube quasienergy quantization allows an identification of part of the spectrum due to stable elliptic islands of the period map with its transparent connection to the classical dynamics it also sets an absolute scale to the understanding of the brillouin zone spectrum of quasienergies in combination with a recent time dependent normal form expansion we show that the applicability and accuracy of the narrow tube quantization is improved its analytic form helps us understand the quasiregular motion centred near periodic responses we numerically consider harmonic and subharmonic elliptic islands of the weakly nonlinear duffing oscillator

on the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials we study the evolution of systems described by internal variables after the introduction of thermodynamic forces and fluxes both the dissipation and dissipation potential are defined then the principle of maximum dissipation pmd and a minimum principle for the dissipation potential are developed in a variational formulation both principles are related to each other several cases are shown where both principles lead to the same evolution equations for the internal variables however also counterexamples are reported where such an equivalence is not valid in this case an extended pmd can be formulated

information maximization clustering based on squared loss mutual information information maximization clustering learns a probabilistic classifier in an unsupervised manner so that mutual information between feature vectors and cluster assignments is maximized a notable advantage of this approach is that it involves only continuous optimization of model parameters which is substantially simpler than discrete optimization of cluster assignments however existing methods still involve nonconvex optimization problems and therefore finding a good local optimal solution is not straightforward in practice in this letter we propose an alternative information maximization clustering method based on a squared loss variant of mutual information this novel approach gives a clustering solution analytically in a computationally efficient way via kernel eigenvalue decomposition furthermore we provide a practical model selection procedure that allows us to objectively optimize tuning parameters included in the kernel function through experiments we demonstrate the usefulness of the pr

narrow tube approximation of semiclassical quasi energies application to the weakly nonlinear duffing oscillator an exact transformation of semiclassical quantization conditions determining quantal quasi energies of time periodic hamiltonian systems is suggested for motion on vortex tubes centred at stable periodic orbits the classical quasi energy of the underlying periodic orbit separates out as a single term which is a lower bound for the quantized quasi energies a particular approximation involving the classical lewis invariant for calculating semiclassical quasi energies near the centre of a perturbed linear response is discussed in some detail

the ermakov lewis invariants for coupled linear oscillators we consider n coupled linear oscillators with time dependent coefficients an exact complex amplitude real phase decomposition of the oscillatory motion is constructed this decomposition is further used to derive n exact constants of motion which generalize the so called ermakov lewis invariant of a single oscillator in the floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing floquet solutions

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

deriving new evolution equations for microstructures via relaxation of variational incremental problems we study material models for rate independent inelasticity in situations where no internal length scale is given and formation of microstructure for the deformation gradients and the internal variables may occur we develop a rational procedure for deriving consistent macroscopic models which allow for the computation of nontrivial effective quantities without resolving the fine scales the method involves the relaxation of variational incremental problems which are derived from an elastic and a plastic potential we use young measures to describe the microscopic distribution of the internal variables as well as the quasiconvexification of the elastic stored energy density with respect to the deformation gradient the resulting model provides a new rate independent model in terms of the deformation and the young measure the approach is based on a derivative free energetic formulation using one functional for elastic energy storage and one for the dissipation distance the latter is derived from a dissipation potential defining the dissipation distance between internal states and hence the wasserstein distance between young measures this approach is strongly linked to an associated time incremental problem which is a minimization problem of the type used for several years now in the engineering literature the update algorithm for the incremental problem is discussed in detail and two simple examples are given

time dependent normal form hamiltonian for dynamical equilibria a new time dependent normal form procedure for dynamical equilibria undergoing parametric excitation of one dimensional 1d hamiltonian systems is developed with the method of lie transforms the expansion is based on the lewis invariant for the linearized motion the time dependent hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit illustrative examples of the formalism are focused on time periodic systems and the dynamics of hamiltonian switching

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

exact quenching phenomenon of undamped driven helmholtz and duffing oscillators abstract a peculiar phenomenon of an exact quenching of integer multiples of the fundamental subharmonic frequency in the periodic response is considered the pure single frequency subharmonic solutions have been found by the exact method of harmonic balance in the undamped oscillator equations with quadratic helmholtz oscillator and cubic duffing oscillator non linear local forces in contrast to known approximate solutions the amplitudes of these exact solutions become arbitrarily large in the limit where the strength of the non linearity vanishes the stability of the subharmonic motions is briefly discussed

multiresolution molecular mechanics a unified and consistent framework for general finite element shape functions we present a general mathematical framework for the newly proposed energy based concurrent atomistic continuum method multiresolution molecular mechanics mmm yang et al 2013 the main features of the generalized framework are 1 consistency with the atomistic framework by directly employing the interatomic potential to calculate force and energy 2 simple procedure for analytically deriving the optimal summation rule for any given finite element shape function employed in the coarse grained region the procedure for obtaining the optimal summation rule is developed based on deriving and then fitting the atomic energy distribution within an element under the constraint of a given shape function to validate the generalized framework test problems including non local harmonic and anharmonic models undergoing tensile shear and bending deformations will be solved using linear bilinear and quadratic elements respectively results obtained using the proposed optimal summation rules for the different element types will be compared with gauss quadrature for accuracy through error structure analyses it is found that the proposed summation rule always outperforms gauss quadrature even when the latter employs more quadrature points than the former it is argued that widely used numerical quadrature techniques such as gauss quadrature are not optimal for coarse grained atomic energy approximation because they do not account for the discrete nature of the atoms in contrast the present summation rule is derived consistently from the underlying atomic energy distribution and thus has better accuracy and smaller computational cost

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

an improved conjugate gradient scheme to the solution of least squares svm the least square support vector machines ls svm formulation corresponds to the solution of a linear system of equations several approaches to its numerical solutions have been proposed in the literature in this letter we propose an improved method to the numerical solution of ls svm and show that the problem can be solved using one reduced system of linear equations compared with the existing algorithm for ls svm the approach used in this letter is about twice as efficient numerical results using the proposed method are provided for comparisons with other existing algorithms

semiclassical narrow tube quantizations of time periodic hamiltonian systems an approximate semiclassical quantization procedure for time periodic hamiltonians is formulated which applies to motion near stable periodic orbits of the system corresponding to elliptic islands of the period map the theory involves canonical transformations and the use of a particular dynamical lewis invariant the basic assumption in the narrow tube approximation is the localization of the quantal wavefunction inside the elliptic islands

a variational formulation of dissipative quasicontinuum methods abstract lattice systems and discrete networks with dissipative interactions are successfully employed as meso scale models of heterogeneous solids as the application scale generally is much larger than that of the discrete links physically relevant simulations are computationally expensive the quasicontinuum qc method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere in previous work beex et al j mech phys solids 64 154 169 2014 the originally conservative qc methodology was generalized to a virtual power based qc approach that includes local dissipative mechanisms in this contribution the virtual power based qc method is reformulated from a variational point of view by employing the energy based variational framework for rate independent processes mielke and roubicek rate independent systems theory and application springer verlag 2015 by construction it is shown that the qc method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential providing solutions equivalent to those of the virtual power based qc formulation the theoretical considerations are demonstrated on three simple examples for them we verify energy consistency quantify relative errors in energies and discuss errors in internal variables obtained for different meshes and two summation rules

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

ebk quantization of quasi energies the semiclassical einstein brillouin keller vortex tube quantization for the quasi energies in time periodic hamiltonians is studied for typical systems when the vortex tubes of the quantizing tubes may have complicated structures in extended phase space numerical application to a periodically driven duffing quartic oscillator yields semiclassical quasi energies in excellent agreement with the exact quantum mechanical ones for the regular states

a quasicontinuum methodology for multiscale analyses of discrete microstructural models summary many studies in different research fields use lattice models to investigate the mechanical behavior of materials full lattice calculations are often performed to determine the influence of localized microscale phenomena on large scale responses but they are usually computationally expensive in this study the quasicontinuum qc method phil mag a 1996 73 1529 1563 is extended towards lattice models that employ discrete elements such as trusses and beams the qc method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations as the qc method has so far only been applied to atomic lattice models the existing summation procedures have been revisited for structural lattice models containing discrete elements this has led to a new qc method that makes use of the characteristic structure of the considered truss network the proposed qc method is to the best of the authors knowledge the only qc method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster qc methods in its present formulation the proposed qc method can only be used for lattice models containing nearest neighbor interactions but with some minor adaptations it can also be used for lattices with next nearest neighbor interactions such as atomic lattices copyright 2011 john wiley sons ltd