formal solutions of semilinear heat equations abstract we study formal power series solutions to the initial value problem for semilinear heat equation t u u f u with polynomial nonlinearity f and prove that they belong to the formal gevrey class g 2 next we give counterexamples showing that the solution in general is not analytic in time at t 0

summability of divergent solutions of the n dimensional heat equation abstract the cauchy problem for n dimensional complex heat equation is considered the borel summability of formal solutions is characterized in terms of analytic continuation with an appropriate growth condition of the spherical mean of the cauchy data

model order reduction in hyperelasticity a proper generalized decomposition approach this paper deals with the extension of proper generalized decomposition methods to non linear problems in particular to hyperelasticity among the different approaches that can be considered for the linearization of the doubly weak form of the problem we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in newton strategies this approach results in an approximation of the problem solution in the form of a series expansion each term of the series is expressed as a finite sum of separated functions the advantage of this approach is the presence of only one tangent operator identical for every term in the series the resulting approach has proved to render very accurate results that can be stored in the form of a meta model in a very compact format this opens the possibility to use these results in real time reaching khz feedback rates or to be used in deployed handheld devices such as smartphones and tablets

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

the latin multiscale computational method and the proper generalized decomposition this paper deals with the synergy between the latin multiscale method and what is called the proper generalized decomposition pgd which is the key of its performances

operators on triangular algebras abstract we study the algebra of differential operators on the triangular algebras and the upper triangular algebras we further identify all the ideals of the algebra of differential operators on the upper triangular algebras

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

derivations of upper triangular matrix rings we give a description of the derivations in tn r the ringof upper triangular matrices over a ring r assuming only the existence of an identity element we show that every derivation is the sum of an inner derivation and another one induced from r

summability of formal solutions to the n dimensional inhomogeneous heat equation the cauchy problem for n dimensional inhomogeneous complex heat equation is considered the borel summability of formal solutions is characterised in terms of analytic continuation with an appropriate growth condition of some function connected with the inhomogeneity and the cauchy data

automorphisms and derivations of upper triangular matrix rings abstract kezlan proved that for a commutative ring c every c automorphism of the ring of upper triangular matrices over c is inner we generalize this result to rings in which all idempotents are central moreover we show that for a semiprime ring a and central subring c every c automorphism of the ring of upper triangular matrices over c is the composite of an inner automorphism and an automorphism induced from a c automorphism of a by the method of proof we re prove results of s p coelho and c p milies and of mathis stating that a derivation of a ring of upper triangular matrices of a c algebra n n matrices over a is a sum of an inner derivation and a derivation induced from a c derivation of a by an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices finally we consider automorphisms of subrings of n n matrices over a commutative ring c where entries over the diagonal are from c and below the diagonal are taken from a nil ideal we prove that all such automorphisms are inner

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

on the multisummability of divergent solutions of linear partial differential equations with constant coefficients abstract we consider the cauchy problem for general linear partial differential equations in two complex variables with constant coefficients we obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the cauchy data

advanced simulation of models defined in plate geometries 3d solutions with 2d computational complexity abstract many models in polymer processing and composites manufacturing are defined in degenerated three dimensional domains involving plate or shell geometries the reduction of models from 3d to 2d is not obvious when complex physics are involved the hypotheses to be introduced for reaching this dimensionality reduction are sometimes unclear and most of possible proposals will have a narrow interval of validity the only getaway is to explore new discretization strategies able to circumvent or at least alleviate the drawbacks related to mesh based discretizations of fully 3d models defined in plate or shell domains an in plane out of plane separated representation of the involved fields within the context of the proper generalized decomposition allows solving the fully 3d model by keeping a 2d characteristic computational complexity moreover the pgd features allow the introduction of many extra coordinates as for example the orientation of the different laminate plies without affecting the solvability of the resulting multidimensional model

summability of formal solutions of linear partial differential equations with divergent initial data abstract we study the cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data we state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent cauchy data we consider both the summability in one variable t with coefficients belonging to some banach space of gevrey series with respect to the second variable z and the summability in two variables t z the results are presented in the general framework of moment pdes

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

jordan derivations and antiderivations on triangular matrices abstract we define an antiderivation from an algebra a into an a bimodule m as a linear map a m such that ab b a b a for all a b a the main result states that every jordan derivation from the algebra of all upper triangular matrices into its bimodule is the sum of a derivation and an antiderivation

on the verification of model reduction methods based on the proper generalized decomposition in this work we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition pgd approximation this estimator which is based on the constitutive relation error enables to capture all error sources i e those coming from space and time numerical discretizations from the truncation of the pgd decomposition etc and leads to guaranteed bounds on the exact error the specificity of the associated method is a double approach i e a kinematic approach and a unusual static approach for solving the parameterized problem by means of pgd this last approach makes straightforward the computation of a statically admissible solution which is necessary for robust error estimation an attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes it thus represents a practical and relevant tool for driving algorithms carried out in pgd being possibly used as a stopping criterion or as an adaptation indicator numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation

an error estimator for separated representations of highly multidimensional models fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics most of these models consist of a partial differential equation defined in a highly multidimensional domain e g schrodinger equation fokker planck equations among many others the main challenge related to these models is their associated curse of dimensionality we proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations also known as finite sums decomposition this technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates the issue related to error estimation has never been addressed this paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques

a model reduction method for the post buckling analysis of cellular microstructures multiscale finite element models are useful in some applications such as biological tissues or foams to determine the effective properties of the solids in this paper a model reduction method is proposed that is applied in the post buckling regime of cellular microstructures the proposed approach combines a perturbation technique with the proper orthogonal decomposition pod the computation of finite series leads to an efficient continuation method which enables handling limit points and instabilities naturally each term of the series is itself developed using pod the size of the problems to be solved is thus drastically reduced computations involving two and three dimensional microstructures in compression enable to evaluate the potential of the technique in view of applications to multiscale analyses

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

multisummability of formal solutions of some linear partial differential equations abstract let u t x n 1 un x tn be a formal power series in one variable t satisfying formally p t x t x u t x f t x t x c c d the purpose of the present paper is to introduce a class of linear partial differential operators and show that u t x is multisummable provided p t x t x belongs to this class

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

multisummability of formal power series solutions of partial differential equations with constant coefficients we study gevrey properties and summability of power series in two variables that are formal solutions of a cauchy problem for general linear partial differential equations with constant coefficients in doing so we extend earlier results in two articles of balser and lutz miyake and schafke for the complex heat equation as well as in a paper of balser and miyake who have investigated the same questions for a certain class of linear pde with constant coefficients subject to some restrictive assumptions moreover we also present an example of a pde where the formal solution of the cauchy problem is not k summable for whatever value of k but instead is multisummable with two levels under corresponding conditions upon the cauchy data that this can occur has not been observed up to now

a reduction method for nonlinear structural dynamic analysis abstract a computational algorithm for predicting the nonlinear dynamic response of a structure is presented the nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a rayleigh ritz analysis the basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing only a few basis updatings are required during the whole time integration the truncation error introduced at every change of basis is pointed out as the cause for a divergence type behaviour and some means for eliminating it are discussed results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations

asymptotic numerical method for problems coupling several nonlinearities abstract the objective of this paper is to assess the efficiency of the asymptotic numerical method to solve problems coupling various nonlinearities the 3d hemispherical stretching of a circular sheet that involves geometrical material and red unilateral contact nonlinearities is chosen as an example an elastoplastic model based on the plasticity deformation theory is adopted the structural discretization is performed by a shell finite element well adapted for problems involving large displacements and large rotations the unilateral contact problem is identified to boundary conditions which are replaced by force displacement relations and solved using a special algorithm comparisons with results obtained by the help of an industrial code establish the interest and the performance of the present method

model order reduction for hyperelastic materials in this paper we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains unlike standard proper orthogonal decomposition methods the proposed algorithm minimizes the use of the newton algorithms in the search of non linear equilibrium paths of elastic bodies the proposed technique is based upon two main ingredients on one side the use of classic proper orthogonal decomposition techniques that extract the most valuable information from pre computed complete models this information is used to build global shape functions in a ritz like framework on the other hand to reduce the use of newton procedures an asymptotic expansion is made for some variables of interest this expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion thus minimizing the updating of the tangent stiffness matrix of the problem the paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic kirchhoff saint venant and neo hookean solids

the borel summable solutions of heat operators on a real analytic manifold abstract we study heat type equations t u u where the operator is given by a sum of squares of commuting real analytic vector fields acting on a real analytic manifold we give necessary and sufficient conditions for convergence and borel summability of formal power series solutions in terms of generalized integral means of the initial data the results are also valid for affine perturbations of