an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

a theoretically supported scalable tfeti algorithm for the solution of multibody 3d contact problems with friction abstract a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3d and proved to be scalable for tresca friction the analysis admits floating bodies the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded the analysis is based on the classical results by farhat et al on scalability of feti for linear problems and on our development of optimal quadratic programming algorithms the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate that our algorithm is effective the effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with coulomb s friction is illustrated on the analysis of a mine support

a numerically scalable domain decomposition method for the solution of frictionless contact problems we present a domain decomposition method with lagrange multipliers for solving iteratively frictionless contact problems this method which is based on the feti method and therefore is named here the feti c method incorporates a coarse contact system that guides the iterative prediction of the active zone of contact we demonstrate numerically that this method is numerically scalable with respect to both the problem size and the number of subdomains copyright 2001 john wiley sons ltd

a parallel multiscale domain decomposition method for the transient dynamic analysis of assemblies with friction the objective of this work is to develop an efficient strategy f or dynamic problems with multiple contacts our approach is based on the multiscale latin method with domain decomposition this is a mixed method which deals simultaneously with the forces and velocities at the interfaces of the different subdomains this strategy has already been applied successfully to a variety of static problems here it is extended to dynamics first we show how the multiscale strategy can be extended to dynamics then we illustrate the capabilities of the method through a 3d academic example and the simulation of a heterogeneous material

an algorithm for the matrix free solution of quasistatic frictional contact problems a contact enforcement algorithm has been developed for matrix free quasistatic finite element techniques matrix free iterative solution algorithms such as non linear conjugate gradients cg and dynamic relaxation dr are desirable for large solid mechanics applications where direct linear equation solving is prohibitively expensive but in contrast to more traditional newton raphson and quasi newton iteration strategies the number of iterations required for convergence is typically of the same order as the number of degrees of freedom of the model it is therefore crucial that each of these iterations be inexpensive to per form which is of course the essence of a matrix free method in applying such methods to contact problems we emphasize here two requirements first that the treatment of the contact should not make an average equilibrium iteration considerably more expensive and second that the contact constraints should be imposed in such a way that they do not introduce spurious energy that acts against the iterative solver these practical concerns are utilized to develop an iterative technique for accurate constraint enforcement that is suitable for non linear conjugate gradient and dynamic relaxation iterative schemes copyright 1999 john wiley sons ltd

proper harmonic maps between asymptotically hyperbolic manifolds generalizing the result of li and tam for the hyperbolic spaces we prove an existence theorem on the dirichlet problem for harmonic maps with c 1 boundary conditions at infinity between asymptotically hyperbolic manifolds

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

fast low rank modifications of the thin singular value decomposition this paper develops an identity for additive modivations of a singular value decomposition svd to reflect updates downdates shifts and edits of the data matrix this sets the stage for fast and ememory efficient sequential algorithms for tracking singular values and subspaces in conjunction with a fast solution for the pseudo inverse of a submatrix of an orthogonal matrix we develop a scheme for computing a thin svd of streaming data in a single pass with linear time complexity a rank r think svd of a p x q matrix can be computed in o pqr time for r less than or equal sqroot min p q linear algebra and its applications

nonlinear model order reduction based on local reduced order bases summary a new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced order bases is presented it is particularly suited for problems characterized by different physical regimes parameter variations or moving features such as discontinuities and fronts instead of approximating the solution of interest in a fixed lower dimensional subspace of global basis vectors the proposed model order reduction method approximates this solution in a lower dimensional subspace generated by most appropriate local basis vectors to this effect the solution space is partitioned into subregions and a local reduced order basis is constructed and assigned to each subregion offline during the incremental solution online of the reduced problem a local basis is chosen according to the subregion of the solution space where the current high dimensional solution lies this is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower dimensional solution space because it is also applicable to the process of hyper reduction the proposed method for nonlinear model order reduction is computationally efficient its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid structure electric interaction problems copyright 2012 john wiley sons ltd

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

a scalable multiscale latin method adapted to nonsmooth discrete media the simulation of discrete systems often leads to large scale problems for instance if they result of a discretization technique or a modeling at a small scale a multiscale analysis may involve an homogenized macroscopic problem as well as a coarse space mechanism to accelerate convergence of the numerical scheme a multilevel domain decomposition technique is used herein as both a numerical strategy to simulate the behaviour of a non smooth discrete media and to provide a macroscopic numerical behaviour of the same system several generic formulations for such systems are discussed in this article a multilevel domain decomposition is tested and several choices of the embedded coarse space are discussed in particular with respect of the emergence of weak interfaces characteristics of the discrete media substructuration the application problem is the quasi static simulation of a large scale tensegrity grid

partially observable nonlinear risk sensitive control problems dynamic programming and verification theorems in this paper we consider continuous time partially observable optimal control problems with exponential of integral cost criteria we derive a rigorous verification theorem when the state and control enter nonlinear in the dynamics in addition we show that the quadratic sensor problem is estimation solvable with respect to a certain cost criterion the framework relies on dynamic programming and the hamilton jacobi theory

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

risk sensitive control and dynamic games for partially observed discrete time nonlinear systems solves a finite horizon partially observed risk sensitive stochastic optimal control problem for discrete time nonlinear systems and obtains small noise and small risk limits the small noise limit is interpreted as a deterministic partially observed dynamic game and new insights into the optimal solution of such game problems are obtained both the risk sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states dynamic programming and associated separated policies a certainty equivalence principle is also discussed the authors results have implications for the nonlinear robust stabilization problem the small risk limit is a standard partially observed risk neutral stochastic optimal control problem

a gauss seidel like algorithm to solve frictional contact problems abstract the purpose of this work is to present some mathematical and numerical results concerning an implicit method devoted to frictional contact problems it has been implemented in a dynamical deep drawing simulation software simem3 renault where unilateral contact and dry friction is assumed between the metal sheet and the tools the method may be viewed as a nonlinear block gauss seidel algorithm a convergence theorem is proved using nonsmooth analysis theory several numerical results are presented to illustrate the behaviour of this algorithm

large deformation frictional contact mechanics continuum formulation and augmented lagrangian treatment absract a complete methodology for the formulation and solution of unilateral contact problem with non associated threshold friction between solids undergoing large deformations is presented the approach includes a continuum contact mechanics model combined with an augmented lagrangian method for treating the contact and friction inequality constraints the main developments are a precise statement of the incremental contact minimization problem the exact computations of the first and second variations of the normal gap and the tangential slip velocity and an extension of the augmented lagrangian formulation for frictional contact problems from the discrete to the continuum framework the corresponding saddle point functional first variation mixed virtual work and second variation exact linearization are given the development are illustrated by numerical simulations of frictional contact problems between rigid elastic and elastoplastic bodies a new severe benchmark for large slip frictional contact elements is also proposed

a finite dimensional risk sensitive control problem a partially observed stochastic control problem with exponential running cost is considered the dynamics are linear and the running cost is quadratic although the control may enter nonlinearly explicit solutions are found to a modified zakai equation and a backward adjoint equation this enables the problem to be expressed in terms of observable finite dimensional dynamics and a separation principle to be applied

classes of nonlinear partially observable stochastic optimal control problems with explicit optimal control laws this paper introduces certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite dimensional state space in some cases the optimal control laws are analogous to linear exponential quadratic gaussian and linear quadratic gaussian tracking problems the problems discussed allow nonlinearities to enter the unobservable dynamics as gradients of potential functions the methodology is based on explicit solutions of a modified duncan mortensen zakai equation

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

uzawa and newton algorithms to solve frictional contact problems within the bi potential framework this paper is concerned with the numerical modeling of three dimensional unilateral contact problems in elastostatics with coulomb friction laws we propose a newton like algorithm to solve the local contact non linear equations within the bi potential framework the piecewise continuous contact tangent matrices are explicitly derived a comparative study is made between the newton algorithm and the previously developed uzawa algorithm a test example is included to demonstrate the developed algorithms and to highlight their performance

the role of information state and adjoint in relating nonlinear output feedback risk sensitive control and dynamic games this paper employs logarithmic transformations to establish relations between continuous time nonlinear partially observable risk sensitive control problems and analogous output feedback dynamic games the first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state the second logarithmic transformation is applied to the risk sensitive cost function using the laplace varadhan lemma in the small noise limit this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game

a multiscale large time increment fas algorithm with time space model reduction for frictional contact problems a multiscale strategy using model reduction for frictional contact computation is presented this new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies this strategy is based on a combination between the latin large time increment method and the fas multigrid solver the latin method is an iterative solver operating on the whole time space domain applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned time space vectors forming the so called reduced basis depict particular scales of the problem it becomes easy to make analogies with multigrid method to take full advantage of multiscale information

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

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

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

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

a scalable tfeti algorithm for two dimensional multibody contact problems with friction a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid of rigid body motions is adapted to the solution of two dimensional multibody contact problems of elasticity with the coulomb friction and proved to be scalable for the tresca friction the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded the analysis is based on the classical results by farhat mandel and roux on scalability of feti with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate a high efficiency of our algorithm in addition its performance is illustrated on analysis of the yielding clamp connection with the coulomb friction

low rank incremental methods for computing dominant singular subspaces computing the singular values and vectors of a matrix is a crucial kernel in numerous scientific and industrial applications as such numerous methods have been proposed to handle this problem in a computationally efficient way this paper considers a family of methods for incrementally computing the dominant svd of a large matrix a specifically we describe a unification of a number of previously independent methods for approximating the dominant svd after a single pass through a we connect the behavior of these methods to that of a class of optimization based iterative eigensolvers on ata an iterative procedure is proposed which allows the computation of an accurate dominant svd using multiple passes through a we present an analysis of the convergence of this iteration and provide empirical demonstration of the proposed method on both synthetic and benchmark data

latin a new view and an extension to wave propagation in nonlinear media summary the latin acronym of large time increment method was originally devised as a non incremental procedure for the solution of quasi static problems in continuum mechanics with material nonlinearity in contrast to standard incremental methods like newton and modified newton latin is an iterative procedure applied to the entire loading path in each latin iteration two problems are solved a local problem which is nonlinear but algebraic and miniature and a global problem which involves the entire loading process but is linear the convergence of these iterations which has been shown to occur for a large class of nonlinear problems provides an approximate solution to the original problem in this paper the latin method is presented from a different viewpoint taking advantage of the causality principle in this new view latin is an incremental method and the latin iterations are performed within each load step similarly to the way that newton iterations are performed the advantages of the new approach are discussed in addition latin is extended for the solution of time dependent wave problems as a relatively simple model for illustrating the new formulation lateral wave propagation in a flat membrane made of a nonlinear material is considered numerical examples demonstrate the performance of the scheme in conjunction with finite element discretization in space and the newmark trapezoidal algorithm in time copyright 2017 john wiley sons ltd

a domain decomposition strategy for nonclassical frictional multi contact problems in this paper we present a numerical strategy to be solve large scale frictional contact problems by domain decomposition methods which are adapted to parallel computers the motivation is given by the study of the mechanical behavior of rolling shutters composed by many hinged slats the numerical treatment of such nonclassical contact problems leads to very large strongly nonlinear nonsymmetric and ill conditioned systems domain decomposition methods are a good alternative to overcome the difficulties of classical sequential solutions we present a nonlinear strategy adapted to problems called multi contact problems

space time proper generalized decompositions for the resolution of transient elastodynamic models in this paper we investigate ability of proper generalized decomposition pgd to solve transient elastodynamic models in space time domain classical methods use time integration schemes and an incremental resolution process we propose here to use standard time integration methods in a non incremental strategy as a result pgd gives a separated representation of the space time solution as a sum of tensorial products of space and time vectors that we interpret as space time modes recent time integration schemes are based on multi field formulations in this case separated representation can be constructed using state vectors in space and same vectors in time however we have experienced bad convergence order using this decomposition furthermore temporal approximation must be the same for all fields thus we propose an extension of classical separated representation for multi field problems this multi field pgd mf pgd uses space and time vectors that are different for each field calculation of decomposition is done using a monolithic approach in space and time potentially allowing the use of different approximations in space and time finally several simulations are performed with the transient elastodynamic problem with one dimension in space different approximations in time are investigated newmark scheme single field time discontinuous galerkin method and two fields time continuous and discontinuous galerkin methods

information transfer between incompatible finite element meshes application to coupled thermo viscoelasticity this article is concerned with information transfer between non matching finite element meshes such a feature is not new in the literature but we focus herein on a geometric approach to transfer solution fields in order to be as application independent as possible moreover the dedicated case concerns the transfer of finite element fields defined at integration points of the meshes and allows iterative exchange of fields in both directions to do so we propose an extension to the mortar technique that fulfills these goals and that does not suffer from a high computational cost the application is the simulation of a strongly coupled thermo viscoelastic problem with phase transition solved with a partitioning technique

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

general finite dimensional risk sensitive problems and small noise limits for a risk sensitive partially observed stochastic control problem the modified zakai equation includes an extra term related to the exponential running cost the finite dimensional solutions of this modified zakai equation are obtained these are analogs of the kalman and benes filters the small noise limits of the finite dimensional risk sensitive problems are then obtained these lead to differential games with deterministic disturbances

bridging proper orthogonal decomposition methods and augmented newton krylov algorithms an adaptive model order reduction for highly nonlinear mechanical problems this article describes a bridge between pod based model order reduction techniques and the classical newton krylov solvers this bridge is used to derive an efficient algorithm to correct on the fly the reduced order modelling of highly nonlinear problems undergoing strong topological changes damage initiation problems tackled via a corrected hyperreduction method are used as an example it is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm even when strong topological changes are involved

conjugate gradient type algorithms for frictional multi contact problems applications to granular materials abstract this paper presents gradient type algorithms to solve frictional multi contact problems written as quasi optimization problems a single loop scheme formally close to the classical conjugate gradient method is proposed with some adaptations of the iterate corrections and gradient projections since the convergence is difficult to prove various tests in the field of granular media are performed with comparison with the non linear gauss seidel scheme

toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver abstract in this paper an efficient a priori model reduction strategy for frictional contact problems is presented we propose to solve this problem by using the finite element method and the non linear latin solver basically this non linear solver assumes a space time separated representation presaging nowadays pgd strategies we extend this family of solvers to frictional engineering applications with reduced subspaces and no prior knowledge about the solution contrary to a posteriori model reduction techniques hereinafter a hybrid a priori a posteriori latin pgd formulation for frictional contact problems is proposed indeed the suggested algorithm may or may not start with an initial guess of the reduced basis and is able to enrich the basis in order to reach a given level of accuracy moreover it provides progressively the solution of the considered problem into a quasi optimal space time separated form compared to the singular value decomposition svd some examples are provided in order to illustrate the efficiency and quasi optimality of the proposed a priori reduced basis latin solver

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

a new approach in non linear mechanics the large time increment method a new algorithm family taking into account the entire loading process in a single large time increment is proposed to compute structures with physical non linearities and is tested on some examples in elastoplasticity the method considerably reduces the number of transfers between local and global levels hence the numerical cost of calculation is also diminished