a proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart young approach the proper generalized decomposition pgd is a methodology initially proposed for the solution of partial differential equations pde defined in tensor product spaces it consists in constructing a separated representation of the solution of a given pde in this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting in particular we introduce a generalization of eckart and young theorem which allows to prove the convergence of the so called progressive pgd for a large class of linear problems defined in tensor product hilbert spaces

chebyshev neural network based model for solving lane emden type equations the objective of this paper is to solve second order non linear ordinary differential equations of lane emden type using chebyshev neural network chnn model these equations are categorized as singular initial value problems artificial neural network ann model is used here to overcome the difficulty of the singularity a single layer neural network is used and the hidden layer is eliminated by expanding the input pattern by chebyshev polynomials here we have used feed forward neural network model and principle of error back propagation homogeneous and non homogeneous lane emden equations are considered to show effectiveness of chebyshev neural network model

model reduction based on proper generalized decomposition for the stochastic steady incompressible navier stokes equations in this paper we consider a proper generalized decomposition method to solve the steady incompressible navier stokes equations with random reynolds number and forcing term the aim of such a technique is to compute a low cost reduced basis approximation of the full stochastic galerkin solution of the problem at hand a particular algorithm inspired by the arnoldi method for solving eigenproblems is proposed for an efficient greedy construction of a deterministic reduced basis approximation this algorithm decouples the computation of the deterministic and stochastic components of the solution thus allowing reuse of preexisting deterministic navier stokes solvers it has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m dimensional reduced basis rather than m coupled problems of the full stochastic galerkin approximation space with m ll m up to one order of magnitude for the problem at hand in this work

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

on a multiscale computational strategy with time and space homogenization for structural mechanics a new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale this strategy which involves homogenization in space as well as in time could replace in several domains of application the standard homogenization techniques which are generally limited to the space domain it is an iterative strategy which calls for the resolution of problems on both a micro fine scale and a macro homogenized scale in this paper we review the bases of this approach and present improved approximation techniques to solve the micro and macro problems

a dynamical adaptive tensor method for the vlasov poisson system abstract a numerical method is proposed to solve the full eulerian time dependent vlasov poisson system the algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step this decomposition is obtained through the use of an efficient modified progressive generalized decomposition pgd method whose convergence is proved we suggest in addition a symplectic time discretization splitting scheme that preserves the hamiltonian properties of the system this scheme is naturally obtained by considering the tensor structure of the approximation the proposed approach is illustrated through time dependent 1d 1d 2d 2d and 3d 3d numerical examples

three dimensional magnetotelluric modeling using difference equations theory and comparisons to integral equation solutions we have developed an algorithm for computing the magnetotelluric response of three dimensional 3 d earth models it is a difference equation algorithm that is based on the integral forms of maxwell s equations rather than the differential forms this formulation does not require approximating derivatives of earth properties or electromagnetic fields as happens when using the second order vector diffusion equation rather one must determine how averages are to be computed side boundary values for the h fields are obtained from putting two dimensional 2 d slices of the model into a larger scale 2 d model and solving for the fields at the 3 d boundary positions to solve the 3 d system of equations we propagate an impedance matrix which relates all the horizontal e fields in a layer to all the horizontal h fields in that same layer up through the earth model applying a plane wave source condition and the side boundary h field values allows us to solve for the unknown fields within the model the r

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

on the existence of a progressive variational vademecum based on the proper generalized decomposition for a class of elliptic parameterized problems abstract in this study we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition pgd the pgd is a novel technique that was developed recently for solving problems with high dimensions and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method this new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the pgd can help in the change to this paradigm the main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum we prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions in particular we illustrate this methodology based on a robot path planning problem this is one of the common tasks when designing the trajectory or path of a mobile robot the construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map

electromagnetic modeling of three dimensional bodies in layered earths using integral equations we have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three dimensional bodies in layered earths the inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions a matrix equation is constructed using the electric tensor green s function appropriate to a layered earth and it is solved for the vector current in each cell subsequently scattered fields are found by integrating electric and magnetic tensor green s functions over the scattering currents efficient evaluation of the tensor green s functions is a major consideration in reducing computation time we find that tabulation and interpolation of the six electric and five magnetic hankel transforms defining the secondary green s functions is preferable to any direct hankel transform calculation using linear filters a comparison of responses over elongate three dimensional 3 d bodies with responses over two dimensional 2 d bodie

advances in three dimensional magnetotelluric modeling using integral equations recent progress in integral equation modeling of three dimensional magnetotelluric responses includes the ability to simulate 3 d structures which outcrop excluding topography which transect layer interfaces and which extend indefinitely in one or more directions the most important factor in achieving this capability is an accurate treatment of the electric surface charge in particular a previous integro difference formulation for evaluating charges has been abandoned in favor of true surface integrations over the source cells with potential differencing across the field cells in the 3 d body the new procedure constitutes a good approximation to galerkin s method while preserving internal consistency in terms of pulse basis functions to verify outcropping structures juxtaposed conductive and resistive prisms at the surface are simulated and compared to 2 d results an elongate version of the 3 d model shows good agreement with both transverse electric and transverse magnetic modes of the 2 d res

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

low rank approximate inverse for preconditioning tensor structured linear systems in this paper we propose an algorithm for the construction of low rank approximations of the inverse of an operator given in low rank tensor format the construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator it provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators these subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank one corrections in order to handle high order tensors approximate projections are computed in low rank hierarchical tucker subsets of the successive subspaces of operators some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step where an optimal rank one correction is searched as the tensor product of operators with the desired properties numerical examples illustrate the ability of this a

the stone weierstrass theorem and its application to neural networks the stone weierstrass theorem and its terminology are reviewed and neural network architectures based on this theorem are presented specifically exponential functions polynomials partial fractions and boolean functions are used to create networks capable of approximating arbitrary bounded measurable functions a modified logistic network satisfying the theorem is proposed as an alternative to commonly used networks based on logistic squashing functions

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

galerkin reduced order modeling scheme for time dependent randomly parametrized linear partial differential equations summary in this paper we consider the problem of constructing reduced order models of a class of time dependent randomly parametrized linear partial differential equations our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates parameter space and time the proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions the unknown basis functions in the decomposition are estimated using an alternating iterative galerkin projection scheme numerical studies on the time dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos based galerkin projection schemes comparison studies are also made against nouy s generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages copyright 2012 john wiley sons ltd

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains abstract a tensor based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains it provides an accurate solution which is explicit with respect to parameters defining the shape of the domain thus allowing efficient a posteriori probabilistic or parametric analyses in the proposed method a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain yielding a weak formulation in a tensor product space product of space functions and parametric functions the paper is limited to the case of neumann conditions on uncertain parts of the boundary the proper generalized decomposition method is then introduced for the construction of a tensor product approximation separated representation of the solution it can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution this tensor based method is made computationally tractable by introducing separated representations of variational forms resulting from separated representations of the parameterized indicator function of the uncertain domain for this purpose a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well posedness of problems associated with approximate indicator functions moreover a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

ideal minimal residual based proper generalized decomposition for non symmetric multi field models application to transient elastodynamics in space time domain abstract it is now well established that separated representations built with the help of proper generalized decomposition pgd can drastically reduce computational costs associated with solution of a wide variety of problems however it is still an open question to know if separated representations can be efficiently used to approximate solutions of hyperbolic evolution problems in space time domain in this paper we numerically address this issue and concentrate on transient elastodynamic models for such models the operator associated with the space time problem is non symmetric and low rank approximations are classically computed by minimizing the space time residual in a natural l 2 sense yet leading to non optimal approximations in usual solution norms therefore a new algorithm has been recently introduced by one of the authors and allows to find a quasi optimal low rank approximation a priori with respect to a target norm we presently extend this new algorithm to multi field models the proposed algorithm is applied to elastodynamics formulated over space time domain with the time discontinuous galerkin method in displacement and velocity numerical examples demonstrate convergence of the proposed algorithm and comparisons are made with classical a posteriori and a priori approaches

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

approximation of constrained problems using the pgd method with application to pure neumann problems in this paper we introduce analyze and compare several approaches designed to incorporate a linear or affine constraint within the proper generalized decomposition framework we apply the considered methods and numerical strategies to two classes of problems the pure neumann case where the role of the constraint is to recover unicity of the solution and the robin case where the constraint forces the solution to move away from the already existing unique global minimizer of the energy functional

general takagi sugeno fuzzy systems with simplified linear rule consequent are universal controllers models and filters abstract takagi sugeno ts fuzzy systems have successfully been employed mainly in a trial and error manner to solve many control and modeling problems but their applications as signal filters remain to be fully explored compared to their nonfuzzy counterparts ts fuzzy controllers and models are difficult to be efficiently constructed because there is a large number of design parameters in the rule consequent the number grows dramatically with the increase of the number of input fuzzy sets and input variables furthermore there exists little published result on relationship between ts fuzzy controllers models filters and their nonfuzzy counterparts in this paper we investigate in relation to some popular nonfuzzy controllers models and filters analytical structure of a general class of multi input single output miso ts fuzzy systems that use arbitrary fuzzy rules with our recently introduced simplified linear rule consequent other components of the fuzzy systems in this study are general arbitrary continuous input fuzzy sets any type of fuzzy logic and and the generalized defuzzifier containing the widely used centroid defuzzifier as a special case we prove that the general miso ts fuzzy systems are 1 nonlinear variable gain controllers when implemented as controllers or 2 nonlinear time varying auto regressive with the extra input arx models when implemented as models or 3 nonlinear infinite impulse response iir or finite impulse response fir filters when implemented as filters furthermore we constructively prove that the general ts fuzzy systems with the simplified linear rule consequent are universal approximators and can approximate any continuous function in closed domain arbitrarily well the practical implication of these results is that these fuzzy systems with much less design parameters are always able to produce solutions to various control modeling and filtering problems we also establish sufficient conditions that can be used to calculate the number of input fuzzy sets and rules needed for achieving prespecified approximation accuracy

effect of randomness on multi frequency aeroelastic responses resolved by unsteady adaptive stochastic finite elements the unsteady adaptive stochastic finite elements uasfe method resolves the effect of randomness in numerical simulations of single mode aeroelastic responses with a constant accuracy in time for a constant number of samples in this paper the uasfe framework is extended to multi frequency responses and continuous structures by employing a wavelet decomposition pre processing step to decompose the sampled multi frequency signals into single frequency components the effect of the randomness on the multi frequency response is then obtained by summing the results of the uasfe interpolation at constant phase for the different frequency components results for multi frequency responses and continuous structures show a three orders of magnitude reduction of computational costs compared to crude monte carlo simulations in a harmonically forced oscillator a flutter panel problem and the three dimensional transonic agard 445 6 wing aeroelastic benchmark subject to random fields and random parameters with various probability distributions

conjugate direction relaxation solutions for 3 d magnetotelluric modeling in recent years there has been a tremendous amount of progress made in three dimensional 3 d magnetotelluric modeling algorithms much of this work has been devoted to the integral equation technique e g hohmann 1975 weidelt 1975 wannamaker et al 1984 wannamaker 1991 this method has contributed significantly to our understanding of electromagnetic field behavior in 3 d models however some of the very earliest work in 3 d modeling concentrated on differential methods e g jones and pascoe 1972 reddy et al 1977 it is generally recognized that differential methods are better suited than integral equation methods to model arbitrarily complex geometries and consequently this area has recently been receiving a great deal of attention e g madden and mackie 1989 xinghua et al 1991 mackie et al 1993 smith 1992 personnal communication differential methods lead to large sparse systems of equations to be solved for the unknown field values it is possible to use relaxation al

novel determination of differential equation solutions universal approximation method in a conventional approach to numerical computation finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations des this paper presents a novel approach to solve des by applying the universal approximation method through an artificial intelligence utility in a simple way in this proposed method neural network model nnm and fuzzy linguistic model flm are applied as universal approximators for any nonlinear continuous functions with this outstanding capability the solutions of des can be approximated by the appropriate nnm or flm within an arbitrary accuracy the adjustable parameters of such nnm and flm are determined by implementing the optimization algorithm this systematic search yields sub optimal adjustable parameters of nnm and flm with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of des the simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear des

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques