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

a comparison of adomian s decomposition method and wavelet galerkin method for solving integro differential equations this paper aims to introduce a comparison of adomian decomposition method and wavelet galerkin method for the solution of integro differential equations from the computational viewpoint the comparison shows that the adomian decomposition method is efficient and easy to use

on variational iterative methods for semilinear problems abstract in this article an iterative method suitable for inverting semilinear problems is presented the method employs a two way parametrization that is able to reduce semilinear boundary value problems into simpler and linear systems governed by the laplacian in contrast to newton picard iterations the method prescribes a well defined procedure for constructing initial iterates and thus does not suffer from lack of global convergence with rectangular domains and periodic boundary conditions the technique also allows fast poisson solvers to be engaged theoretical justifications are provided and supported by several experiments numerical results show that the method yields approximations which are comparable to reference solutions

a new algorithm for matching boundary conditions in decomposition solutions abstract a computationally convenient and simple procedure is presented to match solution to boundary conditions in two point boundary value problems for linear differential and partial differential equations the nonlinear case is discussed elsewhere

application of he s variational iteration method and adomian s decomposition method to the fractional kdv burgers kuramoto equation in this work the fractional kdv burgers kuramoto equation is studied he s variational iteration method vim and adomian s decomposition method adm are applied to obtain its solution comparison with ham is made to highlight the significant features of the employed methods and their capability of handling completely integrable equations

on series solution for second order semilinear parabolic ibvps abstract this paper presents a method to obtain semi analytical solutions to second order semilinear ibvps where the standard adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions in contrast to the standard adomian decomposition method the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations the proposed method is applied to several examples of initial boundary value problems a linear advection diffusion problem a burgers equation and the deterministic kardar parisi zhang kpz equation it is shown that the dependence of the series solution on the truncation order n the number of eigenfunctions m and the diffusion coefficient is rather complex

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

reduced order modeling of parameterized pdes using time space parameter principal component analysis this paper presents a methodology for constructing low order surrogate models of finite element finite volume discrete solutions of parameterized steady state partial differential equations the construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high dimensional discrete solutions using only a few coefficients an incremental greedy approach is developed for efficiently tackling problems with high dimensional parameter spaces for numerical experiments and validation several non linear steady state convection diffusion reaction problems are considered first in one spatial dimension with two parameters and then in two spatial dimensions with two and five parameters in the two dimensional spatial case with two parameters it is shown that a 7 7 coefficient matrix is sufficient to accurately reproduce the expected solution while in the five parameters problem a 13 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy the proposed methodology is expected to find applications to parameter variation studies uncertainty analysis inverse problems and optimal design

cluster based population initialization for differential evolution frameworks this article proposes a procedure to perform an intelligent initialization for population based algorithms the proposed pre processing procedure namely cluster based population initialization cbpi consists of three consecutive stages at the first stage the individuals belonging to a randomly sampled population undergo two subsequent local search algorithms i e a simple local search that performs moves along the axes and rosenbrock algorithm at the second stage the solutions processed by the two local searches undergo the k means clustering algorithm and are grouped into sets on the basis of their euclidean distance at the third stage the best individuals belonging to each cluster are saved into the initial population of a generic optimization algorithm if the population has not been yet filled the other individuals of the population are sampled within the clusters by using a fitness based probabilistic criterion this three stage procedure implicitly performs an initial screening of the problem features in order to roughly estimate the most interesting regions of the decision space the proposed cbpi has been tested on multiple classical and modern differential evolution variants on a wide array of test problems and dimensionality values as well as on a real world problem the proposed intelligent sampling appears to have a significant impact on the algorithmic functioning as it consistently enhances the performance of the algorithms with which it is integrated

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

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

a further consideration of partial solutions in the decomposition method abstract reference 1 showed that in partial differential equation problems involving linear operator terms with respect to x y z t considerable work is saved by solving only one of the linear operator equations to obtain a so called partial solution the heat equation was shown as a simple example which appeared to be a counter example because one partial solution was identically zero the l x u operator equation led to a zero solution while the l t u operator equation led to the correct solution e t sin x this is a case of asymptotic equality as t rather than equality it arises when boundary conditions are special as discussed in 1 in cases where one operator annihilates the series in a finite number of terms one partial solution may not satisfy the given conditions this paper provides further insight into cases where such an improper solution can be obtained and the resolution of such cases by appropriate expansion of the initial term without making a priori assumptions about the solution

new results on the analytic summation of adomian series for some classes of differential and integral equations abstract in this paper we demonstrate that an infinite number of successive integration by parts can be written in a closed form this closed form can be used directly to prove that the analytic summation of adomian series becomes identical to the closed form solution for some classes of differential and integral equations

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

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

reduced dimensional gaussian process emulators of parametrized partial differential equations based on isomap in this paper isomap and kernel isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a gaussian process emulator of parametrized partial differential equations the output space consists of spatial or spatio temporal fields that are functions of multiple input variables for such problems standard multi output gaussian process emulation strategies are computationally impractical and or make restrictive assumptions regarding the correlation structure the method we develop can be applied without modification to any problem involving vector valued targets and vector valued inputs it also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

the restrictions and improvement of the adomian decomposition method the adomian decomposition method is a creative and effective method for exactly solving functional equations of various kinds in this paper the principle of the decomposition method is described and its restrictions are discussed then an improvement is proposed which overcomes the restrictions and complements the ability of the adomian decomposition method some concrete examples are also studied to illustrate the validity of the improvement

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

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

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

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

an efficient differential evolution based algorithm for solving multi objective optimization problems in the present study a modified variant of differential evolution de algorithm for solving multi objective optimization problems is presented the proposed algorithm named multi objective differential evolution algorithm modea utilizes the advantages of opposition based learning for generating an initial population of potential candidates and the concept of random localization in mutation step finally it introduces a new selection mechanism for generating a well distributed pareto optimal front the performance of proposed algorithm is investigated on a set of nine bi objective and five tri objective benchmark test functions and the results are compared with some recently modified versions of de for mops and some other multi objective evolutionary algorithms moeas the empirical analysis of the numerical results shows the efficiency of the proposed algorithm

solution of physical problems by decomposition recent generalizations are discussed and results are presented for the theory and applications of the decomposition method application is made to the duffing equation with an error of 0 0001 in only four terms and less than 10 16 in thirteen terms of the decomposition series application is also made to a dissipative wave equation a matrix riccati equation and advection diffusion nonlinear transport

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

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

a fourier series approach to burgers equation burgers equation is known to have wide application and attempts have been made in the past to solve the non homogeneous form of the equation because of the limitations of the analytical solution for certain values of the parameter nu due to slow convergence the authors discuss a numerical approach which uses the method of lines to solve the homogeneous case and involves finite fourier series the method is attractive as it works efficiently and provides good results for nu 1 0 and results with non increasing energy for nu 0 1

an adaptive multi population differential evolution algorithm for continuous multi objective optimization for evolutionary algorithms the search data during evolution has attracted considerable attention and many kinds of data mining methods have been proposed to derive useful information behind these data so as to guide the evolution search however these methods mainly centered on the single objective optimization problems in this paper an adaptive differential evolution algorithm based on analysis of search data is developed for the multi objective optimization problems in this algorithm the useful information is firstly derived from the search data during the evolution process by clustering and statistical methods and then the derived information is used to guide the generation of new population and the local search in addition the proposed differential evolution algorithm adopts multiple subpopulations each of which evolves according to the assigned crossover operator borrowed from genetic algorithms to generate perturbed vectors during the evolution process the size of each subpopulation is adaptively adjusted based on the information derived from its search results the local search consists of two phases that focus on exploration and exploitation respectively computational results on benchmark multi objective problems show that the improvements of the strategies are positive and that the proposed differential evolution algorithm is competitive or superior to some previous multi objective evolutionary algorithms in the literature

a two step adomian decomposition method the adomian decomposition method adm is a nonnumerical method for solving a wide variety of functional equations and usually gets the solution in a series form in this paper we propose an efficient modification of adm namely two step adomian decomposition method tsadm that will facilitate the calculations we then conduct a comparative study between the tsadm and previous methods with the help of several illustrative examples the results obtained indicate that the tsadm is effective and promising

a comparative study of numerical methods for solving an integro differential equation this paper is devoted to the numerical comparison of methods applied to solve an integro differential equation four numerical methods are compared namely the laplace decomposition method ldm the wavelet galerkin method wgm the laplace decomposition method with the pade approximant ld pa and the homotopy perturbation method hpm

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

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws