numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach the paper deals with the computational homogenization of pressure diffusion processes in a poroelastic medium the underlying physical phenomena are of interest for the interpretation of seismic data with applications in hydrocarbon production and geothermal energy pressure diffusion is assumed to take place on a length scale much smaller than the observer scale thus the macroscopic observer is not able to measure the properties of the poroelastic medium directly but notices an intrinsic viscous attenuation under these circumstances the macro scale can be interpreted as a single phase solid with apparent viscoelastic properties in this paper we establish a numerical upscaling procedure based on a volume averaging concept this enables us to identify the material properties of the viscoelastic substitute model in a numerically efficient manner for this purpose the poroelastic medium on the small scale is modelled in terms of the momentum balance of the biphasic mixture and a coupled diffusion equation we approximate the poroelastic pressure field on the small scale by a linear combination of pressure modes forming a reduced orthogonal basis and being identified by a proper orthogonal decomposition pod technique from the superposition principle the evaluation of the poroelastic continuity equation results in a proper identification of the evolution equations defining the apparent viscoelastic model in comparison to the nested fe2 solution schemes the reduced order approach only requires a small amount of off line precomputations and therefore causes very low numerical costs the proposed method is validated for the simple setup of a layered porous rock with alternating water and gas saturated zones

modal based goal oriented error assessment for timeline dependent quantities in transient dynamics this article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics in particular a new type of quantities of interest referred as timeline dependent quantities is proposed these quantities are scalar time dependent outputs of the transient solution which are better suited to time dependent problems than the standard scalar ones frozen in time the proposed both the standard scalar and the new timeline dependent quantities of interest the key ingredient is the modal based approximation of the associated adjoint problems which allows efficiently computing and storing the adjoint solution the approximated adjoint solution is readily post processed to produce an enhanced solution requiring only one spatial post process for each vibration mode and using the time harmonic hypothesis to recover the time dependence thus the proposed goal oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem the resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem thus the cost of the error estimate at each time step is very low

julia a fresh approach to numerical computing bridging cultures that have often been distant julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing julia is designed to be easy and fast and questions notions generally held to be laws of nature by practitioners of numerical computing beginlist item high level dynamic programs have to be slow item one must prototype in one language and then rewrite in another language for speed or deployment item there are parts of a system appropriate for the programmer and other parts that are best left untouched as they have been built by the experts endlist we introduce the julia programming language and its design a dance between specialization and abstraction specialization allows for custom treatment multiple dispatch a technique from computer science picks the right algorithm for the right circumstance abstraction which is what good computation is really about recognizes what remains the same after dif

duality based two level error estimation for time dependent pdes application to linear and nonlinear parabolic equations we introduce a duality based two level error estimator for linear and nonlinear time dependent problems the error measure can be a space time norm energy norm final time error or other error related functional the general methodology is developed for an abstract nonlinear parabolic pde and subsequently applied to linear heat and nonlinear cahn hilliard equations the error due to finite element approximations is estimated with a residual weighted approximate dual solution which is computed with two primal approximations at nested levels we prove that the exact error is estimated by our estimator up to higher order remainder terms numerical experiments confirm the theory regarding consistency of the dual based two level estimator we also present a novel space time adaptive strategy to control errors based on the new estimator

explicit solution of a stochastic irreversible investment problem and its moving threshold we consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity the firm aims to maximize its expected total discounted real profit net of investment on a finite horizont the capacity is modeled as a controlled ita process where the control is the real investment which is not necessarily a rate but more generally a monotone process the result is a singular stochastic control problem we introduce the associated optimal stopping problem that is the optimal cost of not investing its variational formulation turns out to be a parabolic obstacle problem which we explicitly solve in the case of constant relative risk aversion crra production functions the moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital s replacement cost then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits

optimal boundary surface for irreversible investment with stochastic costs this paper examines a markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production we model market uncertainty and the cost of investment per unit of production capacity as two independent one dimensional regular diffusions and we consider a general convex running cost function the optimization problem is set as a three dimensional degenerate singular stochastic control problem we provide the optimal control as the solution of a reflected diffusion at a suitable boundary surface such boundary arises from the analysis of a family of two dimensional parameter dependent optimal stopping problems and it is characterized in terms of the family of unique continuous solutions to parameter dependent nonlinear integral equations of fredholm type

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

on transformation strains and uniform fields in multiphase elastic media the effect of local eigenstrain and eigenstress fields or transformation fields on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry the residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors general properties of these functions and concentration factors and their relation to the analogous mechanical influence functions and concentration factors are established in part with the help of uniform strain fields in multiphase media specific estimates of the transformation concentration factor tensors are evaluated by the self consistent and mori tanaka methods it is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems their estimates of the mechanical thermal and transformation concentration factor tensors and of the overall stiffness of multiphase media have a similar structure proofs that guarantee that these methods comply with the general properties of the transformation influence functions and provide diagonally symmetric estimates of the overall elastic stiffness are given for two phase and multiphase systems consisting of or reinforced by inclusions of similar shape and alignment one of the possible applications of the results in analysis of overall instantaneous properties and local fields in inelastic composite materials is described in the following paper

on a stochastic irreversible investment problem the productive sector of the economy represented by a single firm employing labor to produce the consumption good is studied in a stochastic continuous time model on a finite time interval the firm must choose the optimal level of employment and capital investment in order to maximize its expected total profits in this stochastic control problem the firm s capacity is modeled as an ito process controlled by a monotone process possibly singular that represents the cumulative real investment it is optimal to invest when the shadow value of installed capital exceeds the capital s replacement cost this threshold is the free boundary of a related optimal stopping problem which we recast as a stopping problem without integral cost similar to the american option problem then under a regularity condition we characterize the free boundary as the unique solution of a nonlinear integral equation

hp discontinuous galerkin time stepping for parabolic problems abstract we consider the hp version of the discontinuous galerkin dg time stepping method for linear parabolic problems with non symmetric elliptic spatial operators we derive new analyticity estimates for the exact solutions by means of semigroup techniques these estimates allow us to show that the hp dg time stepping method can resolve start up singularities at exponential rates of convergence

an a posteriori error estimator for an unsteady advectiondiffusionreaction problem in this work we introduce an a posteriori error estimator of the residual type for the unsteady advectiondiffusionreaction problem for the discretization in time we use an implicit euler scheme and a continuous piecewise linear triangular finite elements for the space together with a stabilized scheme we prove that the approximation error is bounded by above and below by the error estimator using that an adaptive algorithm is proposed analyzed and tested numerically to prove the efficiency of our approach

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

computable exact bounds for linear outputs from stabilized solutions of the advection diffusion reaction equation summary the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection diffusion reaction equation the bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution the new methodology extends the a posteriori error estimates yielding bounds for the standard galerkin formulation to be able to obtain bounds for stabilized formulations this methodology is combined with both hybrid flux and flux free techniques for error assessment the application to stabilized formulations provides sharper estimates than when applied to galerkin methods the best results are found in combination with the flux free technique copyright 2012 john wiley sons ltd

characterization of the optimal boundaries in reversible investment problems this paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good our model allows for general diffusion dynamics on the demand as well as general cost functional the resulting optimization problem leads to a degenerate two dimensional bounded variation singular stochastic control problem for which explicit solution is not available in general and the standard verification approach cannot be applied a priori we use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution in the quadratic cost case we are able to prove a smooth fit c 2 property which gives rise to a full characterization of the optimal boundaries and value function

valuation of investments in real assets with implications for the stock prices a general model for the valuation of natural resource investments is formulated and analyzed within a stochastic control theoretic framework using dynamic programming the value of such an investment with a general payoff function is determined under the assumption that the commodity price process is given by a stochastic differential equation the analysis results in closed form analytic solutions which can easily be computed and exhibits qualitatively different optimal behaviors depending on parameter values implications for stocks and options are also considered

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

a posteriori finite element bounds for output functionals of discontinuous galerkin discretizations of parabolic problems we present a neumann subproblem a posteriori finite element procedure for the efficient calculation of constant free sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous galerkin method in time in space a global coarse mesh and a decoupled fine mesh are used to compute the estimators which are shown to converge to the value of the output obtained for a global coupled fine mesh we first formulate the bound procedure with particular emphasis on the proof of the bounding properties we then provide an illustrative numerical example a problem of heat conduction in a composite material

adaptive component mode synthesis in linear elasticity component mode synthesis cms is a classical method for the reduction of large scale finite element models in linear elasticity in this paper we develop a methodology for adaptive refinement of c

trading a mean reverting asset buy low and sell high this paper is concerned with an optimal trading buy and sell rule the underlying asset price is governed by a mean reverting model the objective is to buy and sell the asset so as to maximize the overall return slippage cost is imposed on each transaction the associated hjb equations quasi variational inequalities are used to characterize the value functions it is shown that the solution to the original optimal stopping problem can be obtained by solving two quasi algebraic equations sufficient conditions are given in the form of a verification theorem a numerical example is reported to demonstrate the results

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

adaptive reduced basis finite element heterogeneous multiscale method an adaptive reduced basis finite element heterogeneous multiscale method rb fe hmm is proposed for elliptic problems with multiple scales the multiscale method is based on the rb fe hmm introduced in a abdulle y bai reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems j comput phys 231 21 2012 7014 7036 it couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains unlike classical numerical homogenization methods the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions the reduced basis obtained by a greedy algorithm in an offline stage in this paper we present a residual based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest for both type of adaptive strategies rigorous a posteriori error estimates are derived and corresponding error estimators are proposed in contrast to the adaptive finite element heterogeneous multiscale method fe hmm there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement up to an offline preliminary stage the rb fe hmm has the same computational complexity as a standard adaptive fem for the effective problem two and three dimensional numerical experiments confirm the efficiency of the rb fe hmm and illustrate the improvements compared to the adaptive fe hmm c 2013 elsevier b v all rights reserved

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

certification of projection based reduced order modelling in computational homogenisation by the constitutive relation error in this paper we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites the upper bound relies on the construction of a reduced model for the stress field upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense the desired bounding property is obtained the lower bound is obtained by defining a hierarchical enriched reduced model for the displacement we show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model

parabolic variational inequality with parameter and gradient constraints abstract this paper concerns a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints the method is to transform a two dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies moreover we analyze the behaviors of the free boundary surfaces the main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter

numerical model reduction with error control in computational homogenization of transient heat flow numerical model reduction nmr is exploited for solving the finite element problem on a representative volume element rve that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the classical method of spectral decomposition a symmetrized version of the space time variational format is adopted for estimating the error from the model reduction in i energy norm and in ii given quantities of interest this technique which was recently developed in the context of the non selfadjoint stationary diffusion convection problem is novel in the present context of nmr by considering the discrete unreduced model as exact we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort the performance of the error estimates is demonstrated via numerical results where the subscale is modeled in both one and three spatial dimensions

a flux free a posteriori error estimator for the incompressible stokes problem using a mixed fe formulation in this contribution we present an a posteriori error estimator for the incompressible stokes problem valid for a conventional mixed fe formulation due to the saddle point property of the problem conventional error estimators developed for pure minimization problems cannot be utilized straight forwardly the new estimator is built up by two key ingredients at first a computed error approximation exactly fulfilling the continuity equation for the error is obtained via local dirichlet problems secondly we adopt the approach of solving local equilibrated flux free problems in order to bound the remaining incompressible error in this manner guaranteed upper and lower bounds of the velocity energy norm of the error as well as goal oriented linear output functionals with respect to a reference overkill mesh are obtained in particular it should be noted that this approach requires no computation of hybrid fluxes furthermore the estimator is applicable to mixed fe formulations using continuous pressure approximations such as the mini and taylor hood class of elements in conclusion a few simple numerical examples are presented illustrating the accuracy of the error bounds c 2010 elsevier b v all rights reserved

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

a posteriori error estimation for the semidiscrete finite element method of parabolic differential equations in this paper three a posteriori error estimators of the error in the semidiscrete finite element solution discrete in space and continuous in time of parabolic partial differential equations are analyzed this approach is based on a posteriori error estimators for the elliptic pdes it is proven that guaranteed resp asymptotically exact a posteriori error estimator for the elliptic problem yields the guaranteed resp asymptotically exact estimator for the parabolic problem

a sequential adaptive strategy in space time with application to consolidation of porous media issues related to space time adaptivity for a class of nonlinear and time dependent problems are discussed the dg k methods are adopted for the time integration and the a posteriori error control is based on the appropriate dual problem in space time one key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions the main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time mesh rather the space mesh and the time step defining each finite space time slab are defined in a truly sequential fashion the proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations its performance is investigated with the aid of a numerical example in 2d

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

long term optimal investment strategies in the presence of adjustment costs we consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment in particular we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project s output commodity we model this economic indicator by means of a general one dimensional ergodic diffusion at any time the project s capacity level can be increased or decreased at given proportional costs the aim is to maximize an ergodic performance criterion that reflects the long term average payoff resulting from the project s management we solve this genuinely two dimensional stochastic control problem by constructing an explicit solution to an appropriate hamilton jacobi bellman equation and by fully characterizing an optimal investment strategy

a model for reversible investment capacity expansion we consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and or demand fluctuations we model market uncertainty by means of a geometric brownian motion and we consider general running payoff functions our model allows for capacity expansion as well as for capacity reduction with each of these actions being associated with proportional costs the resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly we illustrate our results by means of the so called cobb douglas production function the problem that we study presents a model in which the associated hamilton jacobi bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function which may be identically equal to infty thus our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics as well as in other fields

bounds of functional outputs for parabolic problems part ii bounds of the exact solution the paper introduces a methodology to compute upper and lower bounds for linear functional outputs of the exact solutions of parabolic problems in this second part the bounds account for the error both in space and time the assumption stating that the error introduced by the time marching scheme is negligible used in the first part is removed here the bounds are computed starting from an approximation of the exact solution associated with a spatial mesh and a time grid nevertheless the bounds are guaranteed with respect to the exact solution with no reference to any mesh or time discretization

bounds for quantities of interest and adaptivity in the element free galerkin method a novel approach to implicit residual type error estimation in mesh free methods and an adaptive refinement strategy are presented this allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest the proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux equilibration and resulting in a simple implementation that avoids integrals on edges sides of a domain decomposition mesh this is especially interesting for mesh free methods the adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision copyright 2008 john wiley sons ltd

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

local global multiscale model reduction for flows in high contrast heterogeneous media in this paper we study model reduction for multiscale problems in heterogeneous high contrast media our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods see 19 with global model reduction methods such as balanced truncation approaches implemented on a coarse grid local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution in particular our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem global model reduction based on balanced truncation methods is used to identify important global coarse scale modes this provides a substantial cpu savings as lyapunov equations are solved for the coarse system typical local multiscale methods are designed to find an approximation of the solution for any given coarse level inputs in many practical applications a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse level inputs the proposed approaches provide efficient model reduction tools in this direction our numerical results show that only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes one can achieve a balanced and optimal result

optimal control under a dynamic fuel constraint we present a new approach to solve optimal control problems of the monotone follower type the key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint instead of dynamic programming we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the snell envelope of the objective functional s gradient at the optimum the optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in bank and el karoui 2004 ann probab 32 pp 1030 1067 as an illustration we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

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

computational analysis of nonlinear composite structures using the nonuniform transformation field analysis abstract a reduced model for the effective behavior of nonlinear composites such as metal matrix composite materials has been recently proposed by the authors 1 it extends and improves on the transformation field analysis of dvorak 2 by considering nonuniform transformation strains also called plastic modes and is referred to as the nonuniform transformation field analysis the present study is devoted to the implementation of this new homogenized model into a structural computation a brief account on the reduction procedure is given first then the time integration of the model which is required at each integration point of the structural problem is performed by means of an implicit scheme two examples are discussed the response of the homogenized structure is compared to the exact response of the actual heterogeneous structure computed with a very fine mesh it is seen that not only the structural response is accurately captured by the ntfa model but also the local stress field is correctly approximated

a trend following strategy conditions for optimality based on trend following trading strategies that are widely used in the investment world this work provides a set of sufficient conditions that determine the optimality of the traditional trend following strategies when the trends are completely observable a dynamic programming approach is used to verify the optimality under these conditions the value functions are characterized by the associated hjb equations and are shown to be either linear functions or infinity depending on the parameter values the results reveal two counterintuitive facts a trend following may not lead to optimal reward in some cases even when if the investor knows exactly when a trend change occurs b stock volatility is not relevant in trend following when trends are observable

a posteriori error estimation for numerical model reduction in computational homogenization of porous media numerical model reduction is adopted for solving the microscale problem that arizes from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields a reduced basis is obtained for the pressure field using i spectral decomposition sd and ii proper orthogonal decomposition pod this strategy has been used in previous work the main contribution of this article is the extension with an a posteriori estimator for assessing the error in i energy norm and in ii a given quantity of interest the error estimator builds on previous work by the authors the novelty presented in this article is the generalization of the estimator to a coupled problem and more importantly to accommodate the estimator for a pod basis rather than the sd basis guaranteed fully computable and low cost bounds are derived and the performance of the error estimates is demonstrated via numerical results

computing bounds for linear functionals of exact weak solutions to poisson s equation we present a method for poisson s equation that computes guaranteed upper and lower bounds for the values of piecewise polynomial linear functional outputs of the exact weak solution of the infinite dimensional continuum problem with piecewise polynomial forcing the method results from exploiting the lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem localization is achieved by lagrangian relaxation and the bounds are computed by appeal to a local dual problem the proposed method computes approximate lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers the computed bounds hold uniformly for any level of refinement and in the asymptotic convergence regime of the finite element method the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution given a finite element solution and its output adjoint solution the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization the elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators as we demonstrate with a numerical example

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

on error controlled numerical model reduction in fe2 analysis of transient heat flow numerical model reduction is exploited for solving the two scale problem that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the method of spectral decomposition a key feature is the construction of a symmetrized version of the space time variational format by which it is possible to estimate the error from the model reduction in i energy norm and in ii a given quantity of interest in previous work by the authors this strategy has been applied to the solution of an individual representative volume element problem whereas the error transport to the macroscale problem in the finite element squared fe2 approach was ignored in this paper such transport of error is included in the error estimate it is remarkable that it is still possible to obtain guaranteed bounds on the error as compared to the fully resolved discrete finite element problem while using only the reduced basis and with minor extra computational effort the performance of the error estimates is demonstrated via numerical results whereby the subscale is modeled in three dimensions whereas the macroscale problem is either one or two dimensional

a nonconvex singular stochastic control problem and its related optimal stopping boundaries equivalences are known between problems of singular stochastic control ssc with convex performance criteria and related questions of optimal stopping see for example karatzas and shreve siam j control optim 22 1984 pp 856 877 the aim of this paper is to investigate how far connections of this type generalize to a nonconvex problem of purchasing electricity where the classical equivalence breaks down we provide alternative connections to optimal stopping problems we consider a nonconvex infinite time horizon ssc problem whose state consists of an uncontrolled diffusion representing a real valued commodity price and a controlled increasing bounded process representing an inventory we analyze the geometry of the action and inaction regions by characterizing their optimal boundaries unlike the case of convex ssc problems we find that the optimal boundaries may be both reflecting and repelling and it is natural to interpret the problem as one of ssc with discretionary stopping

connections between singular control and optimal switching this paper builds a new theoretical connection between singular control of finite variation and optimal switching problems this correspondence provides a novel method for solving high dimensional singular control problems and enables us to extend the theory of reversible investment sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions consequently our regularity result links singular controls and dynkin games through sequential optimal stopping problems

variationally consistent computational homogenization of transient heat flow a framework for variationally consistent homogenization combined with a generalized macrohomogeneity condition is exploited for the analysis of non linear transient heat conduction within this framework the classical approach of model based first order homogenization for stationary problems is extended to transient problems homogenization is then carried out in the spatial domain on representative volume elements rve which are in practice introduced in quadrature points in standard fashion along with the classical averages a higher order conservation quantity is obtained an iterative fe2 algorithm is devised for the case of non linear diffusion and storage coefficients and it is applied to transient heat conduction in a strongly heterogeneous particle composite parametric studies are carried out in particular with respect to the influence of the internal length associated with the second order conservation quantity copyright c 2009 john wiley sons ltd

computable bounds of functional outputs in linear visco elastodynamics this work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco elastodynamics a novel expression for the error rep resentation is introduced alternative to the previous ones using the cauchy schwarz inequal ity the proposed formulation utilizes symmetrized forms of the error equations to derive error bounds in terms of energy error measures the practical implementation of the method is based on constructing admissible fields for both the original problem and the adjoint problem asso ciated with the quantity of interest here the flux free technique is considered to compute the admissible stress fields the proposed methodology yields estimates with better quality than the ones based on the cauchy schwarz inequality in the studied examples the bound gaps obtained are approximately halved that is the estimated intervals of confidence are reduced

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

the computation of bounds for linear functional outputs of weak solutions to the two dimensional elasticity equations we present a method for the computation of upper and lower bounds for linear functional outputs of the exact solutions to the two dimensional elasticity equations the method can be regarded as a generalization of the well known complementary energy principle the desired output is cast as the supremum of a quadratic linear convex functional over an infinite dimensional domain using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary or dual problem in order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced first the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle the lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation second the solution of the adjoint problem is approximated over the triangulation using a standard galerkin finite element approach a lower bound for the output of interest is computed by repeating the process for the negative of the output reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output the method can be easily generalized to three dimensions however a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions the computed bound gaps are found to converge optimally that is at the same rate as the finite element approximation an attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds using this data set and a simple algorithm the correctness of the computed bounds can be established without recourse to the original code used to compute them in the present paper only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered two examples are given to illustrate the proposed methodology

composite meromorphic functions and growth of the spherical derivative abstract let f and h be transcendental meromorphic and g a transcendental entire function it is shown that if h grows slower than g in a suitable sense then there exists an unbounded sequence zn such that f g zn h zn

a class of singular control problems and the smooth fit principle this paper analyzes a class of singular control problems for which value functions are not necessarily smooth necessary and sufficient conditions for the well known smooth fit principle along with the regularity of the value functions are given explicit solutions for the optimal policy and for the value functions are provided in particular when payoff functions satisfy the usual inada conditions the boundaries between action and continuation regions are smooth and strictly monotonic as postulated and exploited in the existing literature see a k dixit and r s pindyck investment under uncertainty princeton university press princeton nj 1994 m h a davis et al adv in appl probab 19 1987 pp 156 176 t o kobila stochastics stochastics rep 43 1993 pp 29 63 a b abel and j c eberly j econom dynam control 21 1997 pp 831 852 a oksendal finance stoch 4 2000 pp 223 250 j a scheinkman and t zariphopoulou j econom theory 96 2001 pp 180 207 a merhi and m zervos siam j control optim 46 2007 pp 839 876 and l h alvarez a general theory of optimal capacity accumulation under price uncertainty and costly reversibility working paper helsinki center of economic research helsinski finland 2006 illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions

guaranteed a posteriori error estimation for fully discrete solutions of parabolic problems this paper addresses the construction of guaranteed computable estimates for fully discrete solutions of parabolic problems copyright 2003 john wiley sons ltd

an optimal pairs trading rule abstract this paper is concerned with a pairs trading rule the idea is to monitor two historically correlated securities when divergence is underway i e one stock moves up while the other moves down a pairs trade is entered which consists of a pair to short the outperforming stock and to long the underperforming one such a strategy bets the spread between the two would eventually converge in this paper a difference of the pair is governed by a mean reverting model the objective is to trade the pair so as to maximize an overall return a fixed commission cost is charged with each transaction in addition a stop loss limit is imposed as a state constraint the associated hjb equations quasi variational inequalities are used to characterize the value functions it is shown that the solution to the optimal stopping problem can be obtained by solving a number of quasi algebraic equations we provide a set of sufficient conditions in terms of a verification theorem numerical examples are reported to demonstrate the results

exact bounds for linear outputs of the advection diffusion reaction equation using flux free error estimates the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection reaction diffusion equation the proposed approach is an alternative to the standard residual type estimators hybrid flux circumventing the need of flux equilibration following a flux free error estimation strategy the presented estimator provides sharper estimates than the ones provided by both the standard hybrid flux techniques and other flux free techniques

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

nonuniform transformation field analysis the exact description of the overall behavior of composites with nonlinear dissipative phases requires an infinity of internal variables approximate models involving only a finite number of those can be obtained by considering a decomposition of the microscopic anelastic strain field on a finite set of transformation fields the transformation field analysis of dvorak proc r soc lond a 437 1992 311 corresponds to piecewise uniform transformation fields the present theory considers nonuniform transformation fields comparison with numerical simulations shows the accuracy of the proposed model

explicit solutions for an optimal stock selling problem under a markov chain model abstract this paper is concerned with explicit solutions for a classical optimal stock selling problem in contrast to almost all market models treated in the literature the underlying market is solely determined by a two state markov chain such markov chain model is strikingly simple and yet appears capable capturing various market movements ranging from close to brownian motion to no so brownian ones the purpose of this paper is to study the optimal selling rule under such a model and develop a set of analysis techniques useful for related optimal stopping problems in this paper the goal of the problem under consideration is to find an optimal stopping time to sell the stock so as to maximize an expected return explicit solutions to the associated variational inequalities are obtained these solutions are given in terms of a set of threshold levels verification theorems are provided to justify their optimality finally numerical examples are provided to illustrate the results

error estimation including pollution assessment for nonlinear finite element analysis a residual type error estimator for nonlinear finite element analysis is introduced this error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure pollution errors are taken into account by a feedback strategy that is an error estimate based on local computations is used as the input of the pollution analysis this estimator is used in the frame of an adaptive procedure numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions moreover one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms

computing upper and lower bounds for the j integral in two dimensional linear elasticity we present an a posteriori method for computing rigorous upper and lower bounds for the j integral in two dimensional linear elasticity the j integral which is typically expressed as a contour integral is recast as a quadratic continuous functional of the displacement involving only area integration by expanding the quadratic output about an approximate finite element solution the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error the quadratic component is bounded by the energy norm of the error scaled by a continuity constant which is determined explicitly the linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields the method is illustrated with two fracture problems in plane strain elasticity an important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10