two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

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

sampling based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures we define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it the approach is based on dynamic programming equations written for a risk averse formulation of the problem this formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures dual representations of such risk functionals are given and used to derive conditions of coherence in the one period case conditions for convexity and consistency with second order stochastic dominance are also provided the risk averse dynamic programming equations are specialized considering convex combinations of one period extended polyhedral risk measures such as spectral risk measures to implement the proposed policy the approximation of the risk averse recourse functions for stochastic linear programs is discussed in this context we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse p

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

convex computation of the region of attraction of polynomial control systems we address the long standing problem of computing the region of attraction roa of a target set e g a neighborhood of an equilibrium point of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints we show that the roa can be computed by solving an infinite dimensional convex linear programming lp problem over the space of measures in turn this problem can be solved approximately via a classical converging hierarchy of convex finite dimensional linear matrix inequalities lmis our approach is genuinely primal in the sense that convexity of the problem of computing the roa is an outcome of optimizing directly over system trajectories the dual infinite dimensional lp on nonnegative continuous functions approximated by polynomial sum of squares allows us to generate a hierarchy of semialgebraic outer approximations of the roa at the price of solving a sequence of lmi problems with asymptotically vanishing conservatism this sharply contrasts with the existing literature which follows an exclusively dual lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative lmi conditions the approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description the approach is demonstrated on several numerical examples

nonlinear optimal control via occupation measures and lmi relaxations we consider the class of nonlinear optimal control problems ocps with polynomial data i e the differential equation state and control constraints and cost are all described by polynomials and more generally for ocps with smooth data in addition state constraints as well as state and or action constraints are allowed we provide a simple hierarchy of lmi linear matrix inequality relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value under some convexity assumptions the sequence converges to the optimal value of the ocp preliminary results show that good approximations are obtained with few moments

two grid method for linear elasticity on unstructured meshes we propose an abstract two grid algorithm with convergence independent of the coarse space size the abstract algorithm is applied to problems of three dimensional linear elasticity discretized on unstructured meshes with no regularity assumptions we prove uniform convergence with respect to coarse space size domain essential boundary conditions and jumps in young modulus numerical experiments confirm the theory and show that the method works well even if some assumptions of the theory are violated

a moment and sum of squares extension of dual dynamic programming with application to nonlinear energy storage problems abstract we present a finite horizon optimization algorithm that extends the established concept of dual dynamic programming ddp in two ways first in contrast to the linear costs dynamics and constraints of standard ddp we consider problems in which all of these can be polynomial functions second we allow the state trajectory to be described by probability distributions rather than point values and return approximate value functions fitted to these the algorithm is in part an adaptation of sum of squares techniques used in the approximate dynamic programming literature it alternates between a forward simulation through the horizon in which the moments of the state distribution are propagated through a succession of single stage problems and a backward recursion in which a new polynomial function is derived for each stage using the moments of the state as fixed data the value function approximation returned for a given stage is the point wise maximum of all polynomials derived for that stage this contrasts with the piecewise affine functions derived in conventional ddp we prove key convergence properties of the new algorithm and validate it in simulation on two case studies related to the optimal operation of energy storage devices with nonlinear characteristics the first is a small borehole storage problem for which multiple value function approximations can be compared the second is a larger problem for which conventional discretized dynamic programming is intractable

discounted cost markov decision processes on borel spaces the linear programming formulation this paper is concerned with the linear programming formulation of markov decision processes or stochastic dynamic programs with borel state and action spaces and the discounted cost criterion the one stage cost function may be unbounded a linear program and its dual are introduced for which is shown the absence of a duality gap and that a strong duality condition holds these results are used to determine the optimal value and the existence of an optimal policy for the discounted cost problem

symbolic computation of analytic approximate solutions for nonlinear differential equations with boundary conditions the adomian decomposition method adm is one of the most effective methods for constructing analytic approximate solutions in this paper based on the new definitions of the adomian polynomials and the double decomposition method a new algorithm is proposed to construct analytic approximations for nonlinear differential equations with coupled boundary conditions furthermore a maple software package is developed to implement our new algorithm which is user friendly and efficient one only needs to input the system boundary conditions and several necessary parameters then our package will deliver the analytic approximate solution within a few seconds several different types of nonlinear examples are given to illustrate the scope and demonstrate the validity of our software package especially for nonlinear multi order and multi point boundary value problems our package provides a helpful and easy to use tool in science and engineering to deal with the most frequently encountered boundary value problems this maple package is free to download on the website http www cs ecnu edu cn lizb index htm

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

multi stage stochastic optimization applied to energy planning this paper presents a methodology for the solution of multistage stochastic optimization problems based on the approximation of the expected cost to go functions of stochastic dynamic programming by piecewise linear functions no state discretization is necessary and the combinatorial explosion with the number of states the well known curse of dimensionality of dynamic programming is avoided the piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to benders cuts in a stochastic multistage decomposition framework a case study of optimal stochastic scheduling for a 39 reservoir system is presented and discussed

stochastic dual dynamic programming applied to nonconvex hydrothermal models abstract in this paper we apply stochastic dual dynamic programming decomposition to a nonconvex multistage stochastic hydrothermal model where the nonlinear water head effects on production and the nonlinear dependence between the reservoir head and the reservoir volume are modeled the nonconvex constraints that represent the production function of a hydro plant are approximated by mccormick envelopes these constraints are split into smaller regions and the mccormick envelopes are used for each region we use binary variables for this disjunctive programming approach and solve the problem with a decomposition method we resort to a variant of the l shaped method for solving the mip subproblem with binary variables at any stage inside the stochastic dual dynamic programming algorithm a realistic large scale case study is presented

a two grid method for incompressible miscible displacement problems by mixed finite element and eulerian lagrangian localized adjoint methods abstract in this paper we present a scheme for solving two dimensional miscible displacement problems using eulerian lagrangian localized adjoint methods and mixed finite element methods since only the velocity and not the pressure appears explicitly in the concentration equation an eulerian lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation to linearize and decouple the mixed method equations we use a two grid algorithm based on the newton iteration method for this fully discrete problems first we solve the original nonlinear equations on the coarse grid then we solve the linearized problem on the fine grid using newton iteration once it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in this paper

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

a two grid discretization method for decoupling systems of partial differential equations in this paper we propose a two grid finite element method for solving coupled partial differential equations e g the schrodinger type equation with this method the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid it is shown both theoretically and numerically that the resulting solution still achieves asymptotically optimal accuracy

minimum residual adaptive multilevel finite element procedure for the solution of nonlinear stationary problems a multilevel minimum residual type procedure for an adaptive finite element fe solution of a general nonlinear differential problem is analyzed with respect to linearization and adaptive fe approximation errors simple sufficient conditions for the linear and quadratic convergence to the exact solution of the original problem with increasing number of levels are established a nontrivial numerical example illustrates the efficiency of the method

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

a fast two grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half plane we consider the numerical treatment of second kind integral equations on the real line of the form phi s psi s int infty infty k s t z t phi t dt s in mathbb r abbreviated k z in which l 1 z l and bc the space of bounded continuous functions on are assumed known and bc is to be determined we first derive sharp error estimates for the finite section approximation reducing the range of integration to a a via bounds on i k z 1 as an operator on spaces of weighted continuous functions numerical solution by a simple discrete collocation method on a uniform grid on is then analysed in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix vector multiply via the fft to utilise this possibility we propose a modified two grid iteration a feature of which is that the coarse grid matrix is approximated by a banded matrix and analyse convergence and computational cost in cases wherez is not compactly supported a combined finite section and two grid algorithm can be applied and we extend the analysis to this case as an application we consider acoustic scattering in the half plane with a robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied our final result is that ifz related to the boundary impedance in the application takes values in an appropriate compact subsetq of the complex plane then the difference between s and its finite section approximation computed numerically using the iterative scheme proposed is c 1 khlog 1 kh 1 1 2 ka 1 2 in the interval a a 1 forkh sufficiently small wherek is the wavenumber andh the grid spacing moreover this numerical approximation can be computed in c 2 n logn operations wheren 2a h is the number of degrees of freedom the values of the constantsc 1 andc 2 depend only on the setq and not on the wavenumberk or the support ofz

inexact cuts in benders decomposition benders decomposition is a well known technique for solving large linear programs with a special structure in particular it is a popular technique for solving multistage stochastic linear programming problems early termination in the subproblems generated during benders decomposition assuming dual feasibility produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution we describe an inexact cut algorithm prove its convergence under easily verifiable assumptions and discuss a corresponding dantzig wolfe decomposition algorithm the paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling

on the convergence of decomposition methods for multistage stochastic convex programs we prove the almost sure convergence of a class of sampling based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions and uncertainty is modelled by a scenario tree as special cases our results imply the almost sure convergence of stochastic dual dynamic programming cutting plane and partial sampling cupps algorithm and dynamic outer approximation sampling algorithms when applied to problems with general convex cost functions

error estimates on a new nonlinear galerkin method based on two grid finite elements a new nonlinear galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations the new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time with linear finite element discretizations it is proved that the difference between the new nonlinear galerkin solution and the standard galerkin solution in h 1 omega norm is of the order of h 3

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

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 two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

convergence of a fem and two grid algorithms for elliptic problems on disjoint domains in this paper we analyze a fem and two grid fem decoupling algorithms for elliptic problems on disjoint domains first we study the rate of convergence of the fem and in particular we obtain a superconvergence result then with proposed algorithms the solution of the multi component domain problem simple example two disjoint rectangles on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems each on a single component domain on fine meshes the advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy numerical experiments demonstrate the efficiency of the algorithms

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

numerical solution of nonlinear singular boundary value problems in this paper an efficient method based on the simplified reproducing kernel method srkm and least squares method lsm is proposed for solving nonlinear singular boundary value problems the algorithm consists of two steps first we apply srkm to solve a linear equation which contains a set of parameters second the lsm is used to find the optimal parameters error estimation and convergence order for the presented method are also discussed our numerical experiments validate our theoretical findings and demonstrate the performance of the algorithm in terms of simplicity accuracy and efficiency

a multilevel successive iteration method for nonlinear elliptic problems in this paper a multilevel successive iteration method for solving nonlinear elliptic problems is proposed by combining a multilevel linearization technique and the cascadic multigrid approach the error analysis and the complexity analysis for the proposed method are carried out based on the two grid theory and its multilevel extension a superconvergence result for the multilevel linearization algorithm is established which besides being interesting for its own sake enables us to obtain the error estimates for the multilevel successive iteration method the optimal complexity is established for nonlinear elliptic problems in 2 d provided that the number of grid levels is fixed

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

two level picard and modified picard methods for the navier stokes equations abstract iterative methods of picard type for the navier stokes equations are known to converge only for quite small reynolds numbers however we study methods involving just one such iteration at general reynolds numbers for the initial approximation a coarse mesh of width h0 is used the corrected approximation is computed by just one picard or modified picard step on a fine mesh of width h1 for example h1 may be of order o h 0 2 when linear velocity elements are used the resulting method requires the solution of a small system of nonlinear equations on the coarse mesh and only one larger linear system on the fine mesh this two level picard method is proven to converge for fixed reynolds number as h 0 further the fine mesh solution satisfies a quasi optimal error bound the error constants grow as re as for the usual finite element method one very heuristic explanation why one step of the divergent picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower order derivatives thus they are approximated with higher accuracy on the coarse mesh this is linked with a smoothing property of the fine mesh step

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

convergence rates of moment sum of squares hierarchies for optimal control problems we study the convergence rate of moment sum of squares hierarchies of semidefinite programs for optimal control problems with polynomial data it is known that these hierarchies generate polynomial under approximations to the value function of the optimal control problem and that these under approximations converge in the l1 norm to the value function as their degree d tends to infinity we show that the rate of this convergence is o 1 log log d we treat in detail the continuous time infinite horizon discounted problem and describe in brief how the same rate can be obtained for the finite horizon continuous time problem and for the discrete time counterparts of both problems