an algebraic multigrid method for a class of elliptic differential systems the development numerical investigation and analysis of an algebraic multigrid amg method for the numerical solution of a representative class of elliptic differential systems are presented our amg scheme is based on the amg strategy of brandt ruge and stuben however the present approach may be implemented within other amg schemes by means of numerical experiments with model problems arising from some application areas it is demonstrated that the computational performance of the proposed amg scheme is comparable to that of amg applied to a single scalar equation an amg convergence theory is presented that extends convergence results for amg methods for scalar problems to the case of amg applied to elliptic systems some details of our amg implementation are given

an optimization based approach for the design of pde solution algorithms we develop and analyze an optimization based approach for the robust and efficient solution of pde problems consisting of multiple physics operators with fundamentally different mathematical properties our approach relies on three essential steps decomposition of the original problem into subproblems for which robust solution algorithms are available integration of the subproblems into an equivalent pde constrained optimization problem and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the pde constraints this strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components

the numerical solution of the steady state solid fuel ignition model and its optimal control we present a finite difference multigrid technique that efficiently solves optimal control problems associated with the steady state solid fuel ignition model this is a nonlinear indefinite problem which gives rise to singular optimal control systems regarding the direct problem necessary and sufficient conditions for existence of solutions are given and second order convergence of numerical solution errors with respect to the mesh size is proven a robust nonlinear multigrid method is implemented to validate this convergence estimate two optimal control strategies for solid fuel ignition phenomena are considered and compared the corresponding optimality systems are solved to second order accuracy by a multigrid method whose convergence properties are independent of the values of the weights in the cost functionals and of the number of grid points

accurate thermal stress predictions using c0 continuous higher order shear deformable elements the behavior of c 0 continuous higher order shear deformable finite element formulations when applied to thermal stress conditions is analyzed critically it is shown that unlike in the first order formulations the inconsistent initial strain field can also disturb the displacement recovery in the higher order shear deformable elements in this paper the possible errors due to such inconsistent terms are a priori predicted in an analytical sense and validated by using linear and quadratic beam element formulations the paper suggests how the total and the thermal strain components have to be consistently reconstituted to get optimal accuracy from higher order shear deformable finite elements for thermal stress analysis a conflict between the consistency and the continuity requirements is noticed in such formulations the extraneous errors observed in the variationally correct and consistent formulations are attributed to this fact

multilevel algorithms for constrained compact fixed point problems in this paper the authors extend the multilevel algorithm of atkinson and brakhage for compact fixed point problems and the projected newton method of bertsekas to create a fast multilevel algorithm for parabolic boundary control problems having bound constraints on the control results are extended from finite dimension on constraint identification this approach permits both adaptive integration in time and inexact evaluation of the cost functional

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

an optimization based atomistic to continuum coupling method we present a new optimization based method for atomistic to continuum atc coupling the main idea is to cast the latter as a constrained optimization problem with virtual dirichlet controls on the interfaces between the atomistic and continuum subdomains the optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains while the atomistic and continuum force balance equations provide the constraints separation rather then blending of the atomistic and continuum problems and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing atc formulations we present and analyze the method in the context of a one dimensional chain of atoms modeled using a linearized two body potential with next nearest neighbor interactions

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

multigrid optimization in applications abstract iterative techniques are a key methodology for the numerical solution of optimization problems in differential equations in two practical application problems with different characteristics this paper shows how multigrid methods can be applied efficiently to this problem class problem formulations solution approaches as well as numerical results are presented

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

inversion of 3d electromagnetic data in frequency and time domain using an inexact all at once approach we present a general formulation for inverting frequencyor time domain electromagnetic data using an all at once approach in this methodology the forward modeling equations are incorporated as constraints and thus we need to solve a constrained optimization problem where the parameters are the electromagnetic fields the conductivity model and a set of lagrange multipliers this leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought nevertheless experience shows that the constrained problem can be solved faster than the unconstrained one the primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly in this paper we outline the all atonce approach and apply it to electromagnetic problems in both frequency and time domains this is facilitated by a unified representation for forward modeling for these two types of data the optimization problem is solved by finding a stationary point of the lagrangian numerically this leads to a nonlinear system that is solved iteratively using a gauss newton strategy at each iteration a large indefinite matrix is inverted and we discuss how this can be accomplished as a test we invert frequency domain synthetic data from a grounded electrode system that emulates a field csamt survey for the time domain we invert borehole data obtained from a current loop on the surface

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

consistency and recovery in electroelasticity part i standard finite elements developing finite elements for electroelastic analysis requires a special care as the coupling in the discrete equations depends on the matching between the approximations assumed for mechanical and electrical variables to provide a formal basis to this intuitive remark a notion of consistency is established and a rigorous consistency analysis is presented for the standard finite element model based on assumed displacement and electric potential in this way specific analytical requirements are obtained which serve as a guide to select the interpolation functions for primary variables moreover the analysis shows that violating consistency can be reflected as spurious outcomes upsetting the local distributions of secondary variables indeed this undesirable effect is shown to be typical of the standard approach if stress and electric flux density are computed via the constitutive equations to cure the trouble an alternative recovery procedure is devised based on the consistency analysis the procedure is variationally correct and reconstitutes in a consistent manner the distributions of stress and electric flux density

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

perspective on the lagrange jacobi mesh this paper presents a unified treatment of the kinetic energy matrix elements related to a number of lagrange functions associated with the lagrange jacobi mesh the matrix elements can be readily modified for application to problems requiring eigenfunction expansion with lagrange legendre lagrange chebyshev lagrange gegenbauer as well as the lagrange jacobi functions the applicability of and the accuracy attainable with the matrix elements is demonstrated with the solution to the schrodinger equation for confining trigonometric poschl teller potentials the results obtained are within machine accuracy when appropriate choices of the basis functions are used

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

the unexplained accuracy of the lagrange mesh method the lagrange mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a gauss quadrature in order to analyze its accuracy four different lagrange mesh calculations based on the zeros of laguerre polynomials are compared with exact variational calculations based on the corresponding laguerre basis the comparison is performed for three solvable radial potentials the morse harmonic oscillator and coulomb potentials the results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy even in the presence of the centrifugal singularity the same property holds for the approximate wave functions this striking accuracy remains unexplained

multilevel algorithms for ill posed problems in this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations convergence estimates are established for multilevel algorithms applied to tikhonov type regularization methods our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter the algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations

multiscale analysis of heterogeneous domain decomposition methods for time dependent advection reaction diffusion problems abstract domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods we are interested here in the case where there is an accurate but expensive model one should use in the entire domain but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected for the model problem of a time dependent advection reaction diffusion equation in one spatial dimension we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non variational heterogeneous domain decomposition methods we show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

lagrange mesh calculations and fourier transform the lagrange mesh method is a very accurate procedure for computing eigenvalues and eigenfunctions of a two body quantum equation written in the configuration space using a gauss quadrature rule the method only requires the evaluation of the potential at some mesh points the eigenfunctions are expanded in terms of regularized lagrange functions which vanish at all mesh points except one using the peculiarities of the method it is shown that the fourier transform of the eigenfunctions computed in the configuration space can easily be obtained with good accuracy in the physical domain of the momentum space also observables in this space can easily be computed with good accuracy only using matrix elements and eigenfunctions computed in the configuration space

algebraic multigrid based on element interpolation amge we introduce amge an algebraic multigrid method for solving the discrete equations that arise in ritz type finite element methods for partial differential equations assuming access to the element stiffness matrices we have that amge is based on the use of two local measures which are derived from global measures that appear in existing multigrid theory these new measures are used to determine local representations of algebraically smooth error components that provide the basis for constructing effective interpolation and hence the coarsening process for amg here we focus on the interpolation process choice of the coarse grids based on these measures is the subject of current research we develop a theoretical foundation for amge and present numerical results that demonstrate the efficacy of the method

inexact preconditioned conjugate gradient method with inner outer iteration an important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner outer iterations g h golub and m l overton numerical analysis lecture notes in math 912 springer berlin new york 1982 where the preconditioner is solved by an inner iteration to a prescribed precision in this paper we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive definite system and analyze its convergence property we establish a linear convergence result using a local relation of residual norms we also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to high accuracy the analysis is in agreement with observed numerical behavior of the algorithm in particular it suggests a heuristic choice of the stopping threshold for the inner iteration numerical examples are given to show the effectiveness of this choice and to compare the convergence bound

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

a multilevel nonlinear method a multilevel nonlinear method mnm for the numerical solution of nonlinear partial differential equations is developed the mnm algorithm is motivated and analyzed using a simplified model which retains the essential features of the new approach it is thereby shown to combine the advantages of the two classical multigrid approaches to nonlinear problems the analysis is supported by numerical tests for nonlinear differential equations in one and two dimensions

optimization based additive decomposition of weakly coercive problems with applications we present an abstract mathematical framework for an optimization based additive decomposition of a large class of variational problems into a collection of concurrent subproblems the framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions the significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems consequently assuming that stable numerical methods and efficient solvers are available for every subproblem our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily an application of the framework to the oseen equations illustrates its potential

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

semiclassical generalization of the darboux christoffel formula the darboux christoffel formula is a closed form expression for the kernel of the operator that projects onto the first n of a system of one dimensional polynomials orthonormal with respect to some weighting function it is a key element in the theory of gaussian integration and in the theory of discrete variable representation or lagrangian mesh methods for diagonalizing quantum hamiltonians of a few degrees of freedom the one dimensional darboux christoffel formula turns out to have a generalization that is valid in a semiclassical or asymptotic sense for a wider class of orthonormal functions than orthonormal polynomials this class consists of the bound eigenfunctions of one dimensional hamiltonians with time reversal invariance such as kinetic plus potential hamiltonians it also has certain generalizations involving the unbound eigenfunctions of such hamiltonians

a new petrov galerkin smoothed aggregation preconditioner for nonsymmetric linear systems we propose a new variant of smoothed aggregation sa suitable for nonsymmetric linear systems the new algorithm is based on two key generalizations of sa restriction smoothing and local damping restriction smoothing refers to the smoothing of a tentative restriction operator via a damped jacobi like iteration restriction smoothing is analogous to prolongator smoothing in standard sa and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric local damping refers to damping parameters used in the jacobi like iteration in standard sa a single damping parameter is computed via an eigenvalue computation here local damping parameters are computed by considering the minimization of an energy like quantity for each individual grid transfer basis function numerical results are given showing how this method performs on highly nonsymmetric systems

collocation on uniform grids in this paper we derive four sets of sinc like functions defined on a finite interval and obeying different boundary conditions the functions in each set are orthogonal and their nodes are uniformly distributed on the interval we have applied each set to solve a large class of eigenvalue equations with different boundary conditions both on finite intervals and on the real line showing that precise numerical results can be obtained efficiently and rapidly a comparison with results available in the literature is also performed

space mapping for optimal control of partial differential equations solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties e g nonlinearities of the discretized problems in this paper a novel space mapping technique for solving the aforementioned problem class is introduced analyzed and tested the advantage of the space mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing perhaps less nonlinear surrogates the space mapping is based on a regularization approach which in contrast to other space mapping techniques results in a smooth mapping and thus avoids certain irregular situations at kinks a new broyden update formula for the sensitivities of the space map is also introduced this quasi newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space mapping surrogate the overall algorithm employs a trust region framework for global convergence issues involved in the computations are highlighted and a report on a few illustrative numerical tests is given

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

integrals of lagrange functions and sum rules exact values are derived for some matrix elements of lagrange functions i e orthonormal cardinal functions constructed from orthogonal polynomials they are obtained with exact gauss quadratures supplemented by corrections in the particular case of lagrange laguerre and shifted lagrange jacobi functions sum rules provide exact values for matrix elements of 1 x and 1 x2 as well as for the kinetic energy from these expressions new sum rules involving laguerre and shifted jacobi zeros and weights are derived

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

high order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems high mixed order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed for the solution of these systems an efficient and robust multigrid algorithm is presented theoretical and experimental results show the advantages of higher order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang bang control

multigrid methods for parabolic distributed optimal control problems multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated accuracy properties of finite difference approximation are discussed and validated two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies semicoarsening and standard coarsening the resulting multigrid algorithms show robustness with respect to changes of the value of v the weight of the cost of the control is sufficiently small fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters results of numerical experiments are reported that demonstrate sharpness of fourier analysis estimates a multigrid algorithm that solves optimal control problems with box constraints on the control is considered

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

variational collocation on finite intervals in this paper we study a set of functions defined on an interval of finite width which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken we show that these functions can be used within a variational approach to obtain accurate results for a variety of problems we have applied them to the interpolation of functions on finite domains and to the solution of the schrodinger equation and we have compared the performance of the present approach with others

asymptotic properties of receding horizon optimal control problems the asymptotic behavior of receding horizon optimal control problems with terminal cost chosen as a control liapunov function is analyzed for regulator as well as disturbance attenuation problems both the continuous as well as the discrete time cases are treated further the approximation of the continuous time optimal control problem by the discrete time receding horizon problems is studied

lagrange mesh calculations in momentum space the lagrange mesh method is a powerful method to solve eigenequations written in configuration space it is very easy to implement and very accurate using a gauss quadrature rule the method requires only the evaluation of the potential at some mesh points the eigenfunctions are expanded in terms of regularized lagrange functions which vanish at all mesh points except one it is shown that this method can be adapted to solve eigenequations written in momentum space keeping the convenience and the accuracy of the original technique in particular the kinetic operator is a diagonal matrix observables and wave functions in both configuration space and momentum space can also be easily computed with good accuracy using only eigenfunctions computed in the momentum space the method is tested with gaussian and yukawa potentials requiring respectively a small and a large mesh to reach convergence corresponding wave functions in both spaces are compared with each other using the fourier transform

multigrid optimization schemes for solving bose einstein condensate control problems the control of the transport of bose einstein condensates in magnetic microtraps is formulated within the framework of optimal control theory and solved by multigrid optimization schemes the time evolution of the wave function of bose einstein condensates is governed by the gross pitaevskii equation and can be manipulated through variation of a controllable magnetic confinement potential in order to define an optimal control strategy an appropriate cost functional is introduced that must be minimized under the constraint given by the dynamic equation the resulting optimality system consists of two nonlinear schrodinger type equations with opposite time orientation coupled with an elliptic equation for the control function these equations are approximated by using a time splitting pseudospectral method and finite differences to solve the resulting problem a cascadic nonlinear conjugate gradient scheme and a multigrid optimization scheme are considered the convergence properties of these two schemes are investigated theoretically and their computational performance is discussed based on results of numerical experiments it appears that the multigrid optimization scheme provides a robust optimization strategy

consistent force resultant distributions in displacement elements with varying sectional properties force fields computed directly from strains calculated in a displacement type finite element description of a structural element of varying sectional rigidities show extraneous oscillations the origin of these oscillations is traced to the fact that the displacement type finite element procedure determines strains derived from the displacement field in a least squares correct sense and that force resultants computed using these strain fields and the actual sectional rigidities result in unwanted oscillations it is necessary to introduce the concept of redistributed assumed force resultant fields that maintain a consistent relationship13 to the strain fields and also are orthogonal to these strain functions in this paper the hu washizu theorem is invoked to justify the introduction of an orthogonally correct reconstituted assumed force resultant field which will then be free of extraneous oscillations the quadratic isoparametric tapered bar element serves to illustrate the underlying principles 13 it follows that the extremely general hu washizu principle is the most practical procedure of implementing an assumed force resultant assumed strain displacement type formulation to introduce consistency and thereby remove problems associated with field inconsistency such as cause locking in constrained media elasticity and force resultant oscillations due to varying sectional properties

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

v cycle convergence of some multigrid methods for ill posed problems for ill posed linear operator equations we consider some v cycle multigrid approaches that in the framework of bramble pasciak wang and xu 1991 we prove to yield level independent contraction factor estimates consequently we can incorporate these multigrid operators in a full multigrid method that together with a discrepancy principle is shown to act as an iterative regularization method for the underlying infinite dimensional ill posed problem numerical experiments illustrate the theoretical results

lagrange lobatto interpolating polynomials in the discrete variable representation the discrete variable representation dvr is a well known and widely used computational technique in many areas of physics recently the lagrange lobatto basis has attracted increasing attention especially for radial hamiltonians with a singular potential at the origin and finite element dvr constructions however unlike standard dvr functions the lagrange lobatto basis functions are not orthogonal the overlap matrix is usually approximated as the identity using the same quadrature approximation as for the potential based on the special properties of overlap matrix of lagrange lobatto polynomials an explanation of the success of the identity approximation including error bounds is presented results for hydrogen and the more nontrivial potentials of self consistent all electron density functional atomic calculations are also given

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

optimal control formulation for determining optical flow an optimal control formulation for determining optical flow is presented the new framework differs from preceding approaches in that it does not require differentiation of the data and does combine optical flow with image reconstruction it can be considered as a control in the coefficients problem with a cost functional of tracking type a numerical algorithm that solves the optimality system consisting of hyperbolic and elliptic partial differential equations is presented numerical experiments demonstrate the effectiveness of the optimal control approach

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

consistent thermal stress calculation by using a new approach of the enhanced assumed strain method a three field hybrid functional with displacement strain and stress assumptions has been proposed by introducing the enhanced assumed strain concept to the proposed functional a more general finite element formulation for the enhanced assumed strain method has been derived unlike the previous version of the enhanced assumed strain method the stresses can be evaluated from variational formulation within variational framework the present formulation has been applied to the thermal stress problems and it has been addressed that by suitable assumption of the enhanced assumed strains the spurious oscillation of the thermal stresses can be successfully overcome the process of designing the interpolation functions of the enhanced strains which guarantee the field consistency of the strain field is rational and systematic

grid method for computation of generalized spheroidal wave functions based on discrete variable representation we present an efficient and accurate grid method for computations of eigenvalues and eigenfunctions of the generalized spheroidal wave equation different from previous studies our method is based on the expansion of the spheroidal wave function by discrete variable representation basis functions constructed from the associated legendre polynomials the differential operator can be expressed analytically on the grid points which are the zeros of the associated legendre polynomials the resultant potential matrix is simply diagonal and evaluated directly on the same grid the corresponding differential equation is thus converted to an eigenvalue problem of a small matrix whose eigenvalues and eigenvectors are converged very fast the wave functions can then be evaluated accurately at any desired point from the expansion formula with the computed eigenvectors compared to previous methods our method is direct and efficient for any parameter c either small or large

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system the finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered stability of the finite difference optimality system and optimal order error estimates in the discrete l2 norm and in the discrete h1 norm under minimum smoothness requirements on the exact solution are proved sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local fourier analysis a multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system