adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

a framework for variational grid generation conditioning the jacobian matrix with matrix norms functionals for variational grid generation in which the jacobian matrix and matrix norms play a central role are presented all first order geometric qualities of a structured grid length area volume angle between sides aspect ratio grid alignment are determined by the jacobian matrix of the induced mapping between the logical and physical regions grids with desirable quality can be generated by requiring the jacobian matrix or the corresponding metric tensor to have certain properties functionals can be obtained by integrating over the logical or physical domain a power of the norm of a matrix derived from the jacobian matrix a judicious choice of this derived matrix results in a functional whose minima are mappings with desired features many well known grid generators fit in this framework for example the harmonic generator arises when one drives the metric tensor toward the identity the framework provides insight into what grid qualities these functionals describe three new directional control functionals are proposed and tested

efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition a new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented the eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis an efficient method for estimating the coefficients with respect to this basis is proposed the method uses a galerkin based approach by orthogonalizing the residual in the eigenvalue eigenvector equation to the subspace spanned by the basis functions used for approximation in this way the stochastic problem is framed as a system of deterministic non linear algebraic equations this system of equations is solved using a newton raphson algorithm although the proposed approach is not based on statistical sampling the efficiency of the proposed method can be significantly enhanced by initializing the non linear iterative process with a small statistical sample synthesized through a monte carlo sampling scheme the proposed method offers a number of advantages over existing methods based on statistical sampling first it provides an approximation to the complete probabilistic description of the eigensolution second it reduces the computational overhead associated with solving the statistical eigenvalue problem finally it circumvents the dependence of the statistical solution on the quality of the underlying random number generator copyright 2007 john wiley sons ltd

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

a geometric discretization and a simple implementation for variational mesh generation and adaptation we present a simple direct discretization for functionals used in the variational mesh generation and adaptation meshing functionals are discretized on simplicial meshes and the jacobian matrix of the continuous coordinate transformation is approximated by the jacobian matrices of affine mappings between elements the advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional moreover the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form which allows a simple implementation of variational mesh generation and adaptation on computer since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods the result in this work can be used to develop simple implementations of those methods numerical examples are given

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

domain decomposition approaches for mesh generation via the equidistribution principle moving mesh methods based on the equidistribution principle are powerful techniques for the space time adaptive solution of evolution problems solving the resulting coupled system of equations namely the original pde and the mesh pde however is challenging in parallel we propose in this paper several schwarz domain decomposition algorithms for this task we then study in detail the convergence properties of these algorithms applied to the nonlinear mesh pde in one spatial dimension we prove convergence for classical transmission conditions and optimal and optimized variants for the generation of steady equidistributing grids a classical parallel schwarz algorithm is presented and analyzed for the generation of time dependent moving equidistributing grids we conclude our study with numerical experiments

polynomial chaos representation of spatio temporal random fields from experimental measurements two numerical techniques are proposed to construct a polynomial chaos pc representation of an arbitrary second order random vector in the first approach a pc representation is constructed by matching a target joint probability density function pdf based on sequential conditioning a sequence of conditional probability relations in conjunction with the rosenblatt transformation in the second approach the pc representation is obtained by having recourse to the rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target spearman s rank correlation coefficient srcc matrix both techniques are applied to model an experimental spatio temporal data set exhibiting strong non stationary and non gaussian features the data consists of a set of oceanographic temperature records obtained from a shallow water acoustics transmission experiment 1 the measurement data observed over a finite denumerable subset of the indexing set of the random process is treated as a collection of observed samples of a second order random vector that can be treated as a finite dimensional approximation of the original random field a set of properly ordered conditional pdfs that uniquely characterizes the target joint pdf in the first approach and a set of target marginal pdfs and a target srcc matrix in the second approach are estimated from available experimental data digital realizations sampled from the constructed pc representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy the relative advantages and disadvantages of the two proposed techniques are also highlighted

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

anisotropic adaptive simulation of transient flows using discontinuous galerkin methods an anisotropic adaptive analysis procedure based on a discontinuous galerkin finite element discretization and local mesh modification of simplex elements is presented the procedure is applied to transient two and three dimensional problems governed by euler s equation a smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified the mesh metric field in smooth portions of the domain is controlled by a hessian matrix constructed using a variational procedure to calculate the second derivatives the transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain copyright c 2004 john wiley sons ltd

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

an optimization approach to robust nonlinear control design abstract a new procedure of robust control design is presented for nonlinear systems with parametric uncertainty and disturbance this nonlinear optimization based approach can design robust nonlinear controllers of widely different structures and arbitrary complexity by taking the worst case design approach a minimax optimization problem is formulated a number of state of the art optimization procedures are explored as possible candidates for minimaximization it is determined that the tools of non smooth analysis and optimization are the most useful for the efficient solution of practical size worst case design problems included in the paper are the significant details of a special two tier structured minimax programming algorithm implemented in this work the overall algorithm is used in solving two robust nonlinear control design problems the controller designs produced by the minimax optimization algorithm are discussed and the performance of the controllers is studied by simulations

approximate methods for stochastic eigenvalue problems we consider the discretization and solution of eigenvalue problems of elliptic operators with random coefficients for solving the resulting systems of equations we present a new and efficient spectral inverse iteration based on the stochastic galerkin approach with respect to a polynomial chaos basis the curse of dimensionality inherent in normalization over parameter spaces is avoided by a solution of a non linear system of equations defining the galerkin coefficients for reference we also present an algorithm for adaptive stochastic collocation functionality of the algorithms is demonstrated by applying them on four examples of a given model problem convergence of the galerkin based method is analyzed and the results are tested against the collocated reference solutions and theoretical predictions

a new functional for variational mesh generation and adaptation based on equidistribution and alignment conditions abstract a new functional is presented for variational mesh generation and adaptation it is formulated based on combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter the functional is shown to be coercive but not convex a solution procedure using a discrete moving mesh partial differential equation is employed it is shown that the element volumes and altitudes of a mesh trajectory of the mesh equation associated with the new functional are bounded away from zero and the mesh trajectory stays nonsingular if it is so initially numerical examples demonstrate that the new functional performs comparably as an existing one that is also based on the equidistribution and alignment conditions and known to work well but contains an additional parameter

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

jacobian weighted elliptic grid generation variational grid generation techniques are used to derive and analyze a weighted elliptic grid generator that controls the jacobian of the underlying transformation in a least squares sense the euler lagrange equations for the area and volume generators are weighted forms of the well known laplace generator weights are restricted to the class of p matrices to help achieve global invertibility of the map connecting the weights to the jacobian of the map results in an intuitive means of controlling grid spacing area orthogonality and grid line directions examples are given on the unit square to demonstrate point attraction local refinement directional alignment and adaption to a shock

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner