numerical techniques for the transformation to an orthogonal coordinate system aligned with a vector field the authors explore the use of variational grid generation to perform alignment of a grid with a given vector field variational methods have proven to be a powerful class of grid generators but when they are used in alignment difficulties may arise in treating boundaries due to an incompatibility between geometry and vector field in this paper a refinement of the procedure of iterating boundary values is presented it allows one to control the quality of the grid in the face of the above mentioned incompatibility this procedure may be incorporated into any variational alignment algorithm the authors demonstrate its use with respect to a new quasi variational alignment method having a particularly simple structure the latter method is comparable to knupp s method see 7 but avoids use of the winslow equations

mesh generation using vector fields abstract a method for generating structured meshes from vector fields is described the method is based on a weighted variational principle that minimizes a least squares fit to the inverse jacobian matrix the weight is constructed from the vector fields by requiring local alignment of the mesh with the fields and by prescribing local length scales the resulting grid generator is a weighted form of the laplace grid generation equations the method is demonstrated in 2d using a subsurface flow field derived from hydrologic simulations

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

coupling bem with fem by a direct computation of the boundary stiffness matrix the boundary stiffness matrix characterizing in a discretized form the response of the boundary to given dirichlet boundary conditions allows the coupling between bem and fem the paper presents an approach based on the direct computation of the stiffness matrix from the potential function related to a given dirichlet boundary condition the method produces a symmetric stiffness matrix as for the singular galerkin boundary element method but does not need to compute hypersingular integrals in addition the method uses only the nodal values of the boundary potential but does not need discretized values of the normal gradient at the boundary as for usual boundary element methods which reduces the number of matrices to be constructed the regularization of the integrals is achieved by means used for simply singular integrals introduced for the collocation boundary element method an example of application is produced in the case of the solution of laplace equation within a plane domain

an algorithm for aligning a quadrilateral grid with internal boundaries we present a new numerical algorithm that aligns a quadrilateral grid with internal alignment curves iacs these iacs can be used to delineate internal interfaces discontinuities in material properties internal boundaries or major features of a flow field the adaptive iac grid generation algorithm readjusts a predefined reference grid to create a nearby grid where the mesh cell edges are aligned with the iacs on an aligned grid numerical discretizations of partial differential equations can be formulated to satisfy the interfacial relations such as matching fluxes across the discontinuity to reduce the numerical errors introduced by the discontinuity we present examples to demonstrate the effectiveness of the iac grid generation algorithm for multiple embedded interfaces in a quadrilateral grid

second order absorbing boundary conditions for the wave equation a solution for the corner problem the treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view a technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered a solution is proposed for the two dimensional case when second order conditions are used this solution consists of prescribing an adequate corner condition the problem thus obtained is analyzed theoretically and the condition is proved to be optimal the results obtained here are illustrated by numerical simulations some extensions to higher space dimensions and higher order conditions are proposed

general edge asymptotics of solutions of second order elliptic boundary value problems i this is the first of two papers in which we study the singularities of solutions of second order linear elliptic boundary value problems at the edges of piecewise analytic domains in 3 when the opening angle at the edge is variable there appears the phenomenon of crossing of the exponents of singularities for this case we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted sobolev spaces for the regular part of the solution and for the coefficients of the singularities these combinations appear in a natural way as sections of an analytic bundle above the edge their behaviour is described with the help of divided differences of powers of the distance to the edge the class of operators considered includes second order elliptic operators with analytic complex valued coefficients with mixed dirichlet neumann or oblique derivative conditions with our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works in this first part we prove the basic facts in a simplified framework nevertheless the tools we use are essentially the same in the general situation

variational mesh adaptation methods for axisymmetrical problems we study variational mesh adaptation for axially symmetric solutions to two dimensional problems the study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy or regularity and global equidistribution of the monitor function the main results are stated in theorems ref thm4 1 and ref thm4 2 for axially symmetric problems it is natural to choose axially symmetric mesh adaptation to this end it is reasonable to use the monitor function in the form g lambda 1 r mbox boldmath e r mbox boldmath e r t lambda 2 r mbox boldmath e theta mbox boldmath e theta t where mbox boldmath e r and mbox boldmath e theta are the radial and angular unit vectors it is shown that when higher mesh concentration at the origin is desired a choice of lambda 1 and lambda 2 satisfying lambda 1 0 0 by choosing lambda 1 and lambda 2 in contrast numerical results show that the new functional provides better control of the mesh concentration through the monitor function two dimensional numerical results are presented to support the analysis

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

the eigenvalue problem for hermitian matrices with time reversal symmetry this paper describes a computational method for dealing with a class of matrices which arise in quantum mechanics involving time reversal and inversion symmetry the algorithms presented here have greatly reduced the computational effort required to solve this problem and also produce a stable more accurate solution

canonical decompositions of n qubit quantum computations and concurrence the two qubit canonical decomposition su 4 su 2 su 2 su 2 su 2 writes any two qubit unitary operator as a composition of a local unitary a relative phasing of bell states and a second local unitary using lie theory we generalize this to an n qubit decomposition the concurrence canonical decomposition ccd su 2n kak the group k fixes a bilinear form related to the concurrence and in particular any unitary in k preserves the tangle i 1y i ny 2 for n even thus the ccd shows that any n qubit unitary is a composition of a unitary operator preserving this n tangle a unitary operator in a which applies relative phases to a set of ghz states and a second unitary operator which preserves the tangle as an application we study the extent to which a large random unitary may change concurrence the result states that for a randomly chosen a a su 22p the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approach

multigrid convergence for discretizations of singular perturbation problems with grid aligned flow we use fourier analysis to examine multigrid convergence for a variety of discretizations of the two dimensional convection diffusion equation with grid aligned flow velocity emphasis is placed on the problem with a small diffusion coefficient we consider three distinct discretizations of the problem centered differencing standard upwind differencing and a generalization to two dimensions of the el mistikawy and werle discretization emw the analysis covers the effectiveness of block jacobi relaxation for the problem and stability as the diffusion coefficient epsilon to 0 of the discretizations in stark contrast to the problem with general non grid aligned flow we obtain v cycle rates independent of meshwidth h for central differencing as well as for the other discretizations a striking similarity is found between tandard upwinding and the second order accurate generalization of emw the latter is also found to be superior to central differencing in terms of stability superior multigr

monotones and invariants for multi particle quantum states we introduce new entanglement monotones which generalize to the case of many parties those which give rise to the majorization based partial ordering of bipartite states entanglement we give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication these include restrictions which do not follow from any bipartite considerations we derive supermultiplicativity relations between each state s monotones and the monotones for collective processing when the parties share several states we also investigate polynomial invariants under local unitary transformations and show that a large class of these are invariant under collective unitary processing and also multiplicative putting restrictions for example on the exact conversion of multiple copies of one state to multiple copies of another

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

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

r adaptive mesh generation for shell finite element analysis an r adaptive method or moving grid technique relocates a grid so that it becomes concentrated in the desired region this concentration improves the accuracy and efficiency of finite element solutions we apply the r adaptive method to computational mesh of shell surfaces which is initially regular and uniform the r adaptive method given by liao and anderson appl anal 44 1992 285 aggregate the grid in the region with a relatively high weight function without any grid tangling the stress error estimator is calculated in the initial uniform mesh for a weight function however since the r adaptive method is a method that moves the grid shell surface geometry error such as curvature error and mesh distortion error will increase therefore to represent the exact geometry of a shell surface and to prevent surface geometric errors we use the naghdi s shell theory and express the shell surface by a b spline patch in addition using a nine node element which is relatively less sensitive to mesh distortion we try to diminish mesh distortion error in the application of an r adaptive method in the numerical examples it is shown that the values of the error estimator for a cylinder hemisphere and torus in the overall domain can be reduced effectively by using the mesh generated by the r adaptive method also the reductions of the estimated relative errors are demonstrated in the numerical examples in particular a new functional is proposed to construct an adjusted mesh configuration by considering a mesh distortion measure as well as the stress error function the proposed weight function provides a reliable mesh adaptation method after a parameter value in the weight function is properly chosen

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

absorbing boundary conditions for diffusion equations summary we construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients curved artifical boundary and arbitrary convection it relies on the geometric identification of the dirichlet to neumann map and rational interpolation of z 1 2 in the complex plane the boundary conditions are stable accurate and practical for computations

time reversal and n qubit canonical decompositions on pure states of n quantum bits the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin flip the monotone vanishes for n odd but for n even there is an explicit formula for its value on mixed states i e a closed form expression computes the minimum over all ensemble decompositions of a given density for n even a matrix decomposition k1ak2 of the unitary group is explicitly computable and allows for study of the monotone s dynamics the side factors k1 and k2 of this concurrence canonical decomposition ccd are concurrence symmetries so the dynamics reduce to consideration of the a factor this unitary a phases a basis of entangled states and the concurrence dynamics of u are determined by these relative phases in this work we provide an explicit numerical algorithm computing k1ak2 for n odd further in the odd case we lift the monotone to a two argument function the concurrence capacity of according to the double argument lift may be n

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

dirichlet to neumann map for domains with corners and approximate boundary conditions we present in this paper the dirichlet to neumann operator for the wave equation on a straight wedge in r 2 using fourier integral operators as a consequence we recover the classical approximate boundary conditions of orders 1 and 2

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

quantum symmetries and cartan decompositions in arbitrary dimensions decompositions of lie groups are used in systems and control both to analyse dynamics and to design control algorithms for systems with state varying on a lie group in this paper we investigate the relation between cartan decompositions of the unitary group and discrete quantum symmetries to every cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice as an application we describe a new and general method to obtain cartan decompositions of the unitary group of evolutions of multipartite systems from cartan decompositions on the single subsystems the resulting decomposition which we call of the odd even type contains as a special case the concurrence canonical decomposition ccd presented in 6 8 in the context of entanglement theory the ccd is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimensions we present an example of application of the results to control design for quantum systems

model identification for spin networks abstract we consider the problem of determining the unknown parameters of the hamiltonian of a network of spin 1 2 particles in particular we study experiments in which the system is driven by an externally applied electro magnetic field and the expectation value of the total magnetization is measured under appropriate system theoretic assumptions we prove that if it is possible to prepare the system in a known initial state the above experiment allows to identify the parameters of the hamiltonian in the case where the initial state is itself an unknown parameter we characterize all the pairs hamiltonian initial state which give the same value of the magnetization for every form of the driving electro magnetic field the analysis is motivated by recent results on the isospectrality of hamiltonians describing magnetic molecules

a practical guide to direct optimization for planar grid generation abstract in this paper we provide a guide to using the direct optimization formulation of variational grid generation particular emphasis is placed on the smoothness or length functional this is undoubtedly the most important functional in variational grid generation producing smooth grids and ensuring well posedness of the minimization problem when combined with more unruly functionals unfortunately in its most primary form length can produce folded grids when used with nonconvex geometries historically there have been two solutions to this dilemma using an inverse mapping the famous winslow generator or augmenting the functional with others that promote unicity both strategies have the drawback that the resulting minimization problem becomes complicated and expensive as another alternative we introduce a generalized strategy for length which does not use inverse mappings or auxiliary functionals but makes strong use of reference grids this strategy provides flexibility in controlling grid quality and its minimization problems can be solved using a simple multigrid algorithm yielding a robust grid generation scheme with optimal complexity we also survey recent developments in the use of variational grid generation in the form of direct optimization for the alignment problem