multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

a wavelet algorithm for the boundary element solution of a geodetic boundary value problem abstract in this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a part of the surface of the earth this singular equation is the boundary integral equation corresponding to the oblique derivative boundary problem for laplace s equation we introduce special wavelet bases for the spaces of test and trial functions analogously to well known results on wavelet algorithms the stiffness matrices with respect to these bases can be reduced to sparse matrices such that the assembling of the matrices and the iterative solution of the matrix equations quicken though the theoretical results apply only to integral equations with smooth solutions over smooth manifolds we present numerical tests for a geometry as difficult as the surface of the earth

error estimation and adaptation for functional outputs in time dependent flow problems the paper presents an adjoint based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations the algorithm is derived for the unsteady euler equations that are discretized for second order accuracy in both space and time and takes into account the effect of dynamic meshes in addition to error due to temporal resolution the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time step the resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time step the method is demonstrated in the context of both time integrated and instantaneous functionals and the results are compared against traditional adaptation methods

constructing nested bases approximations from the entries of non local operators in this article a method for constructing nested bases approximations to large scale fully populated discretizations of integral operators is introduced the scheme uses only few of the matrix entries for approximating the whole matrix in this sense it is similar to the adaptive cross approximation method however its computational complexity is improved

approximation of integral operators by green quadrature and nested cross approximation we present a fast algorithm that constructs a data sparse approximation of matrices arising in the context of integral equation methods for elliptic partial differential equations the new algorithm uses green s representation formula in combination with quadrature to obtain a first approximation of the kernel function and then applies nested cross approximation to obtain a more efficient representation the resulting mathcal h 2 h2 matrix representation requires mathcal o n k o nk units of storage for an n times n n n matrix where k depends on the prescribed accuracy

efficient preconditioners for boundary element matrices based on grey box algebraic multigrid methods this paper is concerned with the iterative solution of the boundary element equations arising from standard galerkin boundary clement discretizations of first kind boundary integral operators of positive and negative order we construct efficient preconditioners on the basis of so called grey box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices in particular the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense this auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators finally we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners

accelerated block preconditioned gradient method for large scale wave functions calculations in density functional theory an accelerated block preconditioned gradient abpg method is proposed to solve electronic structure problems in density functional theory this iterative algorithm is designed to solve directly the non linear kohn sham equations for accurate discretization schemes involving a large number of degrees of freedom it makes use of an acceleration scheme similar to what is known as rmm diis in the electronic structure community the method is illustrated with examples of convergence for large scale applications using a finite difference discretization and multigrid preconditioning

a comparative study on methods for convergence acceleration of iterative vector sequences we discuss several methods for accelerating the convergence of the iterative solution of nonlinear equation systems commonly in use and point to interrelations between them in particular we investigate two of the most sophisticated schemes namely the anderson mixing and the broyden update both generalized to the consideration of arbitrarily many previous iterations for the broyden method we give a new derivation which is much simpler than that recently proposed by vanderbilt and louie we show that if the additional parameters invented by these authors in order to increase flexibility are used to optimize the convergence of the iteration process they in fact cancel out in addition we prove that in this optimal case the anderson mixing and the broyden update as applied to the inverse jacobian are fully identical thus we come to the conclusion that neither of these schemes is superior moreover we show that broyden update of the inverse jacobian is superior to updating the jacobian itself finally we propose an extension of the anderson mixing which avoids the numerical difficulties all these methods are faced with

a note on the map with the whole space being a scrambled set abstract in this paper we prove that there is no compact system with the whole space being a schweizer smital scrambled set and the huang ye homeomorphism w huang x d ye homeomorphisms with the whole compacta being scrambled sets ergod theory dynam systems 21 2001 77 91 has no schweizer smital pair moreover we present a simple noncompact system whose schweizer smital scrambled set can be the whole space

fast multiresolution algorithms for matrix vector multiplication in this paper the authors present a class of multiresolution algorithms for fast application of structured dense matrices to arbitrary vectors which includes the fast wavelet transform of beylkin coifman and rokhlin and the multilevel matrix multiplication of brandt and lubrecht in designing these algorithms the authors first apply data compression techniques to the matrix and then show how to compute the desired matrix vector multiplication from the compressed form of the matrix in describing this class special attention is paid to an algorithm that is based on discretization by cell averages as it seems to be suitable for discretization of integral transforms with integrably singular kernels

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

the analysis of multigrid algorithms for pseudodifferential operators of order minus one multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one classical multigrid theory deals with the case of differential operators of positive order the pseudodifferential operator gives rise to a coercive form on h sup minus 1 2 omega effective multigrid algorithms are developed for this problem these algorithms are novel in that they use the inner product on h sup minus 1 omega as a base inner product for the multigrid development the authors show that the resulting rate of iterative convergence can at worst depend linearly on the number of levels in these novel multigrid algorithms in addition it is shown that the convergence rate is independent of the number of levels and unknowns in the case of a pseudodifferential operator defined by a single layer potential finally the results of numerical experiments illustrating the theory are presented 19 refs 1 fig 2 tabs

a fast multipole method for the rotne prager yamakawa tensor and its applications we present a fast multipole method fmm for computing sums involving the rotne prager yamakawa tensor the method similar to the approach in tornberg and greengard 2008 26 for the stokeslet decomposes the tensor vector product into a sum of harmonic potentials and fields induced by four different charge and dipole distributions unlike the approach based on the kernel independent fast multipole method ying et al 2004 31 which requires nine scalar fmm calls the method presented here requires only four we discuss its applications to brownian dynamics simulation with hydrodynamic interactions and present some timing results

temporal error control for convection dominated equations in two space dimensions a new time integration strategy for the solution of convection dominated partial differential equations in two space dimensions by the method of lines is presented the strategy aims to ensure that the time integration error is less than the spatial discretisation error this is achieved by making use of the individual contributions of the local spatial discretisation error and the local time integration error to the global error in the numerical solution numerical results are used to illustrate the performance of this strategy

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

coulomb interactions on planar structures inverting the square root of the laplacian we present an adaptive fast multipole method for inverting the square root of the laplacian in two dimensions solving this problem is the dominant computational cost in many applications arising in electrical engineering geophysical fluid dynamics and the study of thin films it corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad tree the amount of work grows linearly with the number of mesh points

the solution of a launch vehicle trajectory problem by an adaptive finite element method we describe a second order discontinuous galerkin finite element method for the solution of an optimal control problem for determining the trajectory of a launch vehicle we derive an a posteriori error estimate that is subsequently implemented as the basis for adaptive error control we demonstrate that the computational error estimate is reliable and accurate while the adaptive error control provides a significant gain in efficiency over uniform discretizations

iterative residual based vector methods to accelerate fixed point iterations fixed point iterations are still the most common approach to dealing with a variety of numerical problems such as coupled problems multi physics domain decomposition or nonlinear problems electronic structure heat transfer nonlinear mechanics methods to accelerate fixed point iteration convergence or more generally sequence convergence have been extensively studied since the 1960 s for scalar sequences the most popular and efficient acceleration method remains the 2 of aitken various vector acceleration algorithms are available in the literature which often aim at being multi dimensional generalizations of the 2 method in this paper we propose and analyse a generic residual based formulation for accelerating vector sequences the question of the dynamic use of this residual based transformation during the fixed point iterations for obtaining a new accelerated fixed point method is then raised we show that two main classes of such iterative algorithms can be derived and that this approach is generic in that various existing acceleration algorithms for vector sequences are thereby recovered in order to illustrate the interest of such algorithms we apply them in the field of nonlinear mechanics on a simplified point wise solver used to perform mechanical behaviour unit testings the proposed test cases clearly demonstrate that accelerated fixed point iterations based on the elastic operator quasi newton method are very useful when the mechanical behaviour does not provide the so called consistent tangent operator moreover such accelerated algorithms also prove to be competitive with respect to the standard newton raphson algorithm when available

stabilized approximation to degenerate transport equations via filtering abstract we analyze a stabilization technique for degenerate transport equations of particular interest are coupled parabolic hyperbolic problems when the diffusion coefficient is zero in part of the domain the unstabilized computed approximations of these problems are highly oscillatory and several techniques have been proposed and analyzed to mitigate the effects of the sub grid errors that contribute to the oscillatory behavior in this paper we modify a time relaxation algorithm proposed in 1 and further studied in 10 our modification introduces the relaxation operator as a post processing step the operator is not time dependent so the discrete relaxation system need only be factored once we provide convergence analysis for our algorithm along with numerical results for several model problems

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

tracking oscillations in the presence of delay induced essential instability hybrid testing which couples physical experiments and computer simulations bidirectionally and in real time is a promising experimental technique in engineering one fundamental problem of this technique are delays in the coupling between simulation and experiment we discuss this issue for a simple prototype hybrid system a pendulum that is vertically excited by coupling it to a simulated linear mass spring damper system under realistic conditions a small delay in the coupling can give rise to an essential instability the linearisation has infinitely many unstable eigenvalues for arbitrarily small delay this type of instability is impossible to compensate for with any of the standard compensation techniques we introduce an approach based on feedback control and newton iterations that is able to overcome this instability the basic idea behind our approach consists of two parts first we change the bidirectional coupling between experiment and computer simulation to a unidirectional coupling and stabilise the experiment with a feedback loop second we place the modified hybrid system into a newton iteration scheme if the iteration converges then the hybrid experiment behaves just as the original emulated system within the experimental accuracy we demonstrate by using a computer simulation for the experimental part that our approach is able to overcome the essential instability in combination with path following methods it allows us to track oscillations and their bifurcations systematically

adaptive timestepping for elastohydrodynamic lubrication solvers the application of variable step time integration techniques in a method of lines approach for the numerical solution of transient elastohydrodynamic lubrication calculations is described spatial discretization of the coupled integro differential equations leads to a system of differential algebraic equations in time at each timestep a large system of nonlinear equations of 104 107 variables is solved by using multigrid methods and multilevel multi integration the application of temporal error control methods is considered with reference to both the differential algebraic formulation of the system to be solved and the multigrid based nonlinear equation solver to be used the error control method used takes account of the magnitude of the spatial error present by using a local time error control based on the magnitude of this error this approach is shown to be beneficial in reducing the computational work required experimental results on a variety of lubrication problems including a challenging rough surface problem are used to display the effectiveness of the methods

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

a fast high order solver for problems of scattering by heterogeneous bodies a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented here a scatterer is described by a continuously or discontinuously varying refractive index n x within a two dimensional 2d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o nlog n operations where n is the number of discretization points a wide variety of numerical examples provided include applications to highly singular geometries high contrast configurations as well as acoustically electrically large problems for which supercomputing resources have been used recently our method provides highly accurate solutions for such problems on small desktop computers in cpu times of the order of seconds

continuous and discontinuous galerkin time stepping methods for nonlinear initial value problems with application to finite time blow up we consider continuous and discontinuous galerkin time stepping methods of arbitrary order as applied to first order initial value ordinary differential equation problems in real hilbert spaces our only assumption is that the nonlinearities are continuous in particular we include the case of unbounded nonlinear operators specifically we develop new techniques to prove general peano type existence results for discrete solutions in particular our results show that the existence of solutions is independent of the local approximation order and only requires the local time steps to be sufficiently small independent of the polynomial degree the uniqueness of local solutions is addressed as well in addition our theory is applied to finite time blow up problems with nonlinearities of algebraic growth for such problems we develop a time step selection algorithm for the purpose of numerically computing the blow up time and provide a convergence result

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

an adaptive finite element procedure for fully coupled point contact elastohydrodynamic lubrication problems this paper presents an automatic locally adaptive finite element solver for the fully coupled ehl point contact problems the proposed algorithm uses a posteriori error estimation in the stress in order to control adaptivity in both the elasticity and lubrication domains the implementation is based on the fact that the solution of the linear elasticity equation exhibits large variations close to the fluid domain on which the reynolds equation is solved thus the local refinement in such region not only improves the accuracy of the elastic deformation solution significantly but also yield an improved accuracy in the pressure profile due to increase in the spatial resolution of fluid domain thus the improved traction boundary conditions lead to even better approximation of the elastic deformation hence a simple and an effective way to develop an adaptive procedure for the fully coupled ehl problem is to apply the local refinement to the linear elasticity mesh the proposed algorithm also seeks to improve the quality of refined meshes to ensure the best overall accuracy it is shown that the adaptive procedure effectively refines the elements in the region s showing the largest local error in their solution and reduces the overall error with optimal computational cost for a variety of ehl cases specifically the computational cost of proposed adaptive algorithm is shown to be linear with respect to problem size as the number of refinement levels grows

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

a fast multi level convolution boundary element method for transient diffusion problems a new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods bem for transient diffusion this approach which is based upon the multi level multi integration concepts of brandt and lubrecht provides a fast accurate and memory efficient time domain method for this entire class of problems conventional bem approaches result in operation counts of order o n 2 for the discrete time convolution over n time steps here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order o n 3 2 for three two dimensional model problems using the multi level convolution bem memory requirements are also significantly reduced while maintaining the same level of accuracy as the conventional time domain bem approach

on scrambled sets for chaotic functions some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions we answer these questions by showing that a scrambled set can have full measure but cannot be residual although a scrambled set can be second category in every interval we also indicate relationships that exist between chaotic functions and transitive functions

higher order adaptive finite element methods for kohn sham density functional theory we present an efficient computational approach to perform real space electronic structure calculations using an adaptive higher order finite element discretization of kohn sham density functional theory dft to this end we develop an a priori mesh adaption technique to construct a close to optimal finite element discretization of the problem we further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite elements in conjunction with gauss lobatto quadrature and a chebyshev acceleration technique for computing the occupied eigenspace the proposed approach has been observed to provide a staggering 100 200 fold computational advantage over the solution of a generalized eigenvalue problem using the proposed solution procedure we investigate the computational efficiency afforded by higher order finite element discretizations of the kohn sham dft problem our studies suggest that staggering computational savings of the order of 1000 fold relative to linear finite elements can be realized for both all electron and local pseudopotential calculations by using higher order finite element discretizations on all the benchmark systems studied we observe diminishing returns in computational savings beyond the sixth order for accuracies commensurate with chemical accuracy suggesting that the hexic spectral element may be an optimal choice for the finite element discretization of the kohn sham dft problem a comparative study of the computational efficiency of the proposed higher order finite element discretizations suggests that the performance of finite element basis is competing with the plane wave discretization for non periodic local pseudopotential calculations and compares to the gaussian basis for all electron calculations to within an order of magnitude further we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources and good scalability of the present implementation up to 192 processors

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

adjoint sensitivity formulation for discontinuous galerkin discretizations in unsteady inviscid flow problems this paper presents an unsteady discrete adjoint algorithm for high order implicit discontinuous galerkin discretizations in time dependent inviscid flow problems the major function of the adjoint approach is to obtain the sensitivity information in a time dependent functional output which in turn is used to drive an unsteady shape optimization process to deliver a minimum of the objective functional a gradient based optimization strategy is investigated in which the sensitivity derivatives of the objective functional with respect to input variables are formulated in the context of high order discontinuous galerkin discretizations while special emphasis is given to the variations and linearizations of curvilinear boundary elements implicit temporal discretizations consisting of a second order backward euler scheme and a fourth order implicit runge kutta scheme are considered exclusively in this work where the corresponding adjoint problem is required to be solved in a backward time integration manner due to the associated transpose operation two numerical examples for the unsteady shape design techniques are presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjoint approach the first involves an inverse shape optimization case by matching a time dependent target pressure profile for a two dimensional periodic vortical gust impinging on a rae 2822 airfoil and the second considers minimization of the acoustic noise produced by subsonic flow over a naca0012 airfoil with a 0 03c thick blunt trailing edge

nonlinear krylov acceleration applied to a discrete ordinates formulation of the k eigenvalue problem we compare a variant of anderson mixing with the jacobian free newton krylov and broyden methods applied to an instance of the k eigenvalue formulation of the linear boltzmann transport equation we present evidence that one variant of anderson mixing finds solutions in the fewest number of iterations we examine and strengthen theoretical results of anderson mixing applied to linear problems

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

adaptive finite element methods for parabolic problems v long time integration we continue our previous work on adaptive finite element methods for parabolic problems now with particular emphasis on long time integration for semidefinite problems

a multi iterate method to solve systems of nonlinear equations we propose an extension of secant methods for nonlinear equations using a population of previous iterates contrarily to classical secant methods where exact interpolation is used we prefer a least squares approach to calibrate the linear model we propose an explicit control of the numerical stability of the method we show that our approach can lead to an update formula in that case we prove the local convergence of the corresponding undamped quasi newton method finally computational comparisons with classical quasi newton methods highlight a significant improvement in terms of robustness and number of function evaluations we also present numerical tests showing the robust behavior of our method in the presence of noise

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

towards efficient backward in time adjoint computations using data compression techniques abstract in the context of a posteriori error estimation for nonlinear time dependent partial differential equations the state of the practice is to use adjoint approaches which require the solution of a backward in time problem defined by a linearization of the forward problem one of the major obstacles in the practical application of these approaches is the need to store or recompute the forward solution to define the adjoint problem and to evaluate the error representation this study considers the use of data compression techniques to approximate forward solutions employed in the backward in time integration the development derives an error representation that accounts for the difference between the standard approach and the compressed approximation of the forward solution this representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data compressed reconstructed solution i e scalar quantities that can be evaluated as the forward problem is integrated this approach is then compared with existing techniques such as checkpointing and time averaged adjoints finally we provide numerical results indicating the potential efficiency of our approach on a transient diffusion reaction equation and on the navier stokes equations these results demonstrate memory compression ratios up to 450 while maintaining reasonable accuracy in the error estimates

a chaotic function with a distributively scrambled set of full lebesgue measure abstract misiurewicz proved that there exists a continuous map of the interval 0 1 onto itself for which there exists a scrambled set of full lebesgue measure in this paper we form a continuous interval map which has a distributively scrambled set of full lebesgue measure in which each point has dense orbit this contains misiurewicz s result since any distributively scrambled set must be scrambled but the converse is not generally true

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

approximation of integral operators by variable order interpolation we employ a data sparse recursive matrix representation so called matrices for the efficient treatment of discretized integral operators we obtain this format using local tensor product interpolants of the kernel function and replacing high order approximations with piecewise lower order ones the scheme has optimal i e linear complexity in the memory requirement and time for the matrix vector multiplication we present an error analysis for integral operators of order zero in particular we show that the optimal convergence h is retained for the classical double layer potential discretized with piecewise constant functions

a fast algorithm for modeling multiple bubbles dynamics this work presents the development of a numerical strategy that combines the fast fourier transform on multipoles fftm method and the boundary element method bem to study the physics of multiple bubbles dynamics in moving boundary problems the recent fftm method can be employed to speedup the resolution of the boundary integral equation however one major drawback of the method is that its efficiency deteriorates quite significantly when the problem is spatially sparsely populated as in the case where multiple bubbles are well separated to overcome this deficiency a new version of fftm with clustering is proposed henceforth called fftm clustering the new algorithm first identifies and groups closely positioned bubbles the original fftm is then used to compute the potential contributions from the bubbles within its own group while contributions from the other separated groups are evaluated via the multipole to local expansions translations operations directly we tested the fftm clustering on several multiple bubble examples to demonstrate its effectiveness over the original fftm method and vast improvement over the standard bem the high efficiency of the fftm clustering method allows us to simulate much larger multiple bubbles dynamics problems within reasonable time some physical behaviors of the multiple bubbles are also presented in this work

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

a note on the grid movement induced by mfe moving grid methods in one space dimension have become popular for solving several kinds of parabolic and hyperbolic partial differential equations pdes involving fine scale structures such as steep moving fronts and emerging steep layers pulses shocks etc in two space dimensions however application of moving grid methods is less trivial than in 1d for some methods e g those based on equidistributing principles it is not even clear how to extend the underlying grid selection procedure to 2d the moving finite element mfe method does not suffer from this drawback its mathematical extension to 2d is trivial however because of the intrinsic coupling between the discretization of the pde and the grid selection the application of mfe as for any other method is not without difficulties in this paper we describe the node movement induced by mfe for various pdes and we indicate some problems concerning the grid structure that can result from this movement

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

wavelet based model for stochastic analysis of beam structures a wavelet based model for stochastic analysis of beam structures is presented in this model the random processes representing the stochastic material and geometric properties are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the process is represented as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients the correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets and the integral eigenvalue problem is converted to a finite dimensional eigenvalue problem the energy principle based finite element approach is used to obtain the equilibrium and boundary conditions neumann expansion of the stiffness matrix is used to write the nodal displacement vector in terms of random coefficients the expectation operator is applied to the nodal displacements and their squares to obtain the mean and standard deviation of the displacements studies show that the results obtained using this method compare well with monte carlo and semianalytical techniques

a posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem we consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary we derive accurate a posteriori error estimates that account for the transfer of error between components of the operator decomposition method as well as the differences between the adjoints of the full problem and the discrete iterative system we use these estimates to guide adaptive mesh refinement in addition we address a loss of order of convergence that results from the decomposition and show that the order of convergence is limited by the accuracy of the transferred gradient information we employ a boundary flux recovery method to regain the expected order of accuracy in an efficient manner

accuracy of three dimensional analysis of regularized singularities summary in computational mechanics the quadrature of discontinuous and singular functions is often required to avoid specialized quadrature procedures discontinuous and singular fields can be regularized however regularization changes the algebraic structure of the solving equations and this can lead to high errors we show how to acquire accurate and consistent results when regularization is carried out a three dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method the accuracy obtained via gaussian quadrature is compared with that obtained by means of cubpack adaptive quadrature fortran tool the use of regularized functions with non compact and compact support is investigated through an error evaluation procedure based on the use of their fourier transform the proposed procedure leads to the remarkable conclusion that regularized delta functions with non compact support exhibit superior performance copyright 2014 john wiley sons ltd

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

must the multilevel sinc transform a fast multilevel algorithm must for evaluating an n sample sinc interpolant at mn points is presented for uniform grids its complexity is 25mn log 1 delta flops for the sinc kernel and 75mn log 1 delta for the sincd kernel where delta is the target evaluation accuracy must is faster than fft and fmm based evaluations for large n and or for large delta it is also applicable to nonuniform grids and to other kernels numerical experiments demonstrating the algorithm s practicality are presented

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

a fractional step θ method for convection diffusion problems abstract in this article we analyze the fractional step method for the time dependent convection diffusion equation in our implementation we completely separate the convection operator from the diffusion operator and stabilize the convective problem using a streamline upwinded petrov galerkin supg method we establish a priori error estimates and show that the optimal value of yields a scheme that is second order in time numerical computations are presented which demonstrate the method and support the theoretical results

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions

a posteriori analysis of an iterative multi discretization method for reaction diffusion systems abstract this paper is concerned with the accurate computational error estimation of numerical solutions of multi scale multi physics systems of reaction diffusion equations such systems can present significantly different temporal and spatial scales within the components of the model indicating the use of independent discretizations for different components however multi discretization can have significant effects on accuracy and stability we perform an adjoint based analysis to derive asymptotically accurate a posteriori error estimates for a user defined quantity of interest these estimates account for leading order contributions to the error arising from numerical solution of each component an error due to incomplete iteration an error due to linearization and for errors arising due to the projection of solution components between different spatial meshes several numerical examples with various settings are given to demonstrate the performance of the error estimators