computing normal form perfect equilibria for extensive two person games this paper presents an algorithm for computing an equilibrium of an extensive two person game with perfect recall the method is computationally efficient by virtue of using the sequence form whose size is proportional to the size of the game tree the equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector if the starting vector represents a pair of completely mixed strategies then the equilibrium is normal form perfect computational experiments compare the sequence form and the reduced normal form and show that only the sequence form is tractable for larger games

convergent adaptive finite elements for the nonlinear laplacian the numerical solution of the homogeneous dirichlet problem for the p laplacian p in 1 infty is considered we propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one while the algorithm is a rather straight forward generalization of those for the linear case p 2 the proof of its convergence is different in particular it does not rely on a strict error reduction

finite element approximations of nonlinear eigenvalue problems in quantum physics in this paper we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics we derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations in particular we analyze the convergence and complexity of an adaptive finite element method in our analysis we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations we also present several numerical examples in quantum physics that support our theory

a posteriori and a priori error analysis for finite element approximations of self adjoint elliptic eigenvalue problems we present a new error analysis for finite element approximations of self adjoint elliptic eigenvalue problems the analysis consists of three steps first we prove a posteriori estimates for the error in the approximate eigenvectors and eigenvalues the error in the eigenvectors is measured both in the l2 and energy norms in these estimates the error is bounded in terms of the mesh size a stability factor and the residual obtained by inserting the approximate eigenpair into the differential equation the stability factor describes relevant stability properties of the continuous problem and we give a precise estimate of its size in terms of the spectrum of the continuous problem the mesh size and the choice of norm next we prove an a priori estimate of the residual in terms of derivatives of the exact eigenvectors and the mesh size finally we obtain precise a priori error estimates by combination of the a posteriori error estimates with the a priori residual estimate the analysis shows that the a posteriori estimates are optimal and may be used for quantitative error estimation and design of adaptive algorithms

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

gradient type optimization methods for electronic structure calculations the density functional theory dft in electronic structure calculations can be formulated as either a nonlinear eigenvalue or a direct minimization problem the most widely used approach for solving the former is the so called self consistent field scf iteration a common observation is that the convergence of scf is not clear theoretically while approaches with convergence guarantees for solving the latter are often not competitive with scf numerically in this paper we study gradient type methods for solving the direct minimization problem by constructing new iterations along the gradient on the stiefel manifold global convergence rate i e convergence to a stationary point from any initial solution as well as local convergence rate can be obtained from the standard theory for gradient methods on manifold directly a major computational advantage is that the computation of linear eigenvalue problems is no longer needed the main costs of our approaches arise from the assembling of the total ene

feedback and adaptive finite element solution of one dimensional boundary value problems this paper examines the concepts of feedback and adaptivity for the finite element method the model problem concernsc 0 elements of arbitrary fixed degree for a one dimensional two point boundary value problem three different feedback methods are introduced and a detailed analysis of their adaptivity is given

l stability of finite element approximations to elliptic gradient equations we examine thel stability of piecewise linear finite element approximationsu to the solutionu to elliptic gradient equations of the form a x u f x u g x wheref is monotonically increasing inu we identify a prioril bounds for the finite element solutionu which we call reduced bounds and which are marginally weaker than those for the original differential equations for the general n dimensionai case we identify new conditions on the mesh such that under the assumption thatf is lipschitz continuous on a finite interval u satisfies the reduced l bounds mentioned above the new n dimensional regularity conditions preclude quasi rectangular meshes moreover we show thatu is stable inl in two dimensions for a discretization mesh on which a x u gives rise to anm matrix whileu is stable for any mesh in one dimension the condition that the discretization of a x u has to be anm matrix still allows the inclusion of the important case of triangulating in a quasi rectangular fashion the results are valid for either the pure neumann problem or the general mixed dirichlet neumann boundary value problem while interfaces may be present the boundary conditions foru are obtained by use of nonexpansive pointwise projection operators

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

ramanujan s lost notebook iv stacks and alternating parity in partitions this paper is centered around the partition function oe n the number of partitions of n in which the parts arranged in ascending order alternate in parity starting with the smallest part odd briefly oe n counts the number of odd even partitions of n the first few examples for the computation of oe n are n relevant partitions of n oe n 1 1 1 2 0 3 3 1 2 2 4 0 5 5 1 4 2 6 1 2 3 1 7 7 1 6 3 4 3 the generating function for oe n turns out to be

convergence and optimal complexity of adaptive finite element eigenvalue computations in this paper an adaptive finite element method for elliptic eigenvalue problems is studied both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved the analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper

a multilevel correction adaptive finite element method for kohn sham equation in this paper an adaptive finite element method is proposed for solving kohn sham equation with the multilevel correction technique in the method the kohn sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes a main feature of the method is that solving large scale kohn sham system is avoided effectively and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method hence the significant acceleration can be obtained on solving kohn sham equation with the proposed multilevel correction technique the performance of the method is examined by a variety of numerical experiments

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

the finite element approximation of the nonlinear poisson boltzmann equation a widely used electrostatics model in the biomolecular modeling community the nonlinear poisson boltzmann equation along with its finite element approximation are analyzed in this paper a regularized poisson boltzmann equation is introduced as an auxiliary problem making it possible to study the original nonlinear equation with delta distribution sources a priori error estimates for the finite element approximation are obtained for the regularized poisson boltzmann equation based on certain quasi uniform grids in two and three dimensions adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge the poisson boltzmann equation does not appear to have been previously studied in detail theoretically and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the poisson boltzmann equation note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear poisson boltzmann equation with delta distribution sources and it also introduces the first provably convergent adaptive method for the equation this last result is currently one of only a handful of existing convergence results of this type for nonlinear problems

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

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

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

higher order adaptive finite element methods for orbital free density functional theory in the present work we study various numerical aspects of higher order finite element discretizations of the non linear saddle point formulation of orbital free density functional theory we first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem we find that a staggered solution procedure where the potential fields are computed consistently for every trial electron density is a robust solution procedure for higher order finite element discretizations we next study the convergence properties of higher order finite element discretizations of orbital free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as coulomb singular potential fields our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite element approximations considered in the present study we finally investigate the computational efficiency afforded by various higher order finite element discretizations which constitutes the main aspect of the present work by measuring the cpu time for the solution of discrete equations on benchmark problems that include large aluminum clusters in these studies we use mesh coarse graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far field electronic fields our studies reveal a significant 100 1000 fold computational savings afforded by the use of higher order finite element discretization alongside providing the desired chemical accuracy we consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite element basis

cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach abstract in this paper a type of cascadic adaptive finite element method for nonlinear eigenvalue problem is proposed based on the complementary approach in this new scheme we only need to do some smoothing steps for the linearized boundary value problems on a series of adaptive finite element spaces and solve some nonlinear eigenvalue problems on a low dimensional space hence the efficiency can be improved since we do not need to solve the nonlinear eigenvalue problems on each adaptive space directly besides the complementary error estimate for nonlinear eigenvalue problem will be introduced in our paper this estimate can not only provide an accurate error estimate for the nonlinear eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions