the algebraic entropy of classical mechanics we describe the lie algebra of classical mechanics modelled on the lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical poisson bracket it is a polynomially graded lie algebra a class we introduce we describe these lie algebras give an algorithm to calculate the dimensions cn of the homogeneous subspaces of the lie algebra of classical mechanics and determine the value of its entropy limn cn1 n it is 1 825 423 774 201 08 a fundamental constant associated with classical mechanics

universal equivalence of partially commutative metabelian lie algebras abstract in this paper we find a criterion for universal equivalence of partially commutative lie algebras whose defining graphs are trees besides we obtain bases for partially commutative metabelian lie algebras

a filtered lanczos procedure for extreme and interior eigenvalue problems when combined with krylov projection methods polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices this general approach can be quite efficient in the situation when a large number of eigenvalues is sought however its competitiveness depends critically on a good implementation this paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric or complex hermitian matrix the technique harnesses the effectiveness of the lanczos algorithm with partial reorthogonalization and the power of polynomial filtering numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed

towards idempotent reduced density matrices via particle hole duality mcweeny s purification and beyond generalizations of mcweeny s purification formula are developed within the formalism of the particle hole duality from the theory of reduced density matrices each of the generalized purification formulas is expressed as a sum of the one particle reduced density matrix 1 rdm and a finite series in the product of the one particle and the one hole rdms a product which vanishes in the limit that the 1 rdm is idempotent two categories of purification formulas are explored i formulas which treat the occupied and the virtual occupation numbers equivalently and ii formulas which treat these occupation numbers differently the latter category includes and extends the purification formulas derived in the context of the 1 2 contracted schrodinger equation d a mazziotti j chem phys 115 8305 2001 while the mcweeny purification minimizes the absolute error in the occupation numbers quadratically the generalized purification formulas exhibit faster than quadratic convergence of the 1 rdm towards idempotency application of these purification formulas in existing algorithms for linear scaling will be explored and discussed including illustrative calculations on sodium wires of length 10 20 30 and 40 atoms

a trace minimization algorithm for the generalized eigenvalue problem an algorithm for computing a few of the smallest or largest eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem ax lambda bx is presented the matrices a and b are assumed to be symmetric and haphazardly sparse with b being positive definite the problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace the rate of convergence of the method is established and a technique for improving it is discussed numerical experiments and comparisons with other methods are presented

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

a proximal gradient method for ensemble density functional theory the ensemble density functional theory e dft is valuable for simulations of metallic systems due to the absence of a gap in the spectrum of the hamiltonian matrices although the widely used self consistent field scf iteration method can be extended to solve the minimization of the total energy functional with respect to orthogonality constraints there is no theoretical guarantee on the convergence of these algorithms in this paper we consider an equivalent model with a single variable and a single spherical constraint by eliminating the dependence on the fractional occupancies a proximal gradient method is developed by keeping the entropy term but linearizing all other terms in the total energy functional convergence to the stationary point is established numerical results using the matlab toolbox kssolv can outperform scf consistently on many metallic systems

free partially commutative structures abstract in this paper we study algebraic structures defined by a presentation of the form a ab ba for a b i we show that these relators are the only one that can be interpreted as both monoid and lie relators we also study the free partially commutative monoids lie algebras and groups in all of these cases we show how to obtain decomposition results for these structures into the corresponding free structures

generation of a section of conjugation classes and lyndon word tree of limited length resume etant donne un alphabet ordonne un mot de lyndon est un mot primitif qui est le plus petit dans sa classe de conjugaison nous donnons ici un algorithme qui construit dans l ordre la liste des mots de lyndon de longueur donnee cet algorithme est optimal en ce sens que le calcul du mot de lyndon suivant de la liste se fait en temps lineaire a partir du mot precedent et sans utilisation d une memoire auxiliaire cet algorithme trouve son application dans la construction d une section des classes de conjugaison d une longueur donnee c est une response a une question posee a l auter par m p schutzenberger

a local level set method for 3d inversion of gravity gradient data abstractwe have developed a local level set method for inverting 3d gravity gradient data to alleviate the inherent nonuniqueness of the inverse gradiometry problem we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain d so that we may convert the original inverse problem into a domain inverse problem because the unknown domain d may take a variety of shapes we parametrized the domain d by a level set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level set function because the convergence of the level set algorithm relied heavily on initializing the level set function to enclose the gravity center of a source body we applied a weighted l1 regularization method to locate such a gravity center so that the level set function can be properly initialized to rapidly compute the gradient of the nonlinear functional arising in the level

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

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

sparse dynamics for partial differential equations we investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis the restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation by reducing or compressing the information needed to represent the solution at every step only the essential dynamics are represented in many cases there are natural bases derived from the differential equations which promote sparsity we find that our method successfully reduces the dynamics of convection equations diffusion equations weak shocks and vorticity equations with high frequency source terms

on the partially commutative shuffle product abstract we give recursive formulas that allow us to compute the partially commutative shuffle product of two partially commutative words

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

integrity of formal power series on a partially commutative alphabet resume nous demontrons que l anneau de series formelles sur un alphabet partiellement commutatif est integre

free lie algebras generalized witt formula and the denominator identity abstract let be a countable abelian semigroup satisfying a suitable finiteness condition and let l l be the free lie algebra generated by a graded vector space v over c in this paper from the denominator identity we derive a dimension formula for the homogeneous subspaces of the free lie algebra l l and discuss numerous applications of our dimension formula to various interesting cases our dimension formula will be expressed in terms of the witt partition functions various expressions of the witt partition functions will give rise to a number of interesting combinatorial identities as a special case we obtain a recursive relation for the coefficients of the elliptic modular function j

the critical forms of the attractors for semigroups and the existence of critical attractors the critical forms of the minimal closed global attractor m and the minimal closed global b attractor m for operator semigroups are given i e if the set x x x x attracts each point x in the phase space x then it must be the unique minimal closed global attractor m and if the set b b x x attracts each bounded set in x then it must be the unique minimal closed global b attractor m on this basis the sufficient conditions for the existence of m and m interrelation between m and m are discussed we introduce some classes of semigroups and study a variety of relations of them and the different classes of semigroups considered by ladyzhenskaya hale and temam when a semigroup9s continuity is not required it is proved that the semigroups with b acp introduced by us coincide with the asymptotically smooth semigroups the semigroups of class ak or b ak respectively and that if these semigroups are b bounded and x x x b then each of them possesses the unique non empty minimal invariant compact global attractor x x x x and the unique non empty minimal invariant compact global b attractor m b b x x m omega o where 0is a small number furthermore the sufficient conditions for m and m are given it is further proved that these conditions which guarantee the existence of the compact invariant global b attractor m are not only sufficient but also necessary so in order to guarantee a semigroup to have such attractor m the conditions which are put on the semigroup cannot be weakened again these results will be useful in studying the existence and construction of attractors

a global attractor for a p x laplacian parabolic problem abstract in this work we study the asymptotic behavior of a p x laplacian parabolic problem of the form u t p x u b u in a bounded smooth domain in r n under dirichlet boundary conditions we prove under suitable assumptions the existence of a compact invariant global b attractor in l 2

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

semilinear limit problems for reaction diffusion equations with variable exponents abstract in this work we study pde limit problems for nonlinear reaction diffusion equations and we study the sensitivity of nonlinear pdes with respect to initial conditions and exponent parameters moreover we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction diffusion equations with spatially variable exponents when the exponents go to 2 in l

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

unconstrained energy functionals for electronic structure calculations the performance of conjugate gradient schemes for minimizing unconstrained energy functionals in the context of condensed matter electronic structure density functional calculations is studied the unconstrained functionals allow a straightforward application of conjugate gradients by removing the explicit orthonormality constraints on the quantum mechanical wave functions however the removal of the constraints can lead to slow convergence in particular when preconditioning is used the convergence properties of two previously suggested energy functionals are analyzed and a new functional is proposed which unifies some of the advantages of the other functionals a numerical example derived from a diamond crystal confirms the analysis

numerical methods for electronic structure calculations of materials the goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems the paper is intended for a diverse scientific computing audience for this reason we assume the reader does not have an extensive background in the related physics our overview focuses on the nature of the numerical problems to be solved their origin and the methods used to solve the resulting linear algebra or nonlinear optimization problems it is common knowledge that the behavior of matter at the nanoscale is in principle entirely determined by the schrodinger equation in practice this equation in its original form is not tractable successful but approximate versions of this equation which allow one to study nontrivial systems took about five or six decades to develop in particular the last two decades saw a flurry of activity in developing effective software one of the main practical variants of the schrodinger equation is based on what is referred to as density functional theory dft the combination of dft with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials this article will emphasize pseudopotential density functional theory but other techniques will be discussed as well

the entropy of graded algebras we define the entropy of a graded algebra a n an by limsupn diman this is related to the notion of entropy in symbolic dynamics and could serve as a natural dimension concept for algebras with exponential growth we study the entropy of quotients and subalgebras of free associative algebras and free lie algebras we also study the behavior of the entropy function under free products and obtain several characterizations of free algebras in terms of entropy

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

a constrained optimization algorithm for total energy minimization in electronic structure calculations a new direct constrained optimization algorithm for minimizing the kohn sham ks total energy functional is presented in this paper the key ingredients of this algorithm involve projecting the total energy functional into a sequence of subspaces of small dimensions and seeking the minimizer of total energy functional within each subspace the minimizer of a subspace energy functional not only provides a search direction along which the ks total energy functional decreases but also gives an optimal step length to move along this search direction numerical examples are provided to demonstrate that this new direct constrained optimization algorithm can be more efficient than the self consistent field scf iteration

limited memory block krylov subspace optimization for computing dominant singular value decompositions in many data intensive applications the use of principal component analysis and other related techniques is ubiquitous for dimension reduction data mining or other transformational purposes such transformations often require efficiently reliably and accurately computing dominant singular value decompositions svds of large and dense matrices in this paper we propose and study a subspace optimization technique for significantly accelerating the classic simultaneous iteration method we analyze the convergence of the proposed algorithm and numerically compare it with several state of the art svd solvers under the matlab environment extensive computational results show that on a wide range of large unstructured dense matrices the proposed algorithm can often provide improved efficiency or robustness over existing algorithms

parabolic and quasiparabolic subgroups of free partially commutative groups let be a finite graph and g be the corresponding free partially commutative group in this paper we study subgroups generated by vertices of the graph which we call canonical parabolic subgroups a natural extension of the definition leads to canonical quasiparabolic subgroups it is shown that the centralisers of subsets of g are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions

eit imaging regularization based on spectral graph wavelets the objective of electrical impedance tomographic reconstruction is to identify the distribution of tissue conductivity from electrical boundary conditions this is an ill posed inverse problem usually solved under the finite element method framework in previous studies standard sparse regularization was used for difference electrical impedance tomography to achieve a sparse solution however regarding elementwise sparsity standard sparse regularization interferes with the smoothness of conductivity distribution between neighboring elements and is sensitive to noise as an effect the reconstructed images are spiky and depict a lack of smoothness such unexpected artifacts are not realistic and may lead to misinterpretation in clinical applications to eliminate such artifacts we present a novel sparse regularization method that uses spectral graph wavelet transforms single scale or multiscale graph wavelet transforms are employed to introduce local smoothness on different scales into the reconstructed images the proposed approach relies on viewing finite element meshes as undirected graphs and applying wavelet transforms derived from spectral graph theory reconstruction results from simulations a phantom experiment and patient data suggest that our algorithm is more robust to noise and produces more reliable images

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

orbital minimization method with l 1 regularization abstract we consider a modification of the orbital minimization method omm energy functional which contains an l 1 penalty term in order to find a sparse representation of the low lying eigenspace of self adjoint operators we analyze the local minima of the modified functional as well as the convergence of the modified functional to the original functional algorithms combining soft thresholding with gradient descent are proposed for minimizing this new functional numerical tests validate our approach in addition we also prove the unanticipated and remarkable property that every local minimum of the omm functional without the l 1 term is also a global minimum

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

microscopic derivation of ginzburg landau theory we give the first rigorous derivation of the celebrated ginzburg landau gl theory starting from the microscopic bardeen cooper schrieffer bcs model close to the critical temperature gl arises as an effective theory on the macroscopic scale the relevant scaling limit is semiclassical in nature and semiclassical analysis with minimal regularity assumptions plays an important part in our proof

compressed modes for variational problems in mathematics and physics this article describes a general formalism for obtaining spatially localized sparse solutions to a class of problems in mathematical physics which can be recast as variational optimization problems such as the important case of schrodinger s equation in quantum mechanics sparsity is achieved by adding an regularization term to the variational principle which is shown to yield solutions with compact support compressed modes linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

symplectic integration with processing a general study the number of conditions to be satisfied by the operators k and p in symplectic integrators with processing given by epe hke p is determined for a hamiltonian of the form h cal a cal b the conditions for k are explicitly written up to order six and used to obtain more efficient methods with fewer evaluations per step than other symplectic integrators special cases in which the number of conditions for the kernel is drastically reduced are also studied it is shown that the kernel completely determines the optimal method one can obtain by processing

the free partially commutative lie algebra bases and ranks abstract in this paper we study the free partially commutative lie k algebra l a defined by a commutation relation on an alphabet a its behavior is very similar to that of the free lie algebra indeed we obtain in particular a partially commutative version of lazard s elimination process which allows us to prove that the k module l a is free and to construct explicitly k bases for it we show also how the classical witt s calculus can be extended to l a

an efficient gauss newton algorithm for symmetric low rank product matrix approximations we derive and study a gauss newton method for computing a symmetric low rank product xx t where x in mathbb r n times k for k n that is the closest to a given symmetric matrix a in mathbb r n times n in frobenius norm when a b t b or bb t this problem essentially reduces to finding a truncated singular value decomposition of b our gauss newton method which has a particularly simple form shares the same order of iteration complexity as a gradient method when k ll n but can be significantly faster on a wide range of problems in this paper we prove global convergence and a q linear convergence rate for this algorithm and perform numerical experiments on various test problems including those from recently active areas of matrix completion and robust principal component analysis numerical results show that the proposed algorithm is capable of providing considerable speed advantages over krylov subspace methods on suitable application problems where hi

localized density matrix minimization and linear scaling algorithms we propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases by adding an entry wise 1 regularization to the free energy of the quantum system based on the fact that the density matrix decays exponentially away from the diagonal for insulating systems or systems at finite temperature the proposed 1 regularized variational method provides an effective way to approximate the original quantum system we provide theoretical analysis of the approximation behavior and also design convergence guaranteed numerical algorithms based on bregman iteration more importantly the 1 regularized system naturally leads to localized density matrices with banded structure which enables us to develop approximating algorithms to find the localized density matrices with computation cost linearly dependent on the problem size

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

the cauchy problem for a class of parabolic equations in weighted variable sobolev spaces existence and asymptotic behavior abstract the paper addresses the questions of existence and asymptotic behavior of solutions to the cauchy problem for the equation u t div d x u p x 2 u a x u q x 2 u f x t u the coefficients d a are nonnegative functions which may vanish on a set of zero measure in r n and a x as x f x t u is globally lipschitz with respect to u the exponents p q r n 1 are given measurable functions we prove that the problem admits at least one weak solution in a weighted sobolev space with variable exponents provided that p ess inf r n p x max 2 n n 2 1 q ess inf r n q x 2 a 2 q x 2 l 1 r n and d s p x s l 1 b r 1 0 with constants max 1 2 n n 2 s min p q and r 1 0 in the case p 2 q x p x a e in r n and f f u there exists a unique strong solution and the problem has a global attractor in l 2 r n

adaptive regularized self consistent field iteration with exact hessian for electronic structure calculation the self consistent field scf iteration has been used ubiquitously for solving the kohn sham ks equation or the minimization of the ks total energy functional with respect to orthogonality constraints in electronic structure calculations although scf with heuristics such as charge mixing often works remarkably well on many problems it is well known that its convergence can be unpredictable and there is no general theoretical analysis on its performance we regularize the scf iteration and establish rigorous global convergence to the first order optimality conditions the hessian of the total energy functional is further exploited by adding the part of the hessian which is not considered in scf our methods can always achieve a highly accurate solution on problems for which scf fails and exhibit a better convergence rate than scf in the kssolv toolbox under the matlab environment

a new first order algorithmic framework for optimization problems with orthogonality constraints in this paper we consider a class of optimization problems with orthogonality constraints the feasible region of which is called the stiefel manifold our new framework combines a function value reduction step with a correction step different from the existing approaches the function value reduction step of our algorithmic framework searches along the standard euclidean descent directions instead of the vectors in the tangent space of the stiefel manifold and the correction step further reduces the function value and guarantees a symmetric dual variable at the same time we construct two types of algorithms based on this new framework the first type is based on gradient reduction including the gradient reflection gr and the gradient projection gp algorithms the other one adopts a columnwise block coordinate descent cbcd scheme with a novel idea for solving the corresponding cbcd subproblem inexactly we prove that both gr gp with a fixed step size and cbcd belong to our algorithmic framework

adaptive local basis set for kohn sham density functional theory in a discontinuous galerkin framework i total energy calculation kohn sham density functional theory is one of the most widely used electronic structure theories in the pseudopotential framework uniform discretization of the kohn sham hamiltonian generally results in a large number of basis functions per atom in order to resolve the rapid oscillations of the kohn sham orbitals around the nuclei previous attempts to reduce the number of basis functions per atom include the usage of atomic orbitals and similar objects but the atomic orbitals generally require fine tuning in order to reach high accuracy we present a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the kohn sham orbitals around the nuclei as well as environmental effects into the basis functions the resulting basis functions are localized in the real space and are discontinuous in the global domain the continuous kohn sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous galerkin dg framework our method is implemented in parallel and the current implementation is able to handle systems with at least thousands of atoms numerical examples indicate that our method can reach very high accuracy less than 1mev with a very small number 4 40 of basis functions per atom