exponentially fitted spline in compression for the numerical solution of singular perturbation problems abstract some exponentially fitted difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression these schemes are second order accurate numerical examples are given in support of the theoretical results

pseudo skeleton approximations with better accuracy estimates abstract we propose a priori accuracy estimates for low rank matrix approximations that use just a small number of the rows and columns this number is greater than the approximation rank unlike the existing methods of pseudo skeleton approximation but the estimates are more accurate than previously known ones this paper generalizes the results of 12 13

high order multiscale discontinuous galerkin methods for the one dimensional stationary schrodinger equation abstract in this paper we extend our previous work on the second order multiscale discontinuous galerkin dg method for one dimensional stationary schrodinger equations with oscillating solutions dong et al 2016 to high order we propose two types of high order multiscale finite element spaces and prove that the resulting dg methods with these spaces converge optimally with respect to the mesh size h in l 2 norm when h is small enough numerically we observe that these two multiscale dg methods achieve at least the second order convergence without any resonance errors when h e and optimal high order convergence when h e where e is the scale of the wave length we also demonstrate their ability to capture highly oscillating solutions of the schrodinger equation in the application of the resonant tunneling diode rtd model

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

stochastic dynamics of invasion and fixation we study evolutionary game dynamics in finite populations we analyze an evolutionary process which we call pairwise comparison for which we adopt the ubiquitous fermi distribution function from statistical mechanics the inverse temperature in this process controls the intensity of selection leading to a unified framework for evolutionary dynamics at all intensities of selection from random drift to imitation dynamics we derive a simple closed formula that determines the feasibility of cooperation in finite populations whenever cooperation is modeled in terms of any symmetric two person game in contrast with previous results the present formula is valid at all intensities of selection and for any initial condition we investigate the evolutionary dynamics of cooperators in finite populations and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations as a function of a given initial number of cooperators showing how this interplay strongly affects the approach to fixation of a given trait in finite populations leading to counterintuitive results at different intensities of selection

stochasticity and evolutionary stability in stochastic dynamical systems different concepts of stability can be obtained in different limits a particularly interesting example is evolutionary game theory which is traditionally based on infinite populations where strict nash equilibria correspond to stable fixed points that are always evolutionarily stable however in finite populations stochastic effects can drive the system away from strict nash equilibria which gives rise to a new concept for evolutionary stability the conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations we show that the two concepts can be derived from the frequency dependent moran process in different limits our results help to determine the appropriate stability concept in large finite populations the general validity of our findings is demonstrated showing that the same results are valid employing vastly different co evolutionary processes

fast monte carlo algorithms for matrices iii computing a compressed approximate matrix decomposition in many applications the data consist of or may be naturally formulated as an m times n matrix a which may be stored on disk but which is too large to be read into random access memory ram or to practically perform superlinear polynomial time computations on it two algorithms are presented which when given an m times n matrix a compute approximations to a which are the product of three smaller matrices c u and r each of which may be computed rapidly let a cur be the computed approximate decomposition both algorithms have provable bounds for the error matrix a a in the first algorithm c columns of a and r rows of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling and the r times n matrix r consists of those r rows of a also after appropriate rescaling then the c times r matrix u may be calculated from c and r for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively it is proven that left a a right xi le min d mathrm rank d le k left a d right xi poly k 1 c left a right f holds in expectation and with high probability for both xi 2 f and for all k 1 ldots mbox rank a thus by appropriate choice of k left a a right 2 le epsilon left a right f also holds in expectation and with high probability this algorithm may be implemented without storing the matrix a in ram provided it can make two passes over the matrix stored in external memory and use o m n additional ram assuming that c and r are constants independent of the size of the input the second algorithm is similar except that it approximates the matrix c by randomly sampling a constant number of rows of c thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a f both algorithms take time which is a low degree polynomial in k 1 epsilon and 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n the proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low rank approximations to a matrix the probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously

fast monte carlo algorithms for finding low rank approximations we consider the problem of approximating a given m n matrix a by another matrix of specified rank k which is smaller than m and n the singular value decomposition svd can be used to find the best such approximation however it takes time polynomial in m n which is prohibitive for some modern applications in this article we develop an algorithm that is qualitatively faster provided we may sample the entries of the matrix in accordance with a natural probability distribution in many applications such sampling can be done efficiently our main result is a randomized algorithm to find the description of a matrix d of rank at most k so that holds with probability at least 1 where v vf is the frobenius norm the algorithm takes time polynomial in k 1 e log 1 only and is independent of m and n in particular this implies that in constant time it can be determined if a given matrix of arbitrary size has a good low rank approximation

fast and robust recursive algorithmsfor separable nonnegative matrix factorization in this paper we study the nonnegative matrix factorization problem under the separability assumption that is there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure pixel assumption we present a family of fast recursive algorithms and prove they are robust under any small perturbations of the input data matrix this family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance

proto value functions a laplacian framework for learning representation and control in markov decision processes this paper introduces a novel spectral framework for solving markov decision processes mdps by jointly learning representations and optimal policies the major components of the framework described in this paper include i a general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators ii a specific instantiation of this approach where global basis functions called proto value functions pvfs are formed using the eigenvectors of the graph laplacian on an undirected graph formed from state transitions induced by the mdp iii a three phased procedure called representation policy iteration comprising of a sample collection phase a representation learning phase that constructs basis functions from samples and a final parameter estimation phase that determines an approximately optimal policy within the linear subspace spanned by the current basis functions iv a specific instantiation of the rpi framework using least squares policy iteration lspi as the parameter estimation method v several strategies for scaling the proposed approach to large discrete and continuous state spaces including the nystrom extension for out of sample interpolation of eigenfunctions and the use of kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored mdps vi finally a series of illustrative discrete and continuous control tasks which both illustrate the concepts and provide a benchmark for evaluating the proposed approach many challenges remain to be addressed in scaling the proposed framework to large mdps and several elaboration of the proposed framework are briefly summarized at the end

rank revealing qr factorizations and the singular value decomposition t chan has noted that even when the singular value decomposition of a matrix a is known it is still not obvious how to find a rank revealing qr factorization rrqr of a if a has numerical rank deficiency this paper offers a constructive proof of the existence of the rrqr factorization of any matrix a of size m x n with numerical rank r the bounds derived in this paper that guarantee the existence of rrqr are all of order f i in comparison with chan s 0 2 it has been known for some time that if a is only numerically rank one deficient then the column permutation l7 of a that guarantees a small rnn in the qr factorization of a n can be obtained by inspecting the size of the elements of the right singular vector of a corresponding to the smallest singular value of a to some extent our paper generalizes this well known result we consider the interplay between two important matrix decompositions the singular value decomposition and the qr factorization of a matrix a in particular we are interested in the case when a is singular or nearly singular it is well known that for any a e r m x n a real matrix with rn rows and n columns where without loss of generality we assume rn n there are orthogonal matrices u and v such that where c is a diagonal matrix with nonnegative diagonal elements we assume that a 2 a2 2 2 on 2 0 the decomposition 0 1 is the singular value decomposition svd of a and the ai are the singular values of a the columns of v are the right singular vectors of a and the columns of u are the left singular vectors of a mathematically in terms of the singular values received december 1 1990 revised february 8 199 1 199 1 mathematics subject classification primary 65f30 15a23 15a42 15a15

discontinuous galerkin methods with plane waves for time harmonic problems a general framework for discontinuous galerkin methods in the frequency domain with numerical flux is presented the main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element the method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two dimensional linearized euler equations it is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high order drp schemes in addition the conditioning of the method is found to be high but not critical in practice it is shown that the ultra weak variational formulation is in fact a subset of the present discontinuous galerkin method a special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles aeroacoustic problems with non uniform flows are also considered and results are presented for the sound radiated from a two dimensional jet

on the existence and computation of rank revealing lu factorizations abstract by exploring properties of schur complements this paper presents bounds on the existence of rank revealing lu factorizations that are comparable with those of rank revealing qr factorizations the new bounds provide substantial improvement over previously derived bounds this paper also proposes two algorithms using gaussian elimination with a block pivoting strategy to select a subset of columns from a given matrix which has a guaranteed relatively large smallest singular value each of these two algorithms is faster than its orthogonal counterpart for dense matrices if implemented appropriately these algorithms are faster than the corresponding rank revealing qr methods even when the orthogonal matrices are not explicitly updated based on these two algorithms an algorithm using only gaussian elimination for computing rank revealing lu factorizations is introduced

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

fast computation of low rank matrix approximations given a matrix a it is often desirable to find a good approximation to a that has low rank we introduce a simple technique for accelerating the computation of such approximations when a has strong spectral features that is when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to a our technique amounts to independently sampling and or quantizing the entries of a thus speeding up computation by reducing the number of nonzero entries and or the length of their representation our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix n to a whose entries are independent random variables with zero mean and bounded variance since with high probability n has very weak spectral features we can prove that the effect of sampling and quantization nearly vanishes when a low rank approximation to a p n is computed we give high probability bounds on the quality of our approximation both in the frobenius and the 2 norm

discontinuous galerkin method based on non polynomial approximation spaces in this paper we develop discontinuous galerkin dg methods based on non polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations pdes the algorithm is based on approximation spaces consisting of non polynomial elementary functions such as exponential functions trigonometric functions etc with the objective of obtaining better approximations for specific types of pdes and initial and boundary conditions it is shown that l2 stability and error estimates can be obtained when the approximation space is suitably selected it is also shown with numerical examples that a careful selection of the approximation space to fit individual pde and initial and boundary conditions often provides more accurate results than the dg methods based on the polynomial approximation spaces of the same order of accuracy

an optimally convergent discontinuous galerkin based extended finite element method for fracture mechanics the extended finite element method xfem enables the representation of cracks in arbitrary locations of a mesh we introduce here a variant of the xfem rendering an optimally convergent scheme its distinguishing features are as follows a the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip the enrichment region b only one and two enrichment functions are added for anti plane shear and planar problems respectively and c the relaxation of the continuity between the enrichment region and the rest of the domain and the adoption of a discontinuous galerkin dg method therein the method is provably stable for any positive value of a stabilization parameter and by weakly enforcing the continuity between the two regions it eliminates blending elements partly responsible for the suboptimal convergence of some early xfems moreover the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined more importantly the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns quadrature strategies required for the optimal convergence are also discussed finally the dg method was modified to retain stability based on an inf sup condition copyright 2009 john wiley sons ltd

optimal regularized low rank inverse approximation abstract in this paper we consider the problem of finding approximate inverses of a given matrix we give an explicit solution to the rank constrained regularized inverse approximation problem and obtain an inverse regularized eckart young like theorem

fixation of strategies with the moran and fermi processes in evolutionary games abstract a model of stochastic evolutionary game dynamics with finite population was built it combines the standard moran and fermi rules with two strategies cooperation and defection we obtain the expressions of fixation probabilities and fixation times the one third rule which has been found in the frequency dependent moran process also holds for our model we obtain the conditions of strategy being an evolutionarily stable strategy in our model and then make a comparison with the standard moran process besides the analytical results show that compared with the standard moran process fixation occurs with higher probabilities under a prisoner s dilemma game and coordination game but with lower probabilities under a coexistence game the simulation result shows that the fixation time in our mixed process is lower than that in the standard fermi process in comparison with the standard moran process fixation always takes more time on average in spatial populations regardless of the game in addition the fixation time decreases with the growth of the number of neighbors

matrices vector spaces and information retrieval the evolution of digital libraries and the internet has dramatically transformed the processing storage and retrieval of information efforts to digitize text images video and audio now consume a substantial portion of both academic and industrial activity even when there is no shortage of textual materials on a particular topic procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking recently developed information retrieval technologies are based on the concept of a vector space data are modeled as a matrix and a user s query of the database is represented as a vector relevant documents in the database are then identified via simple vector operations orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself the purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections

a short note on the error estimates of yuan shu discontinuous galerkin method based on non polynomial approximation spaces in the article yuan and shu 2006 17 yuan and shu have developed discontinuous galerkin dg methods based on non polynomial approximation spaces for solving time dependent problems the authors established l 2 error estimates of the proposed methods and also presented a criterion for the choice of basis functions of the non polynomial spaces to have the same approximation rates as those of polynomial finite element spaces of the same dimension however the verification that some of approximation spaces do satisfy the criterion has been performed only to limited types of approximation spaces in this regards the aim of this short note is to fill the gap we prove that the criterion of yuan and shu can be satisfied by a wide class of basis functions including the well known basis functions such as trigonometric and exponential polynomials

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

sparse grid discontinuous galerkin methods for high dimensional elliptic equations this paper constitutes our initial effort in developing sparse grid discontinuous galerkin dg methods for high dimensional partial differential equations pdes over the past few decades dg methods have gained popularity in many applications due to their distinctive features however they are often deemed too costly because of the large degrees of freedom of the approximation space which are the main bottleneck for simulations in high dimensions in this paper we develop sparse grid dg methods for elliptic equations with the aim of breaking the curse of dimensionality using a hierarchical basis representation we construct a sparse finite element approximation space reducing the degrees of freedom from the standard o h d to o h 1 log 2 h d 1 for d dimensional problems where h is the uniform mesh size in each dimension our method based on the interior penalty ip dg framework can achieve accuracy of o h k log 2 h d 1 in the energy norm where k is the degree of polynomials used error estimates are provided and confirmed by numerical tests in multi dimensions

on the error of estimating the sparsest solution of underdetermined linear systems let a be an n m matrix with m n and suppose that the underdetermined linear system as x admits a sparse solution s0 for which s0 0 1 2 spark a such a sparse solution is unique due to a well known uniqueness theorem suppose now that we have somehow a solution s as an estimation of s0 and suppose that s is only approximately sparse that is many of its components are very small and nearly zero but not mathematically equal to zero is such a solution necessarily close to the true sparsest solution more generally is it possible to construct an upper bound on the estimation error s s0 2 without knowing s0 the answer is positive and in this paper we construct such a bound based on minimal singular values of submatrices of a we will also state a tight bound which is more complicated but besides being tight enables us to study the case of random dictionaries and obtain probabilistic upper bounds we will also study the noisy case that is where x as n moreover we will see that where s0 0 grows to obtain a predetermined guaranty on the maximum of s s0 2 s is needed to be sparse with a better approximation this can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where s0 0 grows

coevolutionary dynamics from finite to infinite populations traditionally frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations however the relationship between deterministic and stochastic approaches remained unclear here we solve this problem by explicitly considering large populations in particular we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics moreover differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and depending on the population size can even invert the direction of the evolutionary process

an exponentially fitted finite difference method for singular perturbation problems in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system the stability of the algorithm is investigated several linear and nonlinear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

non gaussian fluctuations arising from finite populations exact results for the evolutionary moran process the appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations the moran process recently introduced into this context in nowak et al nature london 428 646 2004 defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible in this paper we derive the stationary distribution of the moran process population dynamics for arbitrary 2 ifmmode times else texttimes fi 2 games for the finite size case we show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix in contrast to the common approach to mimic finite size fluctuations by gaussian distributed noise the finite size fluctuations can deviate significantly from a gaussian distribution

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

evolutionary games on cycles with strong selection evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success quantified by the probability that a single mutant takes over a population graph structures compared to the well mixed case can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant properties of the associated mean fixation times can be more intricate especially when selection is strong the intuition is that fixation of a beneficial mutant happens fast in a dominance game that fixation takes very long in a coexistence game and that strong selection eliminates demographic noise here we show that these intuitions can be misleading in structured populations we analyze mean fixation times on the cycle graph under strong frequency dependent selection for two different microscopic evolutionary update rules death birth and birth death we establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size depending on the underlying game we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring death birth update in contrast if selection for an offspring occurs before random removal birth death update then strong selection can remove demographic noise almost entirely

a note on linear function approximation using random projections linear function approximations based on random projections are proposed and justified for a class of fixed point and minimization problems

on maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices abstract the problem of finding a k k submatrix of maximum volume of a matrix a is of interest in a variety of applications for example it yields a quasi best low rank approximation constructed from the rows and columns of a we show that such a submatrix can always be chosen to be a principal submatrix not only for symmetric semidefinite matrices but also for diagonally dominant matrices then we analyze the low rank approximation error returned by a greedy method for volume maximization cross approximation with complete pivoting our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices in particular for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error we also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results

polynomial fitting and interpolation on circular sections we construct weakly admissible polynomial meshes wams on circular sections such as symmetric and asymmetric circular sectors circular segments zones lenses and lunes the construction resorts to recent results on subperiodic trigonometric interpolation the paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

polynomial approximation and cubature at approximate fekete and leja points of the cylinder the paper deals with polynomial interpolation least square approximation and cubature of functions dened on the rectangular cylinder k d 1 1 with d the unit disk the nodes used for these processes are the approximate fekete points afp and the discrete leja points dlp extracted from suitable weakly admissible meshes wams of the cylinder from the analysis of the growth of the lebesgue constants approximation and cubature errors we show that the afp and the dlp extracted from wam are good points for polynomial approximation and numerical integration of functions dened on the cylinder

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

universality of weak selection weak selection which means a phenotype is slightly advantageous over another is an important limiting case in evolutionary biology recently it has been introduced into evolutionary game theory in evolutionary game dynamics the probability to be imitated or to reproduce depends on the performance in a game the influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection in many models of both unstructured and structured populations a key assumption allowing analytical calculations is weak selection which means that all individuals perform approximately equally well in the weak selection limit many different microscopic evolutionary models have the same or similar properties how universal is weak selection for those microscopic evolutionary processes we answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity we find universal higher order expansions which allow a rescaling of the selection intensity with this we can identify specific models which violate linear weak selection results such as the one third rule of coordination games in finite but large populations

efficient denoising algorithms for large experimental datasets and their applications in fourier transform ion cyclotron resonance mass spectrometry modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed in many cases the analysis of spectroscopic data involves the denoising of raw data before any further processing current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length preventing their use on very large datasets taking advantage of recent progress on random projection and probabilistic algorithms we developed a simple and efficient method for the denoising of very large datasets based on the qr decomposition of a matrix randomly sampled from the data this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising this procedure called urqrd uncoiled random qr denoising strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size the efficiency of these numerical tools is demonstrated on experimental data from high resolution broadband fourier transform ion cyclotron resonance mass spectrometry which has applications in proteomics and metabolomics we show that robust denoising is achieved in 2d spectra whose interpretation is severely impaired by scintillation noise these denoising procedures can be adapted to many other data analysis domains where the size and or the processing time are crucial

low rank approximation and decomposition of large matrices using error correcting codes low rank approximation is an important tool used in many applications of signal processing and machine learning recently randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate singular value decompositions of large matrices similar ideas were used to solve least squares regression problems in this paper we show how matrices from error correcting codes can be used to find such low rank approximations and matrix decompositions and extend the framework to linear least squares regression problems the benefits of using these code matrices are the following first they are easy to generate and they reduce randomness significantly second code matrices with mild restrictions satisfy the subspace embedding property and have a better chance of preserving the geometry of an large subspace of vectors third for parallel and distributed applications code matrices have significant advantages over structured random matrices and gaussian random matrices fourth unlike fourier or hadamard transform matrices which require sampling o k log k columns for a rank k approximation the log factor is not necessary for certain types of code matrices in particular 1 epsilon optimal frobenius norm error can be achieved for a rank k approximation with o k epsilon samples fifth fast multiplication is possible with structured code matrices so fast approximations can be achieved for general dense input matrices sixth for least squares regression problem min ax b 2 where a in mathbb r n times d the 1 epsilon relative error approximation can be achieved with o d epsilon samples with high probability when certain code matrices are used

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

application of discontinuous galerkin method to mechanical 2d problem with arbitrary polygonal and very high order finite elements abstract discontinuous galerkin with finite difference rules dgfd is applied to mechanical plane stress state problem the considered domain is discretized by polygonal mesh the polygonal elements can be for example a hexagon pentagon or just quadrangle or triangle they do not have to be convex and a fish mesh where the elements have fish shapes is used when the elements are rectangular then the orthogonality of chebyshev basis functions can be utilized in such a case very high order approximate solution can be obtained in this work the approximation order exceeds 10 and reaches 60 which in the latter case means 3600 numbers of degrees of freedom in a single element the paper is illustrated by a benchmark example in which the exact solution is recovered by dgfd method for various meshes in the other example the stress concentration is easily recovered by very high order version of dgfd method

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

a fast tri factorization method for low rank matrix recovery and completion in recent years matrix rank minimization problems have received a significant amount of attention in machine learning data mining and computer vision communities and these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix and has to be solved iteratively and involves singular value decomposition svd at each iteration therefore those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple svds in this paper we propose a fast tri factorization ftf method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing svds the proposed ftf method can be used to reliably solve a wide range of low rank matrix recovery and completion problems such as robust principal component analysis rpca low rank representation lrr and low rank matrix completion mc we also present three specific models for rpca lrr and mc problems respectively moreover we develop two alternating direction method adm based iterative algorithms for solving the above three problems experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of our ftf method comparing with the state of the art nuclear norm minimization algorithms

convergence analysis of markov chain monte carlo linear solvers using ulam von neumann algorithm the convergence of markov chain based monte carlo linear solvers using the ulam von neumann algorithm for a linear system of the form x hx b is investigated in this paper we analyze the convergence of the monte carlo solver based on the original ulam von neumann algorithm under the conditions that h 1 as well as rho h 1 where rho h is the spectral radius of h we find that although the monte carlo solver is based on sampling the neumann series the convergence of neumann series is not a sufficient condition for the convergence of the monte carlo solver actually properties of h are not the only factors determining the convergence of the monte carlo solver the underlying transition probability matrix plays an important role an improper selection of the transition matrix may result in divergence even though the condition h 1 holds however if the condition h 1 is satisfied we show that there always exist certain transition matrices that guarantee convergence of

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

subset selection for matrices abstract we consider the question of how to delete m k rows from a matrix x r m n so that the resulting matrix a r k n is as non singular as possible bounds for the singular values of a are derived which decrease only algebraically with m and n in addition a number of applications where subset selection is necessary are examined

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

application of discontinuous galerkin method in supersonic and hypersonic gas flows abstract most high order methods often suffer from two and three dimensional unstructured grids particularly for supersonic and hypersonic gas flows we applied a high order mixed modal discontinuous galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows furthermore we provide detailed implementations and validations of two and three dimensional problems examples include double mach reflection problems and supersonic and hypersonic gas flows around a cylinder in addition we provide case studies of a 3d hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows this study can be regarded as an attempt to apply the discontinuous galerkin method to engineering problems