asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems we consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients with these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space and the approximation space for the underlying spatial function space sparse polynomial approximation is employed to obtain the first one while classical finite elements are employed to obtain the latter an error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem a series of detailed numerical experiments supports the conclusions of this analysis

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

elimination and nonlinear equations of rees algebra abstract a new approach is established to computing the image of a rational map whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees in the case the map is everywhere defined this analysis provides free resolutions of graded parts of the rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra a surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion an additional point is that this contribution which of course corresponds to nonlinear equations of the rees algebra can be described in these degrees in terms of non koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by j herzog the third named author and w vasconcelos as a measure of the reach of this torsion analysis we could say that in the case of a general everywhere defined map half of the degrees where the torsion does not vanish are understood

degree and birationality of multi graded rational maps we give formulas and effective sharp bounds for the degree of multi graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties we also extend the jacobian dual criterion to the multi graded setting our approach is based on the study of blow up algebras including syzygies of the ideal generated by the defining polynomials of the rational map a key ingredient is a new algebra that we call thesaturated special fiber ring which turns out to be a fundamental tool to analyze the degree of a rational map we also provide a very effective birationality criterion and a complete description of the equations of the associated rees algebra of a particular class of plane rational maps

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

analysis of dispersion of multinomial responses abstract in analogy to analysis of variance for linear models analyses of dispersion for loglinear models for multinomial responses are constructed the analyses which are based on the entropy and concentration measures are used to construct tests of independence and measures of association

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

the moving curve ideal and the rees algebra the rees algebra of an ideal in a commutative ring is the quotient of a polynomial ring by its ideal of defining relations for a polynomial ring in two variables this ideal was discovered independently by the geometric modeling community where it is called the moving curve ideal we review some properties of the rees algebra and discuss one result and one conjecture concerning the structure of the moving curve ideal and its relation to adjoint curves some parts of the paper are purely expository

fitting ideals and multiple points of surface parameterizations given a birational parameterization f of an algebraic surface s the purpose of this paper is toinvestigate the sets of points on s whose preimage consists in kor more points counting multiplicities these points are described explicitly interms of fitting ideals of some graded parts of the symmetric algebraassociated to the parameterization f to obtain this description weshow that the degree and dimension of a fiber could be computed bycomparing the drop of rank of two explicit representation matrices associated to f

cross entropy dissimilarity measures and characterizations of quadratic entropy a unified approach is given for constructing cross entropy and dissimilarity measures between probability distributions based on a given entropy function or a diversity measure special properties of quadratic entropy introduced by rao 7 are described in particular it is shown that the square root of the jensen difference dissimilarity measure arising out of a quadratic entropy provides a metric on a probability space several characterizations of quadratic entropy are obtained

quantized tensor structured finite elements for second order elliptic pdes in two dimensions we analyze the approximation of the solutions of second order elliptic problems which have point singularities and belong to a countably normed space of analytic functions with a first order h version finite element fe method based on uniform tensor product meshes the fe solutions are well known to converge with algebraic rate at most 1 2 in terms of the number of degrees of freedom and even slower in the presence of singularities we analyze the compression of the fe coefficient vectors represented in the so called quantized tensor train format we prove in a reference square that the resulting fe approximations converge exponentially in terms of the effective number n of degrees of freedom involved in the representation n mathcal o log 5 varepsilon 1 where varepsilon in 0 1 is the accuracy measured in the energy norm numerically we show for solutions from the same class that the entire process of solving the tensor structured galerkin first order fe discretization can achieve accuracy varepsilon in the energy norm with n mathcal o log kappa varepsilon 1 parameters where kappa 3

space time approximation with sparse grids in this article we introduce approximation spaces especially suited for the approximation of solutions of parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time proper truncation then leads to so called space time sparse grid spaces for a uniform discretization of the spatial space of dimension d with o nd degrees of freedom these spaces involve for d 1 also only o nd degrees of freedom for the discretization of the whole space time problem but they provide the same approximation rate as classical space time finite element spaces which need o nd 1 degrees of freedoms this makes these approximation spaces well suited for conventional parabolic and time dependent optimization problems we analyze the approximation properties and the dimension of these sparse grid space time spaces for general stable multiscale bases we then restrict ourselves to an interpolatory multiscale basis i e a hierarchical basis here to be able to handle also complicated spatial domains omega we construct the hierarchical basis from a given spatial finite element basis as follows first we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method then we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach this way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space time sparse grid approach we give numerical results on the convergence rate of the interpolation error of these spaces for various space time problems with two spatial dimensions implementational issues data structures and questions of adaptivity also are addressed to some extent

low rank tensor structure of linear diffusion operators in the tt and qtt formats abstract we consider a class of multilevel matrices arising for example from the discretization of linear diffusion operators in a d dimensional hypercube we derive explicit representations of such matrices in the tensor train tt format introduced recently for the non linear low parametric approximation of multi dimensional vectors with the aim to handle the curse of dimensionality we obtain sharp upper bounds on the tt ranks which are linear or when the diffusion tensor is semiseparable or quasi separable even sublinear in d cf the straightforward quadratic estimate the use of the quantized tensor train qtt decomposition allows to further reduce the number of parameters

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m

fast solution of parabolic problems in the tensor train quantized tensor train format with initial application to the fokker planck equation in this paper we propose two schemes of using the so called quantized tensor train qtt approximation for the solution of multidimensional parabolic problems first we present a simple one step implicit time integration scheme using a solver in the qtt format of the alternating linear scheme als type as the second approach we use the global space time formulation resulting in a large block linear system encapsulating all time steps and solve it at once in the qtt format we prove the qtt rank estimate for certain classes of multivariate potentials and respective solutions in x t variables the log linear complexity of storage and the solution time is observed in both spatial and time grid sizes the method is applied to the fokker planck equation arising from the beads springs models of polymeric liquids

a low rank approach to the computation of path integrals we present a method for solving the reaction diffusion equation with general potential in free space it is based on the approximation of the feynman kac formula by a sequence of convolutions on sequentially diminishing grids for computation of the convolutions we propose a fast algorithm based on the low rank approximation of the hankel matrices the algorithm has complexity of o n r m log m n r 2 m flops and requires o m r floating point numbers in memory where n is the dimension of the integral r n and m is the mesh size in one dimension the presented technique can be generalized to the higher order diffusion processes

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

application of stochastic galerkin fem to the complete electrode model of electrical impedance tomography abstract the aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage the most accurate forward model for impedance tomography is the complete electrode model which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces if the reconstruction task of impedance tomography is recast as a bayesian inference problem it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables respectively in this work we apply a stochastic galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with monte carlo simulations

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

alternating splitting waveform relaxation method and its successive overrelaxation acceleration for the large sparse implicit linear initial value problem we present a block successive overrelaxation scheme for the alternating direction implicit waveform relaxation method to further accelerate its convergence speed and discuss the convergence property of the resulting iteration method in detail numerical implementations about several non hermitian implicit linear initial value problems show that the alternating direction implicit waveform relaxation method is very effective and the block successive overrelaxation technique really accelerates its convergence speed

on the equations of the moving curve ideal of a rational algebraic plane curve given a parametrization of a rational plane algebraic curve c some explicit adjoint pencils on c are described in terms of determinants moreover some generators of the rees algebra associated to this parametrization are presented the main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence

convergence of the parallel chaotic waveform relaxation method for stiff systems in this paper we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models under some different assumptions on the coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given

multigrid methods for implicit runge kutta and boundary value method discretizations of parabolic pdes advanced time discretization schemes for stiff systems of ordinary differential equations odes such as implicit runge kutta and boundary value methods have many appealing properties however the resulting systems of equations can be quite large and expensive to solve many techniques exploiting the structure of these systems have been developed for general odes for spatial discretizations of time dependent partial differential equations pdes these techniques are in general not sufficient and also the structure arising from spatial discretization has to be taken into consideration we show here that for time dependent parabolic problems this can be done by multigrid methods as in the stationary elliptic case the key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously the overall cost is essentially proportional to the cost of integrating a scalar ode for each grid point combination of the multigrid principle with both time stepping and waveform relaxation techniques is described together with a convergence analysis numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients

the implicit equation of a multigraded hypersurface in this article we analyze the implicitization problem of the image of a rational map x pn with x a toric variety of dimension n 1 defined by its cox ring r let i f0 fn be n 1 homogeneous elements of r we blow up the base locus of v i and we approximate the rees algebra reesr i of this blow up by the symmetric algebra symr i we provide under suitable assumptions resolutions z for symr i graded by the divisor group of x cl x such that the determinant of a graded strand det z gives a multiple of the implicit equation for suitable cl x indeed we compute a region in cl x which depends on the regularity of symr i where to choose we also give a geometrical interpretation of the possible other factors appearing in det z a very detailed description is given when x is a multiprojective space

rees algebras of truncations of complete intersections abstract in this paper we describe the defining equations of the rees algebra and the special fiber ring of a truncation i of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m to describe explicitly the rees algebra r i in terms of generators and relations we map another rees ring r m onto it where m is the direct sum of powers of m we compute a grobner basis of the ideal defining r m it turns out that the normal domain r m is a koszul algebra and from this we deduce that in many instances r i is a koszul algebra as well

multiparametric shell eigenvalue problems abstract the eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed here the focus is on the smallest eigenpairs shells of revolution have natural eigenclusters due to symmetries moreover the eigenpairs depend on a deterministic parameter the dimensionless thickness the stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters in the case of random material parameters it is possible that the eigenmodes cross in the stochastic parameter space this interesting phenomenon is demonstrated via numerical experiments finally the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible

preconditioning the augmented lagrangian method for instationary mean field games with diffusion we discuss the application of the augmented lagrangian method to the convex optimization problem of instationary variational mean field games with diffusion the problem is first discretized with space time tensor product piecewise polynomial bases this leads to a sequence of linear problems posed on the space time cylinder that are second order in the temporal variable and fourth order in the spatial variable to solve these large linear problems with the preconditioned conjugate gradients method we propose a preconditioner that is based on a temporal transformation coupled with a spatial multigrid this preconditioner is thus based on standard components and is particularly suitable for parallel computation it is conditionally parameter robust in the sense that the condition number of the preconditioned system is low for sufficiently fine temporal discretizations numerical examples illustrate the method

rational plane curves parameterizable by conics abstract we introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines also known as monoid curves we show that they are the image of monoid curves via suitable quadratic transformations in projective plane we also describe all the possible proper parameterizations of them and a set of minimal generators of the rees algebra associated to these parameterizations extending well known results for curves parameterizable by lines

a schwarz waveform moving mesh method an r refinement moving mesh method is considered for solving time dependent partial differential equations pdes the resulting coupled system consisting of the physical pde and a moving mesh pde is solved by a schwarz waveform relaxation method in particular the computational space time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of pdes on each subdomain dirichlet boundary conditions are used to pass solution information between neighboring regions the efficacy of this approach is demonstrated for some model problems for problems where the solutions evolve on disparate time scales in different regions of the spatial domain this approach demonstrates the significant savings in computational time and effort which are possible

optimized schwarz waveform relaxation for advection reaction diffusion equations in two dimensions optimized schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems they are based on a decomposition in space and an iteration where only subproblems in space time need to be solved each subproblem can be simulated using an adapted numerical method for example with local time stepping or one can even use a different model in different subdomains which makes these methods very suitable also from a modeling point of view for rapid convergence however it is important to use effective transmission conditions between the space time subdomains and for best performance these transmission conditions need to take the physics of the underlying evolution problem into account the optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions we study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions we prove comprehensively best approximation results for transmission conditions of robin and ventcel higher order type which can also be used in the various limits for example for the heat equation since we include in our analysis a positive low frequency limiter both in space and time we give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms and which guarantee asymptotically best performance of the iterative methods we finally show extensive numerical experiments including cases not covered by our analysis for example decompositions with cross points we use q1 finite element discretizations in space and forward and backward euler discretizations in time other discretization could also have been considered since all our analysis is at the continuous level and in all cases we measure performance corresponding to our analysis

bound for the number of one dimensional fibers of a projective morphism abstract given a parameterization p k 2 p k 3 of an algebraic surface s p k 3 we bound the number of one dimensional fibers of the canonical projection of the graph of onto its image

a block krylov subspace implementation of the time parallel paraexp method and its extension for nonlinear partial differential equations a parallel time integration method for nonlinear partial differential equations is proposed it is based on a new implementation of the paraexp method for linear partial differential equations pdes employing a block krylov subspace method for nonlinear pdes the algorithm is based on our paraexp implementation within a waveform relaxation the initial value problem is solved iteratively on a complete time interval nonlinear terms are treated as source terms provided by the solution from the previous iteration at each iteration the problem is decoupled into independent subproblems by the principle of superposition the decoupled subproblems are solved fast by exponential integration based on a block krylov method the new time integration is demonstrated for the one dimensional advection diffusion equation and the viscous burgers equation numerical experiments confirm excellent parallel scaling for the linear advection diffusion problem and good scaling in case the nonlinear burgers equation is simulated

convergence analysis of a periodic like waveform relaxation method for initial value problems via the diagonalization technique we present in this paper a time parallel algorithm for dot u f t u with initial value u 0 u 0 by using the waveform relaxation wr technique and the diagonalization technique with a suitable parameter alpha the wr technique generates a functional sequence u k t via the dynamic iterations dot u k f t u k u k 0 alpha u k t alpha u k 1 t u 0 and at convergence we get u infty t u t each wr iterate represents a periodic like differential equation which is very suitable for applying the diagonalization technique yielding direct parallel in time computation the parameter alpha controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the wr iterations and we perform a detailed analysis for the influence of the parameter alpha on the method we show that the roundoff error is proportional to epsilon 2n 1 max alpha 2 alpha 2 n t vardelta t and epsilon is the machine precision and the convergence factor can be bounded by alpha e tl 1 alpha e tl where l ge 0 is the one sided lipschitz constant of f we also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored our analysis includes the heat and wave equations as special cases numerical results are given to support our findings

on sor waveform relaxation methods waveform relaxation is a numerical method for solving large scale systems of ordinary differential equations on parallel computers it differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously this paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation sor techniques in particular different sor acceleration schemes based on multiplication with a scalar parameter or convolution with a time dependent function are described and theoretically analyzed the theory is applied to a one dimensional and two dimensional model problem and checked against results obtained by numerical experiments

convergence of quasi optimal stochastic galerkin methods for a class of pdes with random coefficients in this work we consider quasi optimal versions of the stochastic galerkin method for solving linear elliptic pdes with stochastic coefficients in particular we consider the case of a finite number n of random inputs and an analytic dependence of the solution of the pde with respect to the parameters in a polydisc of the complex plane c n we show that a quasi optimal approximation is given by a galerkin projection on a weighted anisotropic total degree space and prove a sub exponential convergence rate as a specific application we consider a thermal conduction problem with non overlapping inclusions of random conductivity numerical results show the sharpness of our estimates

residual restarting and richardson iteration for the matrix exponential a well known problem in computing some matrix functions iteratively is the lack of a clear commonly accepted residual notion an important matrix function for which this is the case is the matrix exponential suppose the matrix exponential of a given matrix times a given vector has to be computed we develop the approach of druskin greenbaum and knizhnerman siam j sci comput 19 1998 pp 38 54 and interpret the sought after vector as the value of a vector function satisfying the linear system of ordinary differential equations odes whose coefficients form the given matrix the residual is then defined with respect to the initial value problem for this ode system the residual introduced in this way can be seen as a backward error we show how the residual can be computed efficiently within several iterative methods for the matrix exponential this resolves the question of reliable stopping criteria for these methods further we show that the residual concept can be used to construct new residual based iterative methods in particular a variant of the richardson method for the new residual appears to provide an efficient way to restart krylov subspace methods for evaluating the matrix exponential

computing periodic solutions of linear differential algebraic equations by waveform relaxation we propose an algorithm which is based on the waveform relaxation wr approach to compute the periodic solutions of a linear system described by differential algebraic equations for this kind of two point boundary problems we derive an analytic expression of the spectral set for the periodic wr operator we show that the periodic wr algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1 numerical examples where discrete waveforms are computed with a backward difference formula further illustrate the correctness of the theoretical work in this paper

anova like similarity analysis using expected species shared decomposition generale de la diversite quasiment identique a la decomposition de la variance dans une anova appliquee a l etude du nombre d especes communes dans des echantillons issus de deux populations

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

waveform relaxation with fast direct methods as preconditioner for a restricted class of parabolic pdes one can devise a practical numerical solver with a parallel complexity that is theoretically optimal the method uses a multidimensional fft to decouple the unknowns in the spatial domain into independent scalar odes these are discretized to give recurrence relations in the time dimension solved by parallel cyclic reduction this is the fft cr algorithm we discuss the use of fft cr as a preconditioner to iteratively solve more general parabolic pdes this approach naturally leads to a waveform relaxation scheme waveform relaxation was developed as an iterative method for solving large systems of odes it is the continuous in time analogue of stationary iterative methods for linear algebraic equations using the fft cr solver as a preconditioner preserves most of the potential for concurrency that accounts for the attractiveness of waveform relaxation with simple preconditioners like jacobi or red black gauss seidel while showing an important advantage the convergence rate of the resulting iteration is independent of the mesh size used in the spatial discretization the method can be accelerated by applying an appropriate scaling of the system before preconditioning

convergence analysis of waveform relaxation methods for neutral differential functional systems in this paper the problems of convergence and superlinear convergence of continuous time waveform relaxation method applied to volterra type systems of neutral functional differential equations are discussed under a lipschitz condition with time and delay dependent right hand side imposed on the so called splitting function more suitable conditions about convergence and superlinear convergence of continuous time wr method are obtained we also investigate the initial interval acceleration strategy for the practical implementation of the continuous time waveform relaxation method i e discrete time waveform relaxation method it is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

survey on the theory and applications of μ bases for rational curves and surfaces abstract bases are new representations for rational curves and surfaces which serve as a bridge between their parametric forms and implicit forms geometrically bases are represented by moving lines or moving planes while their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces bases have been proven to be significant in solving many important problems in geometric modeling such as fast implicitization singularity computation reparametrization as well as providing easy inversion formulas for points we review the state of the art results in bases theory and applications for rational curves and surfaces and raise unsolved problems for future research

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

multigrid waveform relaxation on spatial finite element meshes the continuous time case the waveform relaxation method and its multigrid acceleration are studied as solution procedures for the system of ordinary differential equations obtained by finite element discretisation of a linear parabolic initial boundary value problem the convergence properties of the continuous time algorithm are theoretically investigated on finite length and infinite length time intervals in addition quantitative convergence estimates and numerical results are presented for one dimensional and two dimensional model problems

convergence rates of multilevel and sparse tensor approximations for a random elliptic pde partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite dimensional parameter domain for a class of elliptic equations we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations we prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis in some cases the same rate is attained also by a single level approximation using just one spatial discretization for all coefficients we suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting without prior knowledge of this rate numerical computations confirm theoretical results

stabilizing the hierarchical basis by approximate wavelets ii implementation and numerical results this paper is the second part of a work on stabilizing the classical hierarchical basis hb by using wavelet like basis functions implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work which deals with algorithms and their mathematical theory numerical results are presented to confirm the theory established there a comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables

on the closed image of a rational map and the implicitization problem in this paper we investigate some topics around the closed image s of a rational map lambda given by some homogeneous elements f 1 ldots f n of the same degree in a graded algebra a we first compute the degree of this closed image in case lambda is generically finite and f 1 ldots f n define isolated base points in proj a we then relate the definition ideal of s to the symmetric and the rees algebras of the ideal i f 1 ldots f n subset a and prove some new acyclicity criteria for the associated approximation complexes finally we use these results to obtain the implicit equation of s in case s is a hypersurface proj a pp n 2 k with k a field and base points are either absent or local complete intersection isolated points

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

analysis of a new space time parallel multigrid algorithm for parabolic problems we present and analyze a new space time parallel multigrid method for parabolic equations the method is based on arbitrarily high order discontinuous galerkin discretizations in time and a finite element discretization in space the key ingredient of the new algorithm is a block jacobi smoother we present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate we then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties

fast iterative solution of the bethe salpeter eigenvalue problem using low rank and qtt tensor approximation abstract in this paper we propose and study two approaches to approximate the solution of the bethe salpeter equation bse by using structured iterative eigenvalue solvers both approaches are based on the reduced basis method and low rank factorizations of the generating matrices we also propose to represent the static screen interaction part in the bse matrix by a small active sub block with a size balancing the storage for rank structured representations of other matrix blocks we demonstrate by various numerical tests that the combination of the diagonal plus low rank plus reduced block approximation exhibits higher precision with low numerical cost providing as well a distinct two sided error estimate for the smallest eigenvalues of the bethe salpeter operator the complexity is reduced to o n b 2 in the size of the atomic orbitals basis set n b instead of the practically intractable o n b 6 scaling for the direct diagonalization in the second approach we apply the quantized tt qtt tensor representation to both the long eigenvectors and the column vectors in the rank structured bse matrix blocks and combine this with the als type iteration in block qtt format the qtt rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems n o n b hence the overall asymptotic complexity for solving the bse problem by the qtt approximation is estimated by o log n o n o 2 we confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain type molecules while supporting sufficient accuracy

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

the implicitization problem for ϕ pn p1 n 1 abstract we develop in this paper methods for studying the implicitization problem for a rational map p n p 1 n 1 defining a hypersurface in p 1 n 1 based on computing the determinant of a graded strand of a koszul complex we show that the classical study of macaulay resultants and koszul complexes coincides in this case with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions under suitable hypotheses these techniques enable us to obtain the implicit equation up to a power and up to some extra factor we give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of and what are the extra varieties that appear we also give some applications to the problem of computing sparse discriminants

a numerical solver for high dimensional transient fokker planck equation in modeling polymeric fluids in this paper a tensor decomposition approach combined with chebyshev spectral differentiation is presented to solve the high dimensional transient fokker planck equations fpe arising in the simulation of polymeric fluids via multi bead spring mbs model generalizing the authors previous work on the stationary fpe the transient solution is obtained in a single candecomp parafac decomposition cpd form for all times via the alternating least squares algorithm this is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions thereby decoupling it from them as a result the transient solution is obtained without resorting to expensive time stepping schemes a new relaxed approach for imposing the vanishing boundary conditions is proposed improving the quality of the approximation the asymptotic behavior of the temporal basis functions is studied the proposed solver scales very well with the dimensionality of the mbs model numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix riccati differential equation for linear models or monte carlo simulations for nonlinear models

wavelet in time multigrid in space preconditioning of parabolic evolution equations two space time variational formulations of linear parabolic evolution equations are considered one is symmetric and elliptic on the trial space while the other is not in each case a space time petrov galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations the well posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis for the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet in time transformation on possibly nonuniform temporal meshes this transformation approximately block diagonalizes the preconditioner and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel the performance of the preconditioner is documented in a series of numerical experiments

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

monotone waveform relaxation for systems of nonlinear differential algebraic equations we present a monotone waveform relaxation algorithm which produces very tight upper and lower bounds of the transient response of a class of systems described by nonlinear differential algebraic equations daes that satisfy certain lipschitz conditions the choice of initial iteration is critical and we give two methods of finding it we show that the class of systems in which monotone convergence of waveform relaxation is possible is actually larger than previously reported numerical experiments are given to confirm the monotonicity of convergence of the algorithm

on time domain simulation of lossless transmission lines with nonlinear terminations a time domain approach is presented to solve nonlinear circuits with lossless transmission lines mathematically the circuits are described by a special kind of nonlinear differential algebraic equations daes with multiple constant delays in order to directly compute these delay systems in time domain decoupling by waveform relaxation wr is applied to the systems for the relaxation based method we provide a new convergence proof numerical experiments are given to illustrate the novel approach

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

adaptive wavelet methods for elliptic partial differential equations with random operators we apply adaptive wavelet methods to boundary value problems with random coefficients discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain greedy algorithms control the approximate application of the fully discretized random operator and the construction of sparse approximations to this operator we suggest a power iteration for estimating errors induced by sparse approximations of linear operators

syzygies and singularities of tensor product surfaces of bidegree 2 1 let u h op1 p1 2 1 be a basepoint free four dimensional vector space the sections corresponding to u determine a regular map u p1 p1 p3 we study the associated bigraded ideal iu k s t u v from the standpoint of commutative algebra proving that there are exactly six numerical types of possible bigraded minimal free resolution these resolutions play a key role in determining the implicit equation for u p1 p1 via work of buse jouanolou 5 buse chardin 6 botbol 2 and botbol dickensteindohm 3 on the approximation complex z in four of the six cases iu has a linear first syzygy remarkably from this we obtain all differentials in the minimal free resolution in particular this allows us to explicitly describe the implicit equation and singular locus of the image

on the h 1 h1 stability of the l_2 l2 projection onto finite element spaces we study the stability in the h 1 h 1 seminorm of the l 2 l 2 projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove in particular stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven