generalized parametric solutions in stokes flow abstract design optimization and uncertainty quantification among other applications of industrial interest require fast or multiple queries of some parametric model the proper generalized decomposition pgd provides a separable solution a computational vademecum explicitly dependent on the parameters efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored this strategy has been successfully employed in many problems in computational mechanics the application to problems with saddle point structure raises some difficulties requiring further attention this article proposes a pgd formulation of the stokes problem various possibilities of the separated forms of the pgd solutions are discussed and analyzed selecting the more viable option the efficacy of the proposed methodology is demonstrated in numerical examples for both stokes and brinkman models

quantum monte carlo methods recent developments abstract recent developments of the quantum monte carlo methods are reviewed briefly with emphasis on basic ideas namely the transformation of a d dimensional quantum system to the corresponding d 1 dimensional classical system some recent applications to quantum spin systems are summarized in particular numerical evidence of the haldane gap is given the quantum transfer matrix method is also presented for it is also based on the st transformation the quantum monte carlo cam is explained briefly fractal path integrals are also presented namely a general theory of decomposition of exponential operators and symplectic integrators are explained the negative sign problem is also discussed

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

on estimating the separation between a b and c d associated with the generalized sylvester equation axd bxc e abstract in this paper we estimate the following quantity sep a b c d min x f 1 a x d b x c f where a b c m m c d c n n and f is the frobenius norm in terms of the generalized schur decomposition and weiestrass canonical form of the regular matrix pair lower and upper bounds of sep a b c d are derived respectively the results are illustrated by numerical examples

computing real low rank solutions of sylvester equations by the factored adi method abstract we investigate the factored alternating directions implicit adi iteration for large and sparse sylvester equations a novel low rank expression for the associated sylvester residual is established which enables cheap computations of the residual norm along the iteration and which yields a reformulated factored adi iteration the application to generalized sylvester equations is considered as well we also discuss the efficient handling of complex shift parameters and reveal interconnections between the adi iterates w r t those complex shifts this yields a further modification of the factored adi iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage and which produces low rank solution factors consisting of entirely real data certain linear matrix equations such as e g cross gramian sylvester and stein equations are in fact special cases of generalized sylvester equations and we show how specially tailored low rank adi iterations can be deduced from the generalized factored adi iteration

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

a low rank krylov squared smith method for large scale discrete time lyapunov equations abstract the squared smith method is adapted to solve large scale discrete time lyapunov matrix equations the adaptation uses a krylov subspace to generate the squared smith iteration in a low rank form a restarting mechanism is employed to cope with the increase of memory storage of the krylov basis theoretical aspects of the algorithm are presented several numerical illustrations are reported

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

parametric solutions involving geometry a step towards efficient shape optimization abstract optimization of manufacturing processes or structures involves the optimal choice of many parameters process parameters material parameters or geometrical parameters usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model then an appropriate cost function is evaluated and its optimality checked while the optimum is not reached the process parameters should be updated by using an appropriate optimization procedure and then the model must be solved again for the updated process parameters thus a direct numerical solution is needed for each choice of the process parameters with the subsequent impact on the computing time in this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry the main objective of this work is to describe an original approach for computing an off line parametric solution that is a solution able to include information for different parameter values and also allowing to compute readily the sensitivities the curse of dimensionality is circumvented by invoking the proper generalized decomposition pgd introduced in former works which is applied here to compute geometrically parametrized solutions

generalized bracket formulation of constrained dynamics in phase space a generalized bracket formalism is used to define the phase space flow of constrained systems the generalized bracket naturally subsumes the approach to constrained dynamics given by dirac some time ago the dynamical invariant measure and the linear response of systems subjected to holonomic constraints are explicitly derived in light of previous results it is shown that generalized brackets provide a simple and unified view of the statistical mechanics of non hamiltonian phase space flows with a conserved energy

almost symplectic runge kutta schemes for hamiltonian systems symplectic runge kutta schemes for the integration of general hamiltonian systems are implicit in practice one has to solve the implicit algebraic equations using some iterative approximation method in which case the resulting integration scheme is no longer symplectic in this paper the preservation of the symplectic structure is analyzed under two popular approximation schemes fixed point iteration and newton s method respectively error bounds for the symplectic structure are established when n fixed point iterations or n iterations of newton s method are used the implications of these results for the implementation of symplectic methods are discussed and then explored through numerical examples numerical comparisons with non symplectic runge kutta methods and pseudo symplectic methods are also presented

numerical computation and new output bounds for time limited balanced truncation of discrete time systems abstract in this paper balancing based model order reduction mor for large scale linear discrete time time invariant systems in prescribed finite time intervals is studied the first main topic is the development of error bounds regarding the approximated output vector within the time limits the influence of different components in the established bounds will be highlighted after that the second part of the article proposes strategies that enable an efficient numerical execution of time limited balanced truncation for large scale systems numerical experiments illustrate the performance of the proposed techniques

finite dimensional lie poisson approximations to vlasov poisson equations many of the basic equations of conservative continuum mechanics euler vlasov poisson vlasov maxwell mhd etc are hamiltonian systems with respect to lie poisson brackets on dual spaces of infinite dimensional lie algebras the development of lie poisson integrators for finite dimensional lie poisson systems has shown that they are superior in the numerical simulations of these systems especially with regard to long term phenomena this paper shows how to truncate one of these systems the vlasov poisson equation of plasma physics to a finite dimensional lie poisson system this requires replacing the functions on single particle phase space with their poisson bracket lie algebra structure by a finite dimensional lie algebra replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their taylor expansions up to order k unfortunately these truncated taylor expansions do not form a lie algebra since the functions which vanish through order k do not form an ideal under poisson bracket geometrically this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the location from shape coordinates of a lump of density in phase space as far as the poisson bracket is concerned one can then replace the shape coordinates by a finite number of moments the central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a lie group as a product of subgroups the main theorem is first proved by the general theory of poisson reduction then by explicit calculation and lastly by showing that the poisson isomorphism follows from the lift of a natural groupoid isomorphism the groupoid aspect of the theory also provides natural poisson maps useful in the application of ruth type integration techniques which do not seem easily derivable from the general theory of poisson reduction 1994 john wiley sons inc

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

exactly conservative integrators traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants in this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly we illustrate the method by applying it to the three wave truncation of the euler equations the lotka volterra predator prey model and the kepler problem the ideas are discussed in the context of symplectic phase space conserving integration methods as well as nonsymplectic conservative methods we comment on the application of our method to general conservative systems

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

new numerical integrator for the stackel system conserving the same number of constants of motion as the degree of freedom a new numerical integrator for a wide class of separable hamiltonian systems called the stackel system is presented this integrator is designed so as to conserve the same number of constants of motion as the degree of freedom of the stackel system separation of variables of the stackel system is most fundamental for the integrator a combination of canonical transformations and an energy preserving method with a variable step size plays a key role to design such an integrator some typical and important examples of the stackel system are then discretized explicitly they are the three dimensional kepler motion the holt system and the integrable henon heiles system in celestial mechanics

a helmholtz iterative solver for 3d seismic imaging problems a preconditioned iterative solver for the 3d frequency domain wave equation applied to seismic problems is evaluated the preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the undamped wave equation the approximate inverse is computed with one multigrid cycle numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength to evaluate the relevance of this iterative solver the number of floating point operations required for two imaging problems are roughly evaluated this rough estimate indicates that the time domain migration approach is more than one order of magnitude faster the full wave form tomography based on a least squares formulation and a scale separation approach has the same complexity in both domains

on the convergence of inexact newton methods for discrete time algebraic riccati equations abstract in this paper we present a convergence analysis of the inexact newton method for solving discrete time algebraic riccati equations dares for large and sparse systems the inexact newton method requires at each iteration the solution of a symmetric stein matrix equation these linear matrix equations are solved approximatively by the alternating directions implicit adi or smith s methods we give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact newton sequences we show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d stability the theoretical results developed in this paper are an extension to the discrete case of the analysis performed by feitzinger et al 2009 8 for the continuous time algebraic riccati equations in the last section we give some numerical experiments

an exactly conservative integrator for the n body problem the two dimensional n body problem of classical mechanics is a non integrable hamiltonian system for n 2 traditional numerical integration algorithms which are polynomials in the time step typically lead to systematic drifts in the computed value of the total energy and angular momentum even symplectic integration schemes exactly conserve only an approximate hamiltonian we present an algorithm that conserves the true hamiltonian and the total angular momentum to machine precision it is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables we develop the method first for the restricted circular three body problem then for the general two dimensional three body problem and finally for the planar n body problem jacobi coordinates are used to reduce the two dimensional n body problem to an n 1 body problem that incorporates the constant linear momentum and centre of mass constraints for a four body choreography we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators

proper generalized decomposition for parameterized helmholtz problems in heterogeneous and unbounded domains application to harbor agitation abstract solving the helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge this is due to the particular nature of the helmholtz operator and the sensibility of the solution to small variations of the data here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency moreover this is applied to a real engineering problem water agitation inside real harbors for low to mid high frequencies the proper generalized decomposition pgd model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency here its applicability to such a problem is discussed and demonstrated more precisely the contributions of the paper include the pgd implementation into a perfectly matched layer framework to model the unbounded domain and the separability of the operator which is addressed here using an efficient higher order projection scheme then the performance of the pgd in this framework is discussed and improved using the higher order projection and a petrov galerkin approach to construct the separated basis moreover the efficiency of the higher order projection scheme is demonstrated and compared with the higher order singular value decomposition

iterative methods for x axb c alternating direction implicit iterative methods for the solution of matrix equations of the form x axb c are described a convergence analysis based on potential theory shows that iterating in one direction more than the other can give faster convergence than strict alternation of directions computed examples illustrate an application to the restoration of images degraded by noise

computation of nonlinear behavior of hamiltonian systems using lie algebraic methods lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general hamiltonian systems a procedure is presented for the computation of nonlinear effects of arbitrarily high degree and explicit formulas are given through effects of degree 5 expected applications include accelerator design charged particle beam and light optics other problems in the general area of nonlinear dynamics and perhaps with suitable modification the area of s matrix expansions in quantum field theory

adi preconditioned krylov methods for large lyapunov matrix equations abstract in the present paper we propose preconditioned krylov methods for solving large lyapunov matrix equations ax xa t bb t 0 such problems appear in control theory model reduction circuit simulation and others using the alternating direction implicit adi iteration method we transform the original lyapunov equation to an equivalent symmetric stein equation depending on some adi parameters we then define the smith and the low rank adi preconditioners to solve the obtained stein matrix equation we apply the global arnoldi method and get low rank approximate solutions we give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

proper generalized decomposition solution of the parameterized helmholtz problem application to inverse geophysical problems this is the peer reviewed version of the following article signorini m zlotnik s and diez p 2017 proper generalized decomposition solution of the parameterized helmholtz problem application to inverse geophysical problems int j numer meth engng 109 1085 1102 doi 10 1002 nme 5313 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5313 full this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

bayesian and variational bayesian approaches for flows in heterogeneous random media in this paper we study porous media flows in heterogeneous stochastic media we propose an efficient forward simulation technique that is tailored for variational bayesian inversion as a starting point the proposed forward simulation technique decomposes the solution into the sum of separable functions with respect to randomness and the space where each term is calculated based on a variational approach this is similar to proper generalized decomposition pgd next we apply a multiscale technique to solve for each term as in 1 and further decompose the random function into 1d fields as a result our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and also increase the spatial resolution of each term we use the hierarchical solution distributions in a variational bayesian approximation to perform uncertainty quantification in the inverse problem we conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration

on the adi method for sylvester equations this paper is concerned with the numerical solution of large scale sylvester equations ax xb c lyapunov equations as a special case in particular included with c having very small rank for stable lyapunov equations penzl 2000 22 and li and white 2002 20 demonstrated that the so called cholesky factor adi method with decent shift parameters can be very effective in this paper we present a generalization of the cholesky factor adi method for sylvester equations an easily implementable extension of penz s shift strategy for the lyapunov equation is presented for the current case it is demonstrated that galerkin projection via adi subspaces often produces much more accurate solutions than adi solutions

frequency domain acoustic wave modeling and inversion of crosshole data part 1 2 5 d modeling method crosshole experiments usually have sources and receivers confined to a plane and it is assumed that there is negligible variation in the properties of the medium normal to this plane therefore the problem appears two dimensional except for the sources which are 3 d points rather than lines this configuration is denoted as two and one half dimensional we present a frequency domain approach to modeling acoustic wave propagation in such situations which allows correct treatment of point sources but takes advantage of the assumed 2 d nature of the medium to avoid full 3 d simulations the approach uses a fourier transform with respect to the out of plane coordinate to reduce the problem of modeling in 3 d to repeatedly solving a 2 d equation which we accomplish using finite differences the discrete inverse fourier transform from the out of plane wavenumber implies the existence of an infinite number of spurious ghost sources spaced periodically in the out of plane direction these sources generate s

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix