iterative solution of a 3 d scattering problem from arbitrary shaped multidielectric and multiconducting bodies we present an iterative algorithm used for the computation of the scattering from arbitrary shaped bodies made of dielectric and conducting material the harmonic problem is discretized by a hybrid boundary integral method finite element method to solve the discretized linear system a modified version of a minimal residual algorithm has been developed for complex matrices it takes into account the lagrange multiplier constraint in a special way this modified version also allows the linear system to be solved for many right hand sides in an efficient manner

computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation a mathematical formulation and some numerical approximation techniques are described for a system of coupled partial differential and algebraic equations describing multiphase flow transport and interactions of chemical species in the subsurface a parallel simulator parsim has been developed based on these approximation techniques and is being used to study contaminant remediation strategies numerical results for a highly complex geochemistry problem involving strontium disposal in a pit at oak ridge national laboratory are presented

on decoupling inequalities and percolation of excursion sets of the gaussian free field we prove decoupling inequalities for the gaussian free field on mathbb z d d ge 3 as an application we obtain exponential decay with logarithmic correction for d 3 of the connectivity function of excursion sets for large values of the threshold

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

a low cost goal oriented compact proper orthogonal decomposition basis for model reduction of static systems a novel model reduction technique for static systems is presented the method is developed using a goal oriented framework and it extends the concept of snapshots for proper orthogonal decomposition pod to include sensitivity derivatives of the state with respect to system input parameters the resulting reduced order model generates accurate approximations due to its goal oriented construction and the explicit training of the model for parameter changes the model is less computationally expensive to construct than typical pod approaches since efficient multiple right hand side solvers can be used to compute the sensitivity derivatives the effectiveness of the method is demonstrated on a parameterized aerospace structure problem

supernatural quad4 a template formulation abstract a universal template for a 4 node quadrilateral in plane stress is constructed using a combination of old and new techniques the use of natural quantities strongly advocated by j h argyris since his continua and discontinua 1965 exposition is further expanded the qualifier supernatural means that all governing equations kinematic constitutive and equilibrium are expressed in both cartesian and natural forms these two sets are used to build different components of the template with timely help from a computer algebra system a template that include all possible quadrilateral elements that pass the individual element test of bergan and hanssen emerges it yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs a striking example is the construction of a two trapezoid macroelement that is bending exact about one direction for any amount of distortion this concludes a five decade search that begins with the formulation of the wing cover rectangular panel in argyris 1954 seminal serial on energy methods and structural analysis

wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering daubechies family of wavelets combined to the incomplete lower upper ilu factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the partition of unity finite element method pufem after applying the discrete wavelet transform dwt to each dense block in the final matrix and the known right hand side due to the local enrichment by pressure p and shear s plane waves the resulting linear system is solved by the restarted generalized minimum residual method gmres with ilu preconditioners allowing fill in elements in the l and u matrix factors a reordering algorithm of the vertices based on gibbs method is also introduced it leads to a significant reduction of the bandwidth of the wave based finite element fe matrix and enables the ilu preconditioners to be more effective to study the performance of the proposed preconditioners a problem of a horizontal s plane wave scattered by a rigid circular body in an infinite elastic medium is considered the numerical tests show the good performance of the dwt based ilu preconditioners in improving the rate of convergence of gmres for high numbers of approximating p and s plane waves on coarse mesh grids containing multi wavelength sized elements moreover the haar dwt enhances the scalability with respect to the problem size when the number of the nodal points increases of the ilu preconditioner which uses the threshold strategy in the control of the fill in elements despite the high level of the conditioning a good accuracy may be achieved for a discretization level of around 1 9 degrees of freedom per s wavelength which is far below the rule of thumb of 10 nodal points per wavelength adopted for the conventional fe

nonlinear wave and schrodinger equations on compact lie groups and homogeneous spaces we develop linear and nonlinear harmonic analysis on compact lie groups and homogeneous spaces relevant for the theory of evolutionary hamiltonian pdes a basic tool is the theory of the highest weight for irreducible representations of compact lie groups this theory provides an accurate description of the eigenvalues of the laplace beltrami operator as well as the multiplication rules of its eigenfunctions as an application we prove the existence of cantor families of small amplitude time periodic solutions for wave and schrodinger equations with differentiable nonlinearities we apply an abstract nash moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the laplace operator we provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on banach scales of sobolev functions

mode jumping in the von karman equations we study numerical approximations of bifurcating solution curves of the von karman equations with simply supported and clamped boundary conditions respectively of special interest here is the splitting of a double bifurcation point into two simple bifurcation points and the tracing of the associated secondary solution branches which corresponds to the phenomenon of mode jumping in the buckling of a rectangular plate a continuation unsymmetric lanczos algorithm is utilized for curve tracking our numerical results verify the theoretical prediction of schaeffer and golubitsky namely mode jumping occurs when the boundary is partially clamped but not if it is merely simply supported

a comparison of nrbcs for pufem in 2d helmholtz problems at high wave numbers in this work exact and approximate non reflecting boundary conditions nrbcs are implemented with the partition of unity finite element method pufem to solve short wave scattering problems governed by the helmholtz equation in two dimensions by short wave problems we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer various nrbcs are implemented and a comparison of their performance is carried out based on the accuracy of the results ease of implementation and computational cost the aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number

very weak solutions to hypoelliptic wave equations abstract in this paper we study the cauchy problem for the wave equations for hypoelliptic homogeneous left invariant operators on graded lie groups when the time dependent non negative propagation speed is regular holder and distributional for holder coefficients we derive the well posedness in the spaces of ultradistributions associated to rockland operators on graded groups in the case when the propagation speed is a distribution we employ the notion of very weak solutions to the cauchy problem that was already successfully used in similar contexts in 12 and 20 we show that the cauchy problem for the wave equation with the distributional coefficient has a unique very weak solution in an appropriate sense which coincides with classical or distributional solutions when the latter exist examples include the time dependent wave equation for the sub laplacian on the heisenberg group or on general stratified lie groups or p evolution equations for higher order operators on r n or on groups the results already being new in all these cases

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

galerkin projection methods for solving multiple linear systems in this paper we consider using conjugate gradient cg methods for solving multiple linear systems a i x i b i for 1 le i le s where the coefficient matrices a i and the right hand sides b i are different in general in particular we focus on the seed projection method which generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems onto the generated krylov subspace to get the approximate solutions the whole process is repeated until all the systems are solved most papers in the literature t f chan and w l wan it siam j sci comput 18 1997 pp 1698 1721 b parlett it linear algebra appl 29 1980 pp 323 346 y saad it math comp 48 1987 pp 651 662 v simoncini and e gallopoulos it siam j sci comput 16 1995 pp 917 933 c smith a peterson and r mittra it ieee trans antennas and propagation 37 1989 pp 1490 1493 considered only the case where the coefficient matrices a i are the same but the right hand sides are different we extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right hand sides a theoretical error bound is given for the approximation obtained from a projection process onto a krylov subspace generated from solving a previous linear system finally numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method

critical exponent of fujita type for the semilinear damped wave equation on the heisenberg group with power nonlinearity abstract in this paper we consider the cauchy problem for the semilinear damped wave equation on the heisenberg group with power non linearity we prove that the critical exponent is the fujita exponent p fuj q 1 2 q where q is the homogeneous dimension of the heisenberg group on the one hand we will prove the global existence of small data solutions for p p fuj q in an exponential weighted energy space on the other hand a blow up result for 1 p p fuj q under certain integral sign assumptions for the cauchy data by using the test function method

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

a deflated version of the conjugate gradient algorithm we present a deflated version of the conjugate gradient algorithm for solving linear systems the new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin it can also be useful when solving linear systems with multiple right hand sides since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated

conjugate gradient type methods for semilinear elliptic problems with symmetry we study block conjugate gradient methods in the context of continuation methods for bifurcation problems by exploiting symmetry in certain semilinear elliptic differential equations we can decompose the problems into small ones and reduce computational cost on the other hand the associated centered difference discretization matrices on the subdomains are nonsymmetric we symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices these properties allow the discrete pure mode solution paths branching from a multiple bifurcation point 0 m n of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point 0 1 1 of the counterpart of the reduced problem thus the structure of a multiple bifurcation is preserved in discretization while its treatment is reduced to those for simple bifurcation of problems on subdomains in particular we can adapt the continuation lanczos algorithm proposed in 1 to trace simple solution paths sample numerical results are reported

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

numerical methods for treating problems of viscoelastic isotropic solid deformation mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described the range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity nonlinearity and the limitations imposed by the availability of physical data are discussed one of the above models is a volterra integral equation of the second kind in this when the kernel is separable an established technique due to goursat 1933 can be exploited to reformulate the problem as a system of ordinary differential equations this approach will be described for the special case of a linear viscoelastic isotropic homogeneous synchronous constant poisson s ratio solid this method results in a complete decoupling of the space and time dependence in this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required the advantages disadvantages and limitations offered by this and various other schemes for solving problems of viscoelasticity as outlined below are discussed

nonlinear damped wave equations for the sub laplacian on the heisenberg group and for rockland operators on graded lie groups abstract in this paper we study the cauchy problem for the semilinear damped wave equation for the sub laplacian on the heisenberg group in the case of the positive mass we show the global in time well posedness for small data for power like nonlinearities we also obtain similar well posedness results for the wave equations for rockland operators on general graded lie groups in particular this includes higher order operators on r n and on the heisenberg group such as powers of the laplacian or the sub laplacian in addition we establish a new family of gagliardo nirenberg inequalities on a graded lie groups that play a crucial role in the proof but which are also of interest on their own if g is a graded lie group of homogeneous dimension q and a 0 1 r q a and 1 p q r q q a r then we have the following gagliardo nirenberg type inequality u l q g u l a r g s u l p g 1 s for s 1 p 1 q a q 1 p 1 r 1 0 1 provided that a q 1 p 1 r 0 where l a r is the homogeneous sobolev space of order a over l r if a q 1 p 1 r 0 we have p q r q q a r and then the above inequality holds for any 0 s 1

global least squares method gl lsqr for solving general linear systems with several right hand sides abstract in this paper we propose a new method for solving general linear systems with several right hand sides this method is based on global least squares method and reduces the original matrix to the lower bidiagonal form we derive a simple recurrence formula for generating the sequence of approximate solutions x k some theoretical properties of the new method are discussed and we also show that how this method can be implemented for the sylvester equation finally some numerical experiments on test matrices are presented to show the efficiency of the new method

deflated and restarted symmetric lanczos methods for eigenvalues and linear equations with multiple right hand sides a deflated restarted lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors the restarting limits the storage so that finding eigenvectors is practical meanwhile the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting some reorthogonalization is necessary to control roundoff error and several approaches are discussed the eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides experiments are given with large matrices from quantum chromodynamics that have many right hand sides

an eulerian method for transient nonlinear free surface wave problems abstract an eulerian difference method is developed for the transient potential flow of an incompressible fluid with fully nonlinear free surface conditions the free surface coordinate y x t and the velocity potential x y t on the free surface are recognized as the primary unknowns to be solved as an initial value problem from the pair of nonlinear partial differential equations representing the dynamic and the kinematic conditions of the free surface the continuity relation 2 0 for the velocity potential x y t over the flow field below the free surface is recognized as a subsidiary condition to be enforced at all times the field of computation is transformed into a time invariant cartesian region with the free surface x t represented by a coordinate line or surface the iterative solution for x y t p x y t in this fixed field of computation is facilitated by the use of fast fourier transform fft the iterative process converges rapidly in terms of this converged x y t the free surface location x t and its potential x y t are advanced in time results from two planar examples are illustrated the method is equally applicable to problems in three space dimensions possibly involving interactive matching with neighboring flow fields if the initial free surface potential x y t 0 is unknown difficulties may be encountered in data specification for securing a well posed problem for solution

critical friction for wedged configurations abstract a wedged configuration with coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads we analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3 d context the critical friction coefficient w is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses for friction coefficients with w the wedged problem has at least a solution and for the wedged problem is stated in a discrete framework using a mixed finite element approach and the discrete critical friction coefficient h w is introduced as the solution of a global minimization problem involving a non differentiable and non convex functional the existence of h the displacement field of a critical wedged state is proved and a specific numerical method based on a genetic algorithm was developed to compute the critical wedged configurations some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

on some recent adventures into improved finite element cfd methods for convective transport abstract the quest continues for computational fluid dynamics cfd algorithms that are accurate and efficient for convection dominated applications including shocks travelling fronts and wall layers the boundary value optimal galerkin weak statement invariably requires manipulation either in the test space or in an augmented form for the conservation law system to handle the disruptive character introduced by the discretized first order convection term an incredible variety of methodologies have been derived and examined to address this issue in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation the utk cfd research group continues its search on examining the breadth of approaches leading to development of a consistent encompassing theoretical statement exhibiting quality performance included herein are adventures into generalized taylor series lax wendroff methods characteristic euler flux resolutions sub grid embedded high degree lagrange bases with static condensation and assembled stencil fourier analysis optimization for finite element weak statement implementations for appropriate model problems including steady convection diffusion and pure unsteady convection and benchmark navier stokes definitions recent advances have lead to candidate accurate monotone methods with linear basis efficiency this contribution highlights the theoretical developments and presents quantitative documentation of achievable high quality solutions

wave equation for sums of squares on compact lie groups in this paper we investigate the well posedness of the cauchy problem for the wave equation for sums of squares of vector fields on compact lie groups we obtain the loss of regularity for solutions to the cauchy problem in local sobolev spaces depending on the order to which the hormander condition is satisfied but no loss in globally defined spaces we also establish gevrey well posedness for equations with irregular coefficients and or multiple characteristics as in the sobolev spaces if formulated in local coordinates we observe well posedness with the loss of local gevrey order depending on the order to which the hormander condition is satisfied

a comparative study of iterative solvers exploiting spectral information for spd systems when solving the symmetric positive definite spd linear system bf a bf x star bf b with the conjugate gradient method the smallest eigenvalues in the matrix bf a often slow down the convergence consequently if the smallest eigenvalues in bf a could somehow be removed the convergence may be improved this observation is of importance even when a preconditioner is used and some extra techniques might be investigated to further improve the convergence rate of the conjugate gradient on the given preconditioned system several techniques have been proposed in the literature that consist of either updating the preconditioner or enforcing conjugate gradient to work in the orthogonal complement of an invariant subspace associated with the smallest eigenvalues the goal of this work is to compare several of these techniques in terms of numerical efficiency among various possibilities we exploit the partial spectral factorization algorithm presented in m arioli and d ruiz technical report ral tr 2002 021 rutherford appleton laboratory atlas center didcot oxfordshire england 2002 to compute an orthonormal basis of a near invariant subspace of bf a associated with the smallest eigenvalues this eigeninformation is used in combination with different solution techniques in particular we consider the deflated version of conjugate gradient as representative of techniques exploiting the spectral information to update the preconditioner we consider also the approaches that attempt to shift the smallest eigenvalues close to one where most of the eigenvalues of the preconditioned matrix should be located finally we consider an algebraic two grid scheme inspired by ideas from the multigrid philosophy in this paper we describe these various variants and we compare their numerical behavior on a set of model problems from matrix market or arising from the discretization via the finite element technique of some two dimensional 2d heterogeneous diffusion pde problems we discuss their numerical efficiency computational complexity and sensitivity to the accuracy of the eigencalculation

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

quenched invariance principle for simple random walk on clusters in correlated percolation models we prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on mathbb z d d ge 2 with long range correlations introduced in drewitz et al in j math phys 55 8 083307 2014 solving one of the open problems from there this gives new results for random interlacements in dimension d ge 3 at every level as well as for the vacant set of random interlacements and the level sets of the gaussian free field in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime an essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models

analysis of projection methods for solving linear systems with multiple right hand sides we analyze a class of krylov projection methods but mainly concentrate on a specific conjugate gradient cg implementation by smith peterson and mittra ieee transactions on antennas and propogation 37 1989 pp 1490 1493 to solve the linear system ax b where a is symmetric positive definite and b is a multiple of right hand sides this method generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems orthogonally onto the generated krylov subspace to get the approximate solutions the whole process is repeated with another unsolved system as a seed until all the systems are solved we observe in practice a superconvergence behavior of the cg process of the seed system when compared with the usual cg process we also observe that only a small number of restarts is required to solve all the systems if the right hand sides are close to each other these two features together make the method particularly effective in this paper we give theoretical proof to justify these observations furthermore we combine the advantages of this method and the block cg method and propose a block extension of this single seed method

conjugate gradient type methods and preconditioning in this paper we consider various iterative methods for the numerical solution of very large sparse linear systems of equations which arise in the discretization of partial differential equations as the performance of the methods generally depends on the characteristics of the problems to be solved a judicious choice between the methods will require knowledge about the system the aim of this paper is to review the properties of the various iteration methods in order to assist the user in making a deliberate selection

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

global scd algorithm for real positive definite linear systems with multiple right hand sides abstract in the present paper we present the left conjugate direction lcd method for linear systems with multiple right hand sides we define the method by using left and right conjugate direction matrices and petrov galerkin condition the method has no breakdown for real positive definite systems the method reduces to the usual cg type method when a is symmetric positive definite we also show how to apply these methods for solving the lyapunov matrix equation finally a numerical experiment is given to illustrate the result

successive conjugate gradient methods for structural analysis with multiple load cases this paper discusses the use of conjugate gradient method for structural engineering analysis with multiple load cases specifically we introduce a modified conjugate gradient method for the solution of a system of linear equations with multiple right hand sides we show how this method can be used to improve the performance of block conjugate gradient method numerical results are presented to illustrate the effectiveness of the numerical schemes developed in this study

a priori and a posteriori error analysis for a large scale ocean circulation finite element model we consider the finite element solution of the stream function vorticity formulation for a large scale ocean circulation model first we study existence and uniqueness of solution for the continuous and discrete problems under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in h1 seminorm second we introduce and analyze an h adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant we propose an a posteriori anisotropic error indicator based on the recovery of the hessian from the finite element solution which allows us to obtain well adapted meshes the numerical experiments show an optimal order of convergence of the adaptive scheme furthermore this strategy is efficient to eliminate the oscillations around the boundary layer

s step iterative methods for symmetric linear systems abstract in this paper we introduce s step conjugate gradient method for symmetric and positive definite spd linear systems of equations and discuss its convergence in the s step conjugate gradient method iteration s new directions are formed simultaneously from r i ar i a s 1 r i and the preceding s directions all s directions are chosen to be a orthogonal to the preceding s directions the approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions this intuitively means that the progress towards the solution in one iteration of the s step method equals the progress made over s consecutive steps of the one step method this is proven to be true

multiple right hand side techniques for the numerical simulation of quasistatic electric and magnetic fields the simulation of slowly varying transient electric high voltage fields and magnetic fields requires the repeated and successive solution of high dimensional linear algebraic systems of equations with identical or near identical system matrices and different right hand side vectors for these solution processes which are required within implicit time integration schemes and nonlinear quasi newton raphson methods an iterative multiple right hand side mrhs scheme is used which recycles vector subspaces resulting from previous preconditioned conjugate gradient iteration runs the combination of this scheme with a subspace projection extrapolation start value generation scheme is discussed numerical results for three dimensional electric and magnetic field simulations are presented and the efficiency of the new schemes re using eigenvector information from previous iteration processes with different tolerance criteria are compared to those of standard conjugate gradient iterations

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

fem bem coupling for the large body limit in micromagnetics we present and analyze a coupled finite element boundary element method for a model in stationary micromagnetics the finite element part is based on mixed conforming elements for two and three dimensional settings we show well posedness of the discrete problem and present an a priori error analysis for the case of lowest order elements

parallel domain decomposition procedures of improved d d type for parabolic problems two parallel domain decomposition procedures for solving initial boundary value problems of parabolic partial differential equations are proposed one is the extended d d type algorithm which extends the explicit implicit conservative galerkin domain decomposition procedures given in 5 from a rectangle domain and its decomposition that consisted of a stripe of sub rectangles into a general domain and its general decomposition with a net like structure an almost optimal error estimate without the factor h 1 2 given in dawson dupont s error estimate is proved another is the parallel domain decomposition algorithm of improved d d type in which an additional term is introduced to produce an approximation of an optimal error accuracy in l 2 norm

global lsmr gl lsmr method for solving general linear systems with several right hand sides abstract the global solvers are an attractive class of iterative solvers for solving linear systems with multiple right hand sides in this paper first a new global method for solving general linear systems with several right hand sides is presented this method is the global version of the lsmr algorithm presented by fong and saunders 2011 then some theoretical properties of the new method are discussed finally numerical experiments from real applications are used to confirm the effectiveness of the proposed method

application of a block modified chebyshev algorithmto the iterative solution of symmetric linear systemswith multiple right hand side vectors an adaptive richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right hand side vectors this scheme learns about the linear system to be solved by computing inner products of residual matrices during the iterations these inner products are interpreted as block modified moments a block version of the modified chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials the eigenvalues of this block tridiagonal matrix define an interval which determines the choice of relaxation parameters for richardson iteration only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right hand side vectors we outline the changes required

cache issues of algebraic multigrid methods for linear systems with multiple right hand sides this paper is concerned with the solution of a linear system with multiple right hand sides such problems arise from nonlinear time dependent inverse or optimization problems in order to solve these problems efficiently we use block variants of the conjugate gradient method and combine them with cache aware algorithms further we describe the acceleration of algebraic multigrid amg methods by using such cache aware algorithms finally numerical studies are presented that show the high efficiency of the proposed methods

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a new family of global methods for linear systems with multiple right hand sides the global bi conjugate gradient gl bcg method is an attractive matrix krylov subspace method for solving nonsymmetric linear systems with multiple right hand sides but it often show irregular convergence behavior in many applications in this paper we present a new family of global a biorthogonal methods by using short two term recurrences and formal orthogonal polynomials which contain the global bi conjugate residual gl bcr algorithm and its improved version finally numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

updating component reduction bases of static and vibration modes using preconditioned iterative techniques abstract although component mode synthesis cms methods for structural dynamic analysis were introduced in the 1960s one of the current challenges is to apply the methodology in a practical design process in which parametric modifications are applied to the component s for every design change these components need to be remodeled and their reduction bases recomputed thereby seriously hindering the effectiveness of the cms approach to overcome this we propose in this work an approach based on preconditioned iterative methods to efficiently update the reduction basis for the updating of the static modes we use the well known preconditioned conjugate gradient cg method applied in a block form and combined with the projection and reconjugation technique the updating of vibration modes is achieved using the recent inverse free preconditioned krylov subspace ifpks method while techniques for extending its preconditioning and enhancing its initial guess are also proposed both algorithms are started from the nominal solutions and preconditioned using the factorization of the nominal stiffness matrix as a result no factorization of the modified stiffness matrix is needed and optimal use is made of the available information the effectiveness of the proposed updating strategy will be illustrated by an industrial case study

computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics we present a new algorithm that computes eigenvalues and eigenvectors of a hermitian positive definite matrix while solving a linear system of equations with conjugate gradient cg traditionally all the cg iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix which is equivalent theoretically to unrestarted lanczos our algorithm capitalizes on the iteration vectors produced by cg to update only a small window of vectors that approximate the eigenvectors while this window is restarted in a locally optimal way the cg algorithm for the linear system is unaffected yet in all our experiments if the window has more than a properly chosen but small number of vectors it converges to the required eigenvectors at a rate identical to unrestarted lanczos after the solution of the linear system eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems in this case eigenvectors identified in earlier systems can be used to deflate and thus accelerate the convergence of subsequent systems we have used this algorithm with excellent results in lattice quantum chromodynamics applications where hundreds of right hand sides may be needed specifically about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown where the conditioning of the matrix increases as the quark mass reaches a critical value our experiments show almost a constant number of iterations for our method regardless of quark mass and speedups of 8 over original cg for light quark masses

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

the mosaic of high performance domain decomposition methods for structural mechanics part ii formulation enhancements multiple right hand sides and implicit dynamics domain decomposition methods ddm constitute a family of high performance solvers for large scale problems of computational mechanics and of structural mechanics in particular this paper compares popular ddm for structural analysis in both theory and computational performance while it investigates the impact of enhanced mapping operators in their convergence behavior multiple right hand side problems are considered while standard and new ddm solution techniques for implicit dynamics are tested the performed benchmark tests lead to useful conclusions regarding the comparative performance of the methods

smoothing iterative block methods for linear systems with multiple right hand sides in the present paper we present smoothing procedures for iterative block methods for solving nonsymmetric linear systems of equations with multiple right hand sides these procedures generalize those known when solving one right hand linear systems we give some properties of these new methods and then using these procedures we show connections between some known iterative block methods finally we give some numerical examples

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

on chemical distances and shape theorems in percolation models with long range correlations in this paper we provide general conditions on a one parameter family of random infinite subsets of z d to contain a unique infinite connected component for which the chemical distances are comparable to the euclidean distance in addition we show that these conditions also imply a shape theorem for the corresponding infinite connected component by verifying these conditions for specific models we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the gaussian free field as a byproduct we obtain alternative proofs to the corresponding results for random interlacements in the work of cern and popov on the internal distance in the interlacement set electron j probab 17 29 1 25 2012 and while our main interest is in percolation models with long range correlations we also recover results in the spirit of the work of antal and pisztora on the chemical distance for supercritical bernoulli percolation ann probab 24 2 1036 1048 1996 for bernoulli percolation finally as a corollary we derive new results about the chemical diameter of the largest connected component in the complement of the trace of the random walk on the torus

a 0 1 law for the massive gaussian free field we investigate the phase transition in a non planar correlated percolation model with long range dependence obtained by considering level sets of a gaussian free field with mass above a given height h the dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality alongside the critical threshold h for percolation a second parameter h ge h characterizes a strongly subcritical regime we prove that the relevant crossing probabilities converge to 1 polynomially fast below h which firmly suggests that the phase transition is sharp a key tool is the derivation of a suitable differential inequality for the free field that enables the use of a conditional influence theorem

qmr based projection techniques for the solution of non hermitian systems with multiple right hand sides in this work we consider the simultaneous solution of large linear systems of the form ax j b j j 1 k where a is sparse and non hermitian we describe single seed and block seed projection approaches to these multiple right hand side problems that are based on the qmr and block qmr algorithms respectively we use block qmr to solve the block seed system and generate the relevant biorthogonal subspaces approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected block residuals after the initial block seed has converged the process is repeated by choosing a new block seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems we give theory for the single seed case that helps explain the convergence behavior under certain conditions implementation details for both the single seed and block seed algorithms are discussed and advantages of the block seed algorithm in cache based serial and parallel environments are noted the computational savings of our methods over using qmr to solve each system individually are illustrated in two examples

decay estimates for the linear damped wave equation on the heisenberg group abstract this paper is devoted to the derivation of l 2 l 2 decay estimates for the solution of the homogeneous linear damped wave equation on the heisenberg group h n for its time derivative and for its horizontal gradient moreover we consider the improvement of these estimates when further l 1 h n regularity is required for the cauchy data our approach will rely strongly on the group fourier transform of h n and on the properties of the hermite functions that form a maximal orthonormal system for l 2 r n of eigenfunctions of the harmonic oscillator

phase transition in loop percolation we are interested in the clusters formed by a poisson ensemble of markovian loops on infinite graphs this model was introduced and studied in le jan c r math acad sci paris 350 13 14 643 646 2012 ill j math 57 2 525 558 2013 it is a model with long range correlations with two parameters alpha and kappa the non negative parameter alpha measures the amount of loops and kappa plays the role of killing on vertices penalizing kappa 0 or favoring kappa 0 appearance of large loops it was shown in le jan ill j math 57 2 525 558 2013 that for any fixed kappa and large enough alpha there exists an infinite cluster in the loop percolation on mathbb z d in the present article we show a non trivial phase transition on the integer lattice mathbb z d d ge 3 for kappa 0 more precisely we show that there is no loop percolation for kappa 0 and alpha small enough interestingly we observe a critical like behavior on the whole sub critical domain of alpha namely for kappa 0 and any sub critical value of alpha the probability of one arm event decays at most polynomially for d ge 5 we prove that there exists a non trivial threshold for the finiteness of the expected cluster size for alpha below this threshold we calculate up to a constant factor the decay of the probability of one arm event two point function and the tail distribution of the cluster size these rates are comparable with the ones obtained from a single large loop and only depend on the dimension for d 3 or 4 we give better lower bounds on the decay of the probability of one arm event which show importance of small loops for long connections in addition we show that the one arm exponent in dimension 3 depends on the intensity alpha

generalized global conjugate gradient squared algorithm in this paper a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right hand sides is presented the method can be derived by using products of two nearby global bicg polynomials and formal orthogonal polynomials of which global cgs and global bicgstab are just particular cases we also show to apply the method for solving the sylvester matrix equation finally numerical examples are given to illustrate the effectiveness of the proposed method

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

projection techniques for iterative solution of ax b with successive right hand sides we present two projection techniques for computing approximate solutions to linear systems of the form ax n b n for a sequence n 1 2 e g such as arises from time discretization of a partial differential equation the inexpensive approximate solutions can be used as initial guesses for iterative solution of the system resulting in significantly reduced computational expense examples of two and three dimensional incompressible navier stokes calculations are presented in which x represents the pressure and a is a discrete poisson operator in flows containing significant dynamic activity these projection techniques lead to as much as a two fold reduction in solution time

a study of optimal membrane triangles with drilling freedoms abstract this article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms the term optimal is used in the sense of exact inplane pure bending response of rectangular mesh units of arbitrary aspect ratio following a comparative summary of element formulation approaches the construction of an optimal three node triangle using the andes formulation is presented the construction is based upon techniques developed by 1991 in student term projects but taking advantage of the more general framework of templates developed since the optimal element that fits the andes template is shown to be unique if energy orthogonality constraints are enforced two other formulations are examined and compared with the optimal model retrofitting the conventional linear strain triangle element by midpoint migrating and congruential transformations is shown unable to produce an optimal element while rank deficiency is inevitable use of the quadratic strain field of the 1988 allman triangle or linear filtered versions thereof is also unable to reproduce the optimal element moreover these elements exhibit serious aspect ratio lock these predictions are verified on benchmark examples

the block least squares method for solving nonsymmetric linear systems with multiple right hand sides abstract in this paper we present the block least squares method for solving nonsymmetric linear systems with multiple right hand sides this method is based on the block bidiagonalization we first derive two algorithms by using two different convergence criteria the first one is based on independently minimizing the 2 norm of each column of the residual matrix and the second approach is based on minimizing the frobenius norm of residual matrix we then give some properties of these new algorithms finally some numerical experiments on test matrices from harwell boeing collection are presented to show the efficiency of the new method

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

large scale continuation and numerical bifurcation for partial differential equations in this paper the problem of computing bifurcation diagrams for large scale nonlinear parameter dependent steady state systems which arise following the spatial discretization of semilinear pdes is investigated a continuation algorithm which employs a preconditioned version of the recursive projection method rpm is presented the rpm is often expensive when it is used in conjunction with the numerical method of lines preconditioning the jacobian of the underlying fixed point operator results in an algorithm the preconditioned recursive projection method prpm which is capable of efficiently computing equilibrium solution diagrams of large stiff systems for many pde problems the prpm is a fast and effective means of detecting both steady state and hopf bifurcation along a branch of solutions a description of the performance of the prpm when applied to two numerical examples is given

conjugate gradient methods for continuation problems ii we study linear conjugate gradient cg methods for large sparse continuation problems first we show how the cg methods can be incorporated as linear solvers in the context of a continuation method next we indicate how the preconditioned generalized minimal residual algorithm may be used to test for simple and multiple bifurcations without solving the bordered linear systems sample numerical results are reported some concluding remarks concerning the performance of three term cg methods are given

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

a continuation bsor lanczos galerkin method for positive bound states of a multi component bose einstein condensate we develop a continuation block successive over relaxation bsor lanczos galerkin method for the computation of positive bound states of time independent coupled gross pitaevskii equations cgpes which describe a multi component bose einstein condensate bec a discretization of the cgpes leads to a nonlinear algebraic eigenvalue problem naep the solution curve with respect to some parameter of the naep is then followed by the proposed method for a single component bec we prove that there exists a unique global minimizer the ground state which is represented by an ordinary differential equation with the initial value for a multi component bec we prove that m identical ground bound states will bifurcate into m different ground bound states at a finite repulsive inter component scattering length numerical results show that various positive bound states of a two three component bec are solved efficiently and reliably by the continuation bsor lanczos galerkin method

decoupling inequalities and interlacement percolation on g z we study the percolative properties of random interlacements on g where g is a weighted graph satisfying certain sub gaussian estimates attached to the parameters 1 and 2 1 describing the respective polynomial growths of the volume on g and of the time needed by the walk on g to move to a distance we develop decoupling inequalities which are a key tool in showing that the critical level u for the percolation of the vacant set of random interlacements is always finite in our set up and that it is positive when 1 2 we also obtain several stretched exponential controls both in the percolative and non percolative phases of the model even in the case where g d d 2 several of these results are new