a new primal dual weak galerkin finite element method for ill posed elliptic cauchy problems abstract a new numerical method is devised and analyzed for a type of ill posed elliptic cauchy problems by using the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is robust and efficient in the sense that the system arising from the scheme is symmetric well posed and is satisfied by the exact solution if it exists an error estimate of optimal order is established for the corresponding numerical solutions in a scaled residual norm in addition a mathematical convergence is established in a weak l 2 topology for the new numerical method numerical results are reported to demonstrate the efficiency of the primal dual weak galerkin method as well as the accuracy of the numerical approximations

refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

a dynamic thick restarted semi refined able algorithm for computing a few selected eigentriplets of large nonsymmetric matrices able method can be used to compute eigentriplets of large scale nonsymmetric matrices however there is no guarantee for the ritz vectors obtained by this method to converge even if the subspace is good enough furthermore for the sake of storage limitation and amount of computations restarting techniques are often needed in order to deal with these problems we propose a dynamic thick restarted semi refined able algorithm sable in which we use semi refined ritz vectors to approximate the desired eigenvectors the relationship between the novel method and the classical one is given theoretical results show that the new method can circumvent the possible danger that exists in the standard one in some degree numerical experiments are made on real world problems and comparisons are drown on the thick restarted able algorithm table and the dynamic thick restarted semi refined able algorithm sable they show that the latter is often more powerful and attractive than its standard counterpart

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract in this paper yet another iterative procedure namely the minimal error method mem for solving stably the cauchy problem in linear elasticity is introduced and investigated furthermore this method is compared with another two iterative algorithms i e the conjugate gradient cgm and landweber fridman methods lfm previously proposed by marin et al marin l hao d n lesnic d 2002b conjugate gradient boundary element method for the cauchy problem in elasticity quarterly journal of mechanics and applied mathematics 55 227 247 and marin and lesnic marin l lesnic d 2005 boundary element landweber method for the cauchy problem in linear elasticity ima journal of applied mathematics 18 817 825 respectively in the case of two dimensional homogeneous isotropic linear elasticity the inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions the numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method bem for two dimensional homogeneous isotropic linear elastic materials

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

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

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

natural convection in steady solidification finite element analysis of a two phase rayleigh benard problem abstract galerkin finite element approximations and newton s method for solving free boundary problems are combined with computer implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt the newton method gives rapid convergence to steady state velocity temperature and pressure fields and melt solid interface shapes and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability the power of this combination is demonstrated for a two phase rayleigh benard problem composed of melt and and solid in a vertical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt solid interface is always a solution multiple cellular flows bifurcating from the static state are detected and followed as rayleigh number is varied changing the boundary conditions to approach those appropriate for the vertical bridgman solidification system causes imperfections that eliminate the static state the flow structure in the bridgman system is related to those for the rayleigh benard system by a continuous evolution of the boundary conditions

logarithmic convexity for discrete harmonic function and the approximation of the cauchy problem for poisson s equation logarithmic convexity type continuous dependence results for discrete harmonic functions defined as solutions of the standard co piecewise linear approximation to laplace s equation are proved using this result error estimates for a regularization method for approximating the cauchy problem for poisson s equation on a rectangle are obtained numerical results are presented 1 introduction this paper will examine numerical methods for approximating the following cauchy problem for poisson s equation let q 0 1 x 0 1 let f denote the boundary of q let i denote the open segment of f lying on the x axis and let i denote the open segment of f along y 1 then given functions f gj and g2 and positive parameters m e1 and 2 we seek a function u that satisfies

stable determination of a crack from boundary measurements we treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given we prove a theorem of continuous dependence from the data

higher weil petersson volumes of moduli spaces of stable n pointed curves moduli spaces of compact stablen pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the weil petersson volume forms and constitute a version of mumford classes we give various new formulas for the integrals of these forms and their generating functions

a block arnoldi chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices in this paper we describe a block version of arnoldi s method for computing a few eigenvalues with largest or smallest real parts the method is accelerated via chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum a parallel implementation of this method is investigated on the eight processor alliant fx 80 numerical results and comparisons with simultaneous iteration on some harwell boeing matrices are reported

application of bem boundary element method based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries a method employing holograms that conform with arbitrarily shaped sources has been developed for enhancing conventional near field acoustic holography which has been limited to sources with simple geometries e g planar or cylindrical surfaces four holography transformation algorithms have been developed based on acoustic holography theory and the boundary element method bem singular value decomposition svd has been incorporated into the algorithms in order to alleviate the ill posed nature frequently encountered in backward reconstruction of a field a pulsating sphere a cylinder with spherical endcaps and a vibrating piston set in a rigid sphere have been adopted in a numerical simulation for verifying the algorithms satisfactory agreement has been achieved between the holographically transformed results and the analytical solutions

a refined shift and invert arnoldi algorithm for large unsymmetric generalized eigenproblems abstract the shift and invert arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and or the associated eigenvectors of a large unsymmetric matrix pair but there is no guarantee for the approximate eigenvectors ritz vectors obtained by this method to converge even though the subspace is good enough in order to correct this problem a refined shift and invert arnoldi method is proposed that uses certain refined ritz vectors to approximate the desired eigenvectors the refined ritz vectors can be computed cheaply and reliably by small sized singular value decompositions it is shown that the refined method converges a refined shift and invert arnoldi algorithm is developed and several numerical examples are reported comparisons are drawn on the refined algorithm and the shift and invert arnoldi algorithm indicating that the former is considerably more efficient than the latter

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

growth of weil petersson volumes and random hyperbolic surfaces of large genus in this paper we investigate the geometric properties of random hyperbolic surfaces of large genus we describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces first we study the asymptotic behavior of weil petersson volume vg n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the weil petersson measure as g

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

an implicitly restarted lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets e kokiopoulou et al developed an implicitly restarted lanczos bidiagonalization method named irlanb for computing the smallest singular triplets they use the wanted harmonic ritz values and the corresponding refined harmonic ritz vectors to approximate the desired smallest singular triplets however they use the unwanted harmonic ritz values as shifts named the harmonic shifts to implicitly restart their algorithm therefore although they replace the harmonic ritz vectors by the refined harmonic ritz vectors which are always better than the harmonic ritz vectors to improve the convergence of irlanb the subspace after implicitly restarting is the same as that without this replacement in this paper using the information of the refined harmonic ritz vectors we present a new shift strategy named the refined harmonic shifts to replace the harmonic shifts used in irlanb and obtain a new method called irrlanb the refined harmonic shifts are better than the harmonic shifts and can be computed cheaply and reliably the numerical experiments are reported to indicate that irrlanb is superior than irlanb

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

block arnoldi and davidson methods for unsymmetric large eigenvalue problems we present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices namely the block arnoldi method and an adaptation of the davidson method to unsymmetric matrices we give some theoretical results concerning the convergence and discuss implementation aspects of the two methods finally some results of numerical tests on a variety of matrices in which we compare these two methods are reported

an efficient numerical scheme for the biharmonic equation by weak galerkin finite element methods on polygonal or polyhedral meshes this paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak galerkin wg finite element methods the wg finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual h 2 semi norm weak partial derivatives and their approximations called discrete weak partial derivatives are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra the discrete weak partial derivatives serve as building blocks for the wg finite element method the resulting matrix from the wg method is symmetric positive definite and parameter free an error estimate of optimal order is derived in an h 2 equivalent norm for the wg finite element solutions error estimates in the usual l 2 norm are established yielding optimal order of convergence for all the wg finite element algorithms except the one corresponding to the lowest order i e piecewise quadratic elements some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme

on the use of harmonic ritz pairs in approximating internal eigenpairs the goal of this paper is to increase our understanding of harmonic rayleigh ritz for real symmetric matrices we do this by discussing different though related topics a priori error analysis a posteriori error analysis a comparison with refined rayleigh ritz and the selection of a suitable harmonic ritz vector

on the reconstruction of the vibro acoustic field over the surface enclosing an interior space using the boundary element method the vibrational velocity sound pressure and acoustic power on the vibrating boundary comprising an enclosed space are reconstructed by the boundary element method based on the measured field pressures the singular value decomposition is used to obtain the inverse solution in the least square sense and to express the acoustic modal expansion between the measurement and source fields in general such an inverse operation has been considered an ill posed problem having a divergence phenomenon involved with extremely small measurement errors the ill conditioned nature of the acoustic inverse problem is caused by the singularity of the transfer matrix which produces nonradiating wave components in order to minimize the singularity and to also reduce the number of measurement points optimal measurement positions are determined by the effective independence method regularization methods are used to stabilize the reconstructed field by suppressing nonradiating components resulting in the singular transfer

deflated block krylov subspace methods for large scale eigenvalue problems we discuss a class of deflated block krylov subspace methods for solving large scale matrix eigenvalue problems the efficiency of an arnoldi type method is examined in computing partial or closely clustered eigenvalues of large matrices as an improvement we also propose a refined variant of the arnoldi type method comparisons show that the refined variant can further improve the arnoldi type method and both methods exhibit very regular convergence behavior

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

tchebychev acceleration technique for large scale nonsymmetric matrices the acceleration by tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed a simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue new criteria are established to identify the optimal ellipse of the eigenspectrum the algorithm is fast reliable and does not require a search for all possible ellipses which enclose the spectrum the procedure is applicable to nonsymmetric linear systems as well

estimates of weil petersson volumes via effective divisors we study the asymptotics of the weil petersson volumes of the moduli spaces of compact riemann surfaces of genus g with n punctures for fixed n as g

an iterative block arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems a modified m step arnoldi algorithm proposed by jia and elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version in the new method we use new modified approximate eigenvectors obtained from a linear combination of ritz vectors and the wasted m 1 th block basis vector whose residual norms are satisfied with some p 1 dimensional minimization problems the resulting block arnoldi algorithm is not only better than the standard m step one both in theory and in practice but also cheaper than the standard m 1 step one the relationships among the residual norms of ritz pairs and those of new approximate eigenpairs are analyzed theoretical results show that the new method can overcome the drawback of nonconvergence which exists in the conventional one in some extent we carry out some numerical experiments which exhibit that the new algorithm is more efficient and works better than its counterparts

an alternating iterative algorithm for the cauchy problem associated to the helmholtz equation in this paper the iterative algorithm proposed by kozlov et al comput maths math phys 31 1991 45 for obtaining approximate solutions to the ill posed cauchy problem for the helmholtz equation is analysed the technique is then numerically implemented using the boundary element method bem the numerical results confirm that the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularising criterion is also proposed

a refined harmonic lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices the harmonic lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix a we prove that for good enough projection subspaces harmonic ritz values converge if the columns of a are strongly linearly independent on the other hand harmonic ritz values may miss some desired singular values when the columns of a are almost linearly dependent furthermore harmonic ritz vectors may converge irregularly and may even fail to converge based on the refined projection principle for large matrix eigenproblems due to the first author we propose a refined harmonic lanczos bidiagonalization method that takes the rayleigh quotients of the harmonic ritz vectors as approximate singular values and extracts the best approximate singular vectors called the refined harmonic ritz approximations from the given subspaces in the sense of residual minimizations the refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the rayleigh quotients converge an implicitly restarted refined harmonic lanczos bidiagonalization algorithm irrhlb is developed we study how to select the best possible shifts and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted harmonic lanczos bidiagonalization algorithm irhlb we propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately numerical experiments are reported that compare irrhlb with five other algorithms based on the lanczos bidiagonalization process it appears that irrhlb is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets

a thick restarted block arnoldi algorithm with modified ritz vectors for large eigenproblems the block arnoldi method is one of the most commonly used techniques for large eigenproblems in this paper we exploit certain modified ritz vectors to take the place of ritz vectors in the thick restarted block arnoldi algorithm and propose a modified thick restarted block arnoldi algorithm for large eigenproblems we then consider how to periodically combine the refined subspace iterative method with the modified thick restarting block arnoldi algorithm for computing a few dominant eigenpairs of a large matrix the resulting algorithm is called a subspace block arnoldi algorithm numerical experiments show the efficiency of our new algorithms

a primal dual weak galerkin finite element method for fokker planck type equations this paper presents a primal dual weak galerkin finite element method for a class of second order elliptic equations of fokker planck type the method is based on a variational form where all the

stabilized finite element methods for nonsymmetric noncoercive and ill posed problems part ii hyperbolic equations in this paper we consider stabilized finite element methods for hyperbolic transport equations without coercivity abstract conditions for the convergence of the methods are introduced and these conditions are shown to hold for three different stabilized methods the galerkin least squares method the continuous interior penalty method and the discontinuous galerkin method we consider both the standard stabilization methods and the optimization based method introduced in e burman siam j sci comput 35 2013 pp a2752 a2780 the main idea of the latter is to write the stabilized method in an optimization framework and select the discrete function for which a certain cost functional in our case the stabilization term is minimized some numerical examples illustrate the theoretical investigations

numerical solution of the cauchy problem for steady state heat transfer in two dimensional functionally graded materials the application of the method of fundamental solutions to the cauchy problem for steady state heat conduction in two dimensional functionally graded materials fgms is investigated the resulting system of linear algebraic equations is ill conditioned and therefore regularization is required in order to solve this system of equations in a stable manner this is achieved by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

a refined iterative algorithm based on the block arnoldi process for large unsymmetric eigenproblems when the matrix in question is unsymmetric the approximate eigenvectors or ritz vectors obtained by orthogonal projection methods including arnoldi s method and the block arnoldi method cannot be guaranteed to converge in theory even if the corresponding approximate eigenvalues or ritz values do in order to circumvent this danger a new strategy has been proposed by the author for arnoldi s method the strategy used is generalized to the block arnoldi method in this paper it discards ritz vectors and instead computes refined approximate eigenvectors by small sized singular value decompositions it is proved that the new strategy can guarantee the convergence of refined approximate eigenvectors if the corresponding ritz values do the resulting refined iterative algorithm is realized by the block arnoldi process numerical experiments show that the refined algorithm is much more efficient than the iterative block arnoldi algorithm