balanced incomplete factorization in this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard lu or ldl t factors direct factors and their inverses inverse factors at the same time algorithmically we derive this method from the approach based on the sherman morrison formula r bru j cerdan j marin and j mas siam j sci comput 25 2003 pp 701 715 in contrast to the robust incomplete decomposition rif algorithm m benzi and m t ma numer linear algebra appl 10 2003 pp 385 400 the direct and inverse factors here directly influence each other throughout the computation consequently the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way for the symmetric positive definite case we derive the theory and present an algorithm for computing the incomplete ldl t factorization and we discuss experimental results we call this new approximate ldl t factorization the balanced incomplete factorization bif our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods moreover the internal coupling of the computation of direct and inverse factors results in much shorter setup times times to compute approximate decomposition than rif a method of a similar and very high level of robustness we also derive and present the theory for the general nonsymmetric case but do not discuss its implementation

a numerical comparison of chebyshev methods for solving fourth order semilinear initial boundary value problems in solving semilinear initial boundary value problems with prescribed non periodic boundary conditions using implicit explicit and implicit time stepping schemes both the function and derivatives of the function may need to be computed accurately at each time step to determine the best chebyshev collocation method to do this the accuracy of the real space chebyshev differentiation spectral space preconditioned chebyshev tau real space chebyshev integration and spectral space chebyshev integration methods are compared in the l 2 and w 2 2 norms when solving linear fourth order boundary value problems and in the l 0 t l 2 and l 0 t w 2 2 norms when solving initial boundary value problems we find that the best chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to fourier spectral discretization

ordering anisotropy and factored sparse approximate inverses we consider ordering techniques to improve the performance of factored sparse approximate inverse preconditioners concentrating on the ainv technique of m benzi and m t r u ma several practical existing unweighted orderings are considered along with a new algorithm minimum inverse penalty mip that we propose we show how good orderings such as these can improve the speed of preconditioner computation dramatically and also demonstrate a fast and fairly reliable way of testing how good an ordering is in this respect our test results also show that these orderings generally improve convergence of krylov subspace solvers but may have difficulties particularly for anisotropic problems we then argue that weighted orderings which take into account the numerical values in the matrix will be necessary for such systems after developing a simple heuristic for dealing with anisotropy we propose several practical algorithms to implement it while these show promise we conclude a better heuristic is required for robustness

gmres for the differentiation operator we investigate using the gmres method with the differentiation operator this operator is unbounded and thus does not fall into the framework of existing krylov subspace theory we establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane these conditions are used to determine when gmres converges this algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals even when the kernel of oscillations is unknown

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

fast preconditioned conjugate gradient algorithms for wiener hopf integral equations in this paper the authors study circulant approximations of finite sections of a wiener hopf integral equation on the half line such circulant operators are defined by periodic kernel functions they approximate finite sections of the wiener hopf operator within a sum of a small operator and an operator with fixed finite rank constructions are given of two such circulant operators and their use as preconditioners for the conjugate gradients algorithm is explained numerical examples are included to illustrate the results

an improved algorithm for the evaluation of cauchy principal value integrals of oscillatory functions and its application abstract a new interpolatory type quadrature rule is proposed for the numerical evaluation of cauchy principal value integrals of oscillatory kind 1 1 f x x e i x d x where 1 1 the method is based on an interpolatory procedure at clenshaw curtis points and the singular point and the fast computation of the modified moments with cauchy type singularity based on this result a new method is presented for the computation of the oscillatory integrals with logarithmic singularities too these methods enjoy fast implementation and high accuracy convergence rates on are also provided numerical examples support the theoretical analyses

numerical solution of stochastic linear heat conduction problem by using new algorithms we consider a stochastic linear heat conduction problem so it is reduced to a special integral equation of the second kind our aim in this paper is to give stable numerical algorithms for approximating solution of integral equation and heat conduction

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

circulant preconditioners for ill conditioned boundary integral equations from potential equations in this paper we consider solving potential equations by the boundary integral equation approach the equations so derived are fredholm integral equations of the first kind and are known to be ill conditioned their discretized matrices are dense and have condition numbers growing like o n where n is the matrix size we propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners these are convolution operators with periodic kernels and hence can be inverted efficiently by using fast fourier transforms we prove that the preconditioned systems are well conditioned and hence the convergence rate of the method is linear numerical results for two types of regions are given to illustrate the fast convergence 1998 john wiley sons ltd

numerical evaluation of highly oscillatory bessel transforms abstract in this paper we mainly focus on numerical evaluation of highly oscillatory bessel transforms based on the multiple integral the schemes for computing this class of the transform are presented by means of the bounds of special functions we give the corresponding errors of the schemes to demonstrate the efficiency and accuracy of the proposed methods some test examples are shown

computing the hilbert transform and its inverse we construct a new method for approximating hilbert transforms and their inverse throughout the complex plane both problems can be formulated as riemann hilbert problems via plemelj s lemma using this framework we re derive existing approaches for computing hilbert transforms over the real line and unit interval with the added benefit that we can compute the hilbert transform in the complex plane we then demonstrate the power of this approach by generalizing to the half line combining two half lines we can compute the hilbert transform of a more general class of functions on the real line than is possible with existing methods

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

on evaluation of bessel transforms with oscillatory and algebraic singular integrands in this paper we study efficient methods for computing the integrals of the form 0 1x a 1 x bf x j v wx dx where a b v w are the given constants and w 1 j v is the bessel function of the first kind and of order v f is a sufficiently smooth function on 0 1 firstly we express the moments in a closed form with the aid of special functions secondly we induce the filon type method based on the taylor interpolation polynomial at two endpoints and the hermite interpolation polynomial at clenshaw curtis points on evaluating the highly oscillatory bessel integrals with algebraic singularity theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if w is sufficiently large

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

a physically motivated sparse cubature scheme with applications to molecular density functional theory we present a novel approach for performing multi dimensional integration of arbitrary functions the method starts with smolyak type sparse grids as cubature formulae on the unit cube and uses a transformation of coordinates based on the conditional distribution method to adapt those formulae to real space our method is tested on integrals in one two three and six dimensions the three dimensional integration formulae are used to evaluate atomic interaction energies via the gordon kim model the six dimensional integration formulae are tested in conjunction with the nonlocal exchange correlation energy functional proposed by lee and parr this methodology is versatile and powerful we contemplate application to frozen density embedding next generation molecular mechanics force fields kernel type exchange correlation energy functionals and pair density functional theory

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

mass and momentum conservation of the least squares spectral collocation method for the navier stokes equations from the literature it is known that the least squares spectral element method lssem for the stationary stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum furthermore it is known that the least squares spectral collocation method lsscm leads to superior conservation of mass and momentum for the stationary stokes equations in the present paper we consider mass and momentum conservation of the lsscm for time dependent stokes and navier stokes equations we observe that the lsscm leads to improved conservation of mass and momentum for these problems furthermore the lsscm leads to the well known time dependent profiles for the velocity and the pressure profiles to obtain these results we use only a few elements each with high polynomial degree avoid normal equations for solving the overdetermined linear systems of equations and introduce the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with the least squares spectral collocation scheme to discretize the internal flow problems

adaptive reduced basis generation for reduced order modeling for the solution of stochastic nondestructive evaluation problems a novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented more specifically the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty in order to achieve this computational efficiency the approach combines reduced basis reduced order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters provided the distributions are known the reduced order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique to then create the reduced order basis the capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty the first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing then the second case study considered characterization of an unknown temperature dependent thermal conductivity of a solid from simulated thermal testing overall the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

on the stability hyperellipsoid of a schur polynomial abstract the robust schur stability of a polynomial with uncertain coefficients is investigated using a linear fractional transformation the stability hyperellipsoid of a schur polynomial is determined an effective algorithm is supplied for finding an optimal transformation parameter such that the resulting stability hyperellipsoid encloses entirely the largest hypersphere

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

on the stability radius of a schur polynomial abstract the robust schur stability of a polynomial with uncertain coefficients will be investigated the stability hypersphere for such polynomials will be determined in terms of tshebyshev polynomials

exponential convergence of gauss jacobi quadratures for singular integrals over simplices in arbitrary dimension galerkin discretizations of integral operators in mathbb r d require the evaluation of integrals int s 1 int s 2 f x y dydx where s 1 s 2 are d dimensional simplices and f has a singularity at x y in a chernov t von petersdorff and c schwab m 2 an math model numer anal 45 2011 pp 387 422 we constructed a family of hp quadrature rules q n with n function evaluations for a class of integrands f allowing for algebraic singularities at x y possibly nonintegrable with respect to either dx or dy hypersingular kernels and gevrey delta smooth for x ne y this is satisfied for kernels from broad classes of pseudodifferential operators we proved that q n achieves the exponential convergence rate mathcal o exp rn gamma with the exponent gamma 1 2d delta 1 in this paper we consider a special singularity x y alpha with real alpha which appears frequently in appplication and prove that an improved converg

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

preconditioners for wiener hopf equations with high order quadrature rules we consider solving the wiener hopf equations with high order quadrature rules by preconditioned conjugate gradient pcg methods we propose using convolution operators as preconditioners for these equations we will show that with the proper choice of kernel functions for the preconditioners the resulting preconditioned equations will have clustered spectra and therefore can be solved by the pcg method with superlinear convergence rate moreover the discretization of these equations by high order quadrature rules leads to matrix systems that involve only toeplitz or diagonal matrix vector multiplications and hence can be computed efficiently by ffts numerical results are given to illustrate the fast convergence of the method and the improvement on accuracy by using higher order quadrature rule we also compare the performance of our preconditioners with the circulant integral operators

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

the kink phenomenon in fejer and clenshaw curtis quadrature the fejer and clenshawcurtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions when n the number of points in the integration rule increases the error does not decay to zero evenly but does so in two distinct stages for n less than a critical value the error behaves like o 2n where is a constant greater than 1 for these values of n the accuracy of both the fejer and clenshawcurtis rules is almost indistinguishable from that of the more celebrated gausslegendre quadrature rule for larger n however the error decreases at the rate o n i e only half as fast as before convergence curves typically display a kink where the convergence rate cuts in half in this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon

numerical methods based on rational variable substitution for wiener hopf equations of the second kind this paper considers numerical methods for wiener hopf equations of the second kind y t 0 k t s y s ds g t 0 t by applying rational variable substitution to integrals on the semi infinite interval 0 and using the well known clenshaw curtis quadrature to the resulted integral we get a clenshaw curtis rational ccr quadrature rule we then apply the ccr quadrature to wiener hopf equations the reduction of singularities in the transformed equation is considered numerical examples are given to illustrate the efficiency of the numerical methods proposed in this paper

operational quadrature methods for wiener hopf integral equations we study the numerical solution of wiener hopf integral equations by a class of quadrature methods which lead to discrete wiener hopf equations with quadrature weights constructed from the fourier transform of the kernel or rather from the laplace transforms of the kernel halves as the analytical theory of wiener hopf equations is likewise based on the fourier transform of the kernel this approach enables us to obtain results on solvability and stability and error estimates for the discretization the discrete wiener hopf equations are solved by using an approximate wiener hopf factorization obtained with fft numerical experiments with the milne equation of radiative transfer are included

simulation based optimal bayesian experimental design for nonlinear systems the optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction particularly in situations where experiments are time consuming and expensive to conduct we propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation based models in particular we focus on finding sets of experiments that provide the most information about targeted sets of parameters our framework employs a bayesian statistical setting which provides a foundation for inference from noisy indirect and incomplete data and a natural mechanism for incorporating heterogeneous sources of information an objective function is constructed from information theoretic measures reflecting expected information gain from proposed combinations of experiments polynomial chaos approximations and a two stage monte carlo sampling method are used to evaluate the expected information gain stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high dimensional settings these algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics

refining an approximate inverse direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades the advances in graph theoretic algorithms have not received enough attention from the iterative methods community for example we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner for very sparse preconditioners however a reformulation of the algorithm with outer products can exploit even more zeros to good advantage we also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings the article ends by proposing a block version of the algorithm for further gains in efficiency and robustness

a fast two grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half plane we consider the numerical treatment of second kind integral equations on the real line of the form phi s psi s int infty infty k s t z t phi t dt s in mathbb r abbreviated k z in which l 1 z l and bc the space of bounded continuous functions on are assumed known and bc is to be determined we first derive sharp error estimates for the finite section approximation reducing the range of integration to a a via bounds on i k z 1 as an operator on spaces of weighted continuous functions numerical solution by a simple discrete collocation method on a uniform grid on is then analysed in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix vector multiply via the fft to utilise this possibility we propose a modified two grid iteration a feature of which is that the coarse grid matrix is approximated by a banded matrix and analyse convergence and computational cost in cases wherez is not compactly supported a combined finite section and two grid algorithm can be applied and we extend the analysis to this case as an application we consider acoustic scattering in the half plane with a robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied our final result is that ifz related to the boundary impedance in the application takes values in an appropriate compact subsetq of the complex plane then the difference between s and its finite section approximation computed numerically using the iterative scheme proposed is c 1 khlog 1 kh 1 1 2 ka 1 2 in the interval a a 1 forkh sufficiently small wherek is the wavenumber andh the grid spacing moreover this numerical approximation can be computed in c 2 n logn operations wheren 2a h is the number of degrees of freedom the values of the constantsc 1 andc 2 depend only on the setq and not on the wavenumberk or the support ofz

integral equations on the half line a modified finite section approximation on considere la solution approchee des equations integrales de la forme y t 0 k t s y s ds f t comportant des noyaux du type de wiener hopf on etablit la convergence uniforme de la solution approchee de section finie vers la solution on traite un exemple numerique

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

clenshaw curtis and gauss legendre quadrature for certain boundary element integrals following a recent article by trefethen siam review 50 2008 pp 67 87 the use of clenshaw curtis quadrature rather than gauss legendre quadrature for nearly singular integrals which arise in the boundary element method has been investigated when these quadrature rules are used in association with the sinh transformation the authors have concluded after considering asymptotic estimates of the truncation errors for certain proto type functions arising in this context that gauss legendre quadrature should continue to be the preferred quadrature rule

representation of solutions to the one dimensional schrodinger equation in terms of neumann series of bessel functions abstract a new representation of solutions to the equation y q x y 2 y is obtained for every x the solution is represented as a neumann series of bessel functions depending on the spectral parameter due to the fact that the representation is obtained using the corresponding transmutation operator a partial sum of the series approximates the solution uniformly with respect to which makes it especially convenient for the approximate solution of spectral problems the numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy

stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics the solution of linear systems arising in the finite element analysis of shells and solids by the preconditioned conjugate gradient method is considered stabilized and block versions of the ainv factorized approximate inverse preconditioner are presented and tested on a variety of difficult problems comparisons with other preconditioning methods are also included

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

hybrid preconditioned algorithms for boundary hypersingular integral equations three preconditioners for solving boundary hypersingular integral equations of the first kind are proposed the first and the second preconditioners are based on circulant operators the third is a hybrid of iterative substructuring techniques and circulant operators although this class of preconditioners are composition of circulant operators but we have shown that the condition number of the third preconditioner dose not exceed the condition number of two preconditioners numerical results are included

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

on the convergence rate of clenshaw curtis quadrature for integrals with algebraic endpoint singularities abstract in this paper we are concerned with clenshaw curtis quadrature for integrals with algebraic endpoint singularities an asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of chebyshev polynomials numerical examples are provided to confirm our analysis

real polynomial chebyshev approximation by the caratheodory fejer method a new method is presented for near best approximation of a real function f on tau tau by a polynomial of degree m the method is derived by transplanting the given problem to the unit disk then applying the caratheodory fejer theorem the resulting near best approximation is constructed from the principal eigenvalue and eigenvector of a hankel matrix of chebyshev coefficients of f it is well known that as tau to 0 the mth partial sum of the chebyshev series of f agrees with the best approximation to a relative error o tau in contrast our approximation is shown to differ from best by at most o tau 2m 3 a similar result is given for approximation on 1 1 as m to infty such high order agreement is of both practical and theoretical importance in particular it establishes a real analogue of the phenomenon that on the complex unit disk best approximation error curves tend to closely approximate circles several numerical examples are presented

fast evaluation of quadrature formulae on the sphere recently a fast approximate algorithm for the evaluation of expansions in terms of standard l 2 s 2 orthonormal spherical harmonics at arbitrary nodes on the sphere s 2 has been proposed in s kunis and d potts fast spherical fourier algorithms j comput appl math 161 75 98 2003 the aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules we give a formulation in matrix vector notation and an explicit factorisation of the spherical fourier matrix based on the former algorithm starting from this we obtain the corresponding factorisation of the adjoint spherical fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere we provide results of numerical tests showing the stability of the obtained algorithm using as examples classical gaus legendre and clenshaw curtis quadrature rules as well as the healpix pixelation scheme and an equidistribution

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

the fast implementation of higher order hermite fejer interpolation the numerical computation of barycentric hermite interpolation may suffer from devastating inaccuracies in the absence of overflow or underflow in this paper we propose a new fast implementation of the second barycentric formula for higher order hermite fejer interpolation at gauss jacobi or jacobi gauss lobatto pointsystems our method relies upon the sturm liouville equations for jacobi polynomials whereby the barycentric weights can be efficiently evaluated and the overall computational complexity is linear with respect to the number of grids in particular the exponentially increasing common factors in the barycentric weights cancel out from the second barycentric formula which yields a superiorly stable method for computing the simplified barycentric weights

krylov deferred correction accelerated method of lines transpose for parabolic problems in this paper a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented unlike traditional method of lines mol the new krylov deferred correction kdc accelerated method of lines transpose mol t first discretizes the temporal direction using gaussian type nodes and spectral integration and symbolically applies low order time marching schemes to form a preconditioned elliptic system which is then solved iteratively using newton krylov techniques such as newton gmres or newton bicgstab method each function evaluation in the newton krylov method is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers preliminary numerical experiments show that the kdc accelerated mol t technique is unconditionally stable can be spectrally accurate in both temporal and spatial directions and allows optimal time stepsizes in long time simulations

construction of preconditioners for wiener hopf equations by operator splitting abstract in this paper we propose a new type of preconditioners for solving finite section wiener hopf integral equations i a x g by the preconditioned conjugate gradient algorithm we show that for an integer u 1 the operator i a can be decomposed into a sum of operators i p u v for 0 v 1 u v i p u v 1 thus the way the preconditioners are constructed is very similar to the approach used in the additive schwarz method for elliptic problems as for the convergence rate we prove that the spectra of the resulting preconditioned operators 1 u v i p u v 1 i a are clustered around 1 and thus the algorithm converges sufficiently fast finally we discretize the resulting preconditioned equations by rectangular rule numerical results show that our methods converges faster than those preconditioned by using circulant integral operators

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

implementing the complex integral method with the transformed clenshaw curtis quadrature abstract gauss laguerre quadrature plays an important role in implementing the numerical steepest decent method for computing highly oscillatory integrals however it consumes too much time when the analytic region of the integrand is narrow in this paper we analyze the convergence rate of the transformed clenshaw curtis quadrature and show that this method also shares the property that the higher the oscillation the better the calculation moreover it is efficient to compute highly oscillatory integral with a nearly singularity numerical tests are performed to verify our given results

an efficient quadrature for 2 5d boundary element calculations abstract in recent years the boundary element method has become a widely used tool for calculating the mitigation effects of noise barriers however since for large structures calculations in 3d become very inefficient most of the standard implementations are only in 2d this means that the noise source is implicitly assumed to be given by a coherent line source which is not realistic in most cases by using a fourier transform with respect to a spatial coordinate along the length of the structure it is possible to reduce the 3d problem to several 2d problems with distinct wavenumbers which allows the simulation of more realistic noise sources and which is typically referred to as 2 5d bem to that end it is necessary to numerically calculate a fourier like integral over all the 2d solutions in this work an efficient way to calculate this integral is given building on existing approaches using clenshaw curtis filon quadrature and demodulation combined with an adaptive order selection scheme as bem calculations are costly the main focus of the method introduced lies on avoiding too many of these calculations the efficiency of the method is illustrated using two different examples a reflecting cylinder and an l shaped noise barrier

a structural diagnosis of some ic orderings we present a novel analysis of the potential effectiveness of a matrix ordering for the incomplete cholesky factorization ic in terms of just the sparsity structure by looking at the structure of the approximate inverse implicitly created by ic we can help to explain the success of reverse cuthill mckee orderings the problems ic 0 has under red black orderings that disappear when extra fill is included and where fill must be added to make fill reducing orderings such as minimum degree effective

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

a user friendly method for computing indefinite integrals of oscillatory functions for indefinite integrals q f x 1 x f t e i t d t x 1 1 torii and the first author hasegawa and torii 1987 developed a quadrature method of clenshaw curtis c c type its improvement was made and combined with sidi s m w transformation by sidi and the first author hasegawa and sidi 1996 to compute infinite oscillatory integrals the improved method per se however has not been elucidated in its attractive features which here we reveal with new results and its detailed algorithm a comparison with a method of c c type for definite integrals q f 1 due to dominguez et al 2011 suggests that a smaller number of computations is required in our method this is achieved by exploiting recurrence and normalization relations and their associated linear system we show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension for f z analytic on and inside an ellipse in the complex plane z the error of the approximation to q f x of the improved method is shown to be bounded uniformly numerical examples illustrate the stability and performance of the method

nystrom clenshaw curtis quadrature for integral equations with discontinuous kernels a new highly accurate numerical approximation scheme based on a gauss type clenshaw curtis quadrature for fredholm integral equations of the second kind x t ab k t s x s ds y t whose kernel k t s is either discontinuous or not smooth along the main diagonal is presented this scheme is of spectral accuracy when k t s is infinitely differentiable away from the diagonal t s relation to the singular value decomposition is indicated application to integro differential schrodinger equations with nonlocal potentials is given

uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins abstract in this work we propose an uncertainty quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes which we model as a coupled time dependent non linear monodimensional depth only system of pdes with uncertain parameters while in previous works formaggia et al 2013 porta et al 2014 we assumed a simplified depositional history with only one material in this work we consider multi layered basins in which each layer is characterized by a different material and hence by different properties this setting requires several improvements with respect to our earlier works both concerning the deterministic solver and the stochastic discretization on the deterministic side we replace the previous fixed point iterative solver with a more efficient newton solver at each step of the time discretization on the stochastic side the multi layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters that need an appropriate treatment for surrogate modeling techniques such as sparse grids to be effective we propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space the reference coordinate system is built upon exploiting physical features of the problem at hand we employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations we showcase the capabilities of our numerical methodologies through two synthetic test cases in particular we show that our methodology reproduces with high accuracy multi modal probability density functions displayed by target state variables e g porosity

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

new interpolatory quadrature formulae with chebyshev abscissae abstract we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of any one of the four chebyshev polynomials of degree n plus one of the points 1 or 1 in particular we derive explicit formulae for the weights and examine their positivity we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness f these quadrature formulae in addition we establish their convergence for riemann integrable functions on 1 1 as well as for functions having a monotonic singularity at 1 or 1

error bounds for gaussian quadrature and weighted l 1 polynomial approximation error bounds for gaussian quadrature are given in terms of the number of quadrature points and smoothness properties of the function whose integral is being approximated an intermediate step invol

the convergence of nystrom methods for wiener hopf equations we consider second kind integral equations on the half line where the integral operator is a compact perturbation of a convolution operator it is shown that these may be solved numerically by nystrom methods based on composite quadrature rules provided the underlying mesh is graded to correctly match the behaviour of the solution we prove the same rates of convergence that occur when the methods are applied to equations on finite intervals numerical examples are given

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

a trapezoidal rule error bound unifying the euler maclaurin formula and geometric convergence for periodic functions the error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non periodicity at the boundary using a contour integral we derive a unified bound for the combined error from both sources for analytic integrands the bound gives the euler maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another similar results are also given for the midpoint rule

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

quadrature methods for integral equations of the second kind over infinite intervals convergence results are proved for a class of quadrature methods for integral equations of the form y t fit o k t s y s ds an important special case is the nystrom method in which the integral term is approximated by an ordinary quadrature rule for all of the methods considered here the rate of convergence is the same apart from a constant factor as that of the quadrature approximation to the integral term

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy