Mario Arioli

LUM Jean Monnet, Libera Università Mediterranea di Bari | LUM Bari · Department of Economics

The Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the para...

We present an iterative method based on a generalization of the Golub-Kahan bidiagonalization for solving indefinite matrices with a 2 \(\times \) 2 block structure. We focus in particular on our recent implementation of the algorithm using the parallel numerical library PETSc. Since the algorithm is a nested solver, we investigate different choice...

We present a scalability study of Golub-Kahan bidiagonalization for the parallel iterative solution of symmetric indefinite linear systems with a 2 × 2 block structure. The algorithms have been implemented within the parallel numerical library PETSc. Since a nested inner-outer iteration strategy may be necessary, we investigate different choices fo...

Kinematic relationships between degrees of freedom, also named multi-point constraints, are frequently used in structural mechanics. In this paper, the Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure. The condition number of the precondition...

We present an iterative method based on a generalization of the Golub-Kahan bidiagonalization for solving indenite matrices with a 2x2 block structure. We focus in particular on our recent implementation of the algorithm using the parallel numerical library PETSc. Since the algorithm is a nested solver, we investigate dierent choices for parallel i...

This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be avai...

This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be avai...

Interpolation spaces and their applications from a numerical linear algebra perspective - Part I describes the theoretical tools

Interpolation spaces and their applications from a numerical linear algebra perspective: Part II : several applications of the Interpolation Spaces

We study the numerical solution of boundary and initial value problems for differential equations posed on graphs or networks. The graphs of interest are quantum graphs, i.e., metric graphs endowed with a differential operator acting on functions defined on the graph's edges with suitable side conditions. We describe and analyse the use of linear f...

In this work we analyse the Steklov-Poincaré (or interface Schur complement) matrix arising in a domain decomposition method in the presence of anisotropy. Our problem is formulated such that three types of anisotropy are being considered: refinements with high aspect ratios, uniform refinements of a domain with high aspect ratio and anisotropic di...

We study the numerical solution of boundary and initial value problems for dieren-tial equations posed on graphs or networks. The graphs of interest are quantum graphs, i.e., metric graphs endowed with a dierential operator acting on functions defined on the graph's edges with suitable side conditions. We describe and analyze the use of linear fini...

In this work we analyse the Steklov-Poincar ́e (or interface Schur complement) matrix arising in a domain decomposition method in the presence of anisotropy. Our problem is formulated such that three types of anisotropy are being considered: refinements with high aspect ratios, uniform refinements of a domain with high aspect ratio and anisotropic...

We study the solution of the linear least-squares problem minx ∥b−Ax∥2 where the matrix A ∈ IRm×n (m ≥ n) has rank n and is large and sparse. We assume that A is available as a matrix, not an operator. The preconditioning of this problem is difficult because the matrix A does not have the properties of differential problems that make standard preco...

We study the solution of the linear least-squares problem minx ∥b−Ax∥_2 where the matrix A ∈ IR^{m×n} (m ≥ n) has rank n and is large and sparse. We assume that A is available as a matrix, not an operator. The preconditioning of this problem is difficult because the matrix A does not have the properties of differential problems that make standard p...

The preconditioning of linear least-squares problems is a hard task. The linear model underpinning least-squares problems, that is the overdetermined matrix defining it, does not have the properties of differential problems that make standard preconditioners effective. Incomplete Cholesky techniques applied to the normal equations do not produce a...

The Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step o...

We present a new approach for preconditioning the interface Schur complement arising in the domain decomposition of second-order scalar elliptic problems. The preconditioners are discrete interpolation norms recently introduced in Arioli & Loghin (2009, Discrete interpolation norms with applications. SIAM J. Numer. Anal., 47, 2924–2951). In particu...

The Golub-Kahan bidiagonalization algorithm has been widely used in solving least-squares problems and in the computation of the SVD of rectangular matrices. Here we propose an algo- rithm based on the Golub-Kahan process for the solution of augmented systems that minimizes the norm of the error and, in particular, we propose a novel estimator of t...

The Conjugate Gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from least-squares problems. Taking into account the results of Hestenes and Stiefel (1952), Golub and Meurant (1997), and Strakos and Tichy (2002), which make it possible to approximate the energy norm of the error during...

We consider a family of practical stopping criteria for linear solvers for adaptive finite element methods for symmetric elliptic problems. A contraction property between two consecutive levels of refinement of the adaptive algorithm is shown when a family of smallness criteria for the corresponding linear solver residuals are assumed on each level...

It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss...

When solving partial differential elliptic equations in variational form on a domain W by approximating the solution with a finite-element method, parallelism is usually achieved by Domain Decomposition techniques. In Fig.1, we illustrate this on a simple square domain W=(-1,1)´(-1,1), where we decompose the whole domain in small squares and then w...

We describe norm representations for interpolation spaces generated by finite- dimensional subspaces of Hilbert spaces. These norms are products of integer and non-integer powers of the Grammian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the...

19 Mathematicians are particularly attracted by the deep connection between this problem and several eso-teric fields of functional analysis and variational calculus. Digital images are commonly presented as matrices of scalars for grey-scale images or vectors for colour images. These matrices are seen as the values of a distribution (generalized f...

We consider the triangular factorization of matrices in single-precision arithmetic and show how these factors can be used to obtain a backward stable solution. Our aim is to obtain double-precision accuracy even when the system is ill-conditioned. We examine the use of iterative refinement and show by example that it may not converge. We then show...

A strict adherence to threshold pivoting in the direct solution of symmetric indefinite problems can result in substantially more work and storage than forecast by a sparse analysis of the symmetric problem. One way of avoiding this is to use static pivoting where the data structures and pivoting sequence generated by the analysis are respected and...

We consider here the linear least squares problem miny∈RnAy − b� 2, where b ∈ Rm and A ∈ Rm×n is a matrix of full column rank n, and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using the Frobenius or the spectral norm for A and the Euclidean norm for b. In this paper, we are conce...

A strict adherence to threshold pivoting in the direct solution of symmetric indefinite problems can result in substantially more work and storage than forecast by a sparse analysis of the symmetric problem. One way of avoiding this is to use static pivoting where the data structures and pivoting sequence generated by the analysis are respected and...

We study stopping criteria that are suitable in the solution by Krylov space based methods of linear and non linear systems of equations arising from the mixed and the mixed-hybrid finite-element approximation of saddle point problems. Our approach is based on the equivalence between the Babuska and Brezzi conditions of stability which allows us to...

Mixed-hybrid finite element discretization of Darcy’s law, and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddle-point problems. In this paper we consider solution techniques based on the computation of a null-space basis of the whole or of a part of the left lower off-diagon...

We use the null space algorithm approach to solve the augmented systems produced by the mixed finite-element approximation of Darcy’s laws. Taking into account the properties of the graph representing the triangulation, we adapt the null space technique proposed by M. Arioli and L. Baldini [SIAM J. Matrix Anal. Appl. 23, No. 2, 425–442 (2001; Zbl 1...

This work extends the results of Arioli (2002), Arioli, Noulard and Russo (2001) on stopping criteria for iterative solution methods for nite element problems to the case of nonsymmetric positive-de nite problems. We show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to converge...

We consider the solution of ill-conditioned symmetric and positive definite large sparse linear systems of equations. These arise, for instance, when using some symmetrising preconditioning technique for solving a general (possibly unsymmetric) ill-conditioned linear system, or in domain decomposition of a numerically di#cult elliptic problem. We a...

The conjugate gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results by Golub and Meurant (1997), Meurant (1977), Meurant (1999a), and Strakos and Tichy (2003) whi...

A Null Space algorithm is considered to solve the augmented system produced by the mixed finite element approximation of Darcy's Law. The method based on the combination of a Gaussian factorisation technique for sparse matrices with an iterative Krylov solver. The computational efficiency of the method relies on the use of spanning trees to compute...

The Conjugate Gradient method has always been successfully used in solving the symmetric and positive de nite systems obtained by the nite element approximation of selfadjoint elliptic partial dierential equations. Taking into account recent results by Golub and Meurant (1997), Meurant (1997), Meurant (1999a), and Strakos and Tichy (2002) which mak...

A Null Space algorithm is considered to solve the augmented system produced by the mixed nite element approximation of Darcy's Law. The method is based on the combination of a Gaussian factorisation technique for sparse matrices with an iterative Krylov solver.

A null space algorithm is considered to solve the augmented system produced by the mixed finite-element approximation of Darcy's Law. The method is based on the combination of an orthogonal factorization technique for sparse matrices with an iterative Krylov solver. The computational efficiency of the method relies on a suitable stopping criterion...

We present a roundoff error analysis of a null space method for solving quadratic programming minimization problems. This method combines the use of a direct LU factorization of the Constraints with an iterative solver on the corresponding null space. Numerical experiments are presented which give evidence of the good performance of the algorithm o...

We use the null space algorithm approach to solve the augmented systems produced by the mixed nite-element approximation of Darcy's laws. Taking into account the properties of the graph representing the triangulation, we adapt the null space technique proposed by Arioli and Baldini (2001), where an iterative-direct hybrid method is described. In pa...

We show that, when solving a linear system with an iterative method, it is necessary to measure the error in the space in which the residual lies. We present examples of linear systems which emanate from the finite element discretization of elliptic partial differential equations, and we show that, when we measure the residual in H −1(Ω), we obtain...

Mixed-hybrid nite element discretization of the Darcy's law and the continuity equation that describe the potential uid ow problem in porous media leads to symmetric inde nite linear systems with a particular block structure. In this paper we consider an approach for a solution of such systems based on the computation of a null-space basis of the w...

We present a roundoff error analysis of a null space method for solving quadratic programming minimization problems. This method combines the use of a direct QR factorization of the constraints with an iterative solver on the corresponding null space. Numerical experiments are presented which give evidence of the good performances of the algorithm...

In the solution of magnetostatic problems the use of a mixed formulation based on both the magnetic and magnetic displacement fields is particularly appropriate as it allows us to impose the physical conditions exactly and to maintain the continuity properties of the two fields, together with an efficient treatment of boundary conditions. The discr...

In most practical cases, the convergence of the GMRES method applied to a linear algebraic systemAx=b is determined by the distribution of eigenvalues ofA. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper the previous work of Greenbaum et al. is extended in the followin...

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices. Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method. These techniques approximate the solution of a large sparse linear system of...

We analyze the Pade method for computing the exponential of a real matrix. More precisely, we study the roundoff error introduced by the method in the general case and in three special cases: (1) normal matrices; (2) essentially nonnegative matrices (a(ij) greater than or equal to 0, i not equal j); (3) matrices A such that A = D-1 BD, with D diago...

We analyze the stability of the Cooley-Tukey algorithm for the Fast Fourier Transform of ordern=2 k and of its inverse by using componentwise error analysis.We prove that the components of the roundoff errors are linearly related to the result in exact arithmetic. We describe the structure of the error matrix and we give optimal bounds for the tota...

We consider acceleration techniques for the block Cimmino iterative method for solving general sparse systems. For iteration matrices that are not too ill conditioned, the conjugate gradient algorithm is a good method for accelerating the convergence. For ill-conditioned problems, the use of preconditioning techniques can improve the situation, but...

. P. Francois and J. M. Muller, (1990), Faut-il faire confiance aux ordinateurs ?, Rapport 90-03, LIP, ENS Lyon. P. Francois, (1989), Contribution `a l"etude de m'ethodes de controle automatique de l'erreur d'arrondi : la m'ethodologie SCALP, Ph. D. dissertation, Institut National Polytechnique de Grenoble. V. Frayss' e, (1992), Reliability of comp...

. We study the parallel implementations of a block iterative method in heterogeneous computing environments for solving linear systems of equations. The method is a generalization of the row-projection Cimmino method where blocks are obtained by partitioning the original linear system of equations. The method is referred to as the Block Cimmino met...

We present a parallel scheduler for distributing work to a group of processors in a heterogeneous computing environment. Some of the processors in the heterogeneous computing environment can be clustered to take advantage of particular communication networks. Here, the scheduler has been used in the implementation of a parallel block iterative solv...

The roundoff error analysis of several algorithms commonly used to compute the Fast Cosine Transform and the derivatives using the Chebyshev pseudospectral method are studied. We derive precise expressions for the algorithmic error, and using them we give new theoretical upper bounds and produce a statistical analysis. The results are compared with...

This paper shows how a theory for backward error analysis can be used to derive a family of stopping criteria for iterative methods and considers particular members of this family. Some theoretical justification is given for why these methods should work well and experimental evidence is presented to justify these claims.

A block version of Cimmino's algorithm for solving general sets of consistent sparse linear equations is described. The case of matrices in block tridiagonal form is emphasized because it is assumed that the general case can be reduced to this form by permutations. It is shown how the basic method can be accelerated by using the conjugate gradient...

A statistical approach to the study of the stability of a stationary iterative method for solving a linear system x=Px+q is studied. An asymptotic stability factor is introduced. The relations between this stability measure, the spectral radius of the iteration matrix, and the condition number of the system are studied. The special case when the it...

We consider acceleration techniques for the block Cimmino iterative method for solving general sparse systems. For iteration matrices which are not too ill conditioned, the conjugate gradient algorithm is a good method for accelerating the convergence. For ill conditioned problems, the classical conjugate gradient algorithm does poorly because of c...

Variants of the p4 algorithm of Hellerman and Rarick and the p5 algorithm of Erisman, Grimes, Lewis, and Poole, used for generating a bordered block triangular form for the in-core solution of sparse sets of linear equations, are considered. A particular concern is with maintaining numerical stability. Methods for ensuring stability and the extra c...