Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical techniques.

Structured inverse eigenvalue problems

We consider 2 × 2 block indefinite linear systems whose (2, 2) block is zero. Such systems arise in many applications. We discuss two techniques that are based on modifying the (1, 1) block in a way that makes the system easier to solve. The main part of the paper focuses on an augmented Lagrangian approach: a technique that modifies the (1,1) block without changing the system size. The choice of the parameter involved, the spectrum of the linear system, and its condition number are discussed, and some analytical observations are provided. A technique of deflating the (1,1) block is then introduced. Finally, numerical experiments that validate the analysis are presented.

/pdf/on-solving-block-structured-indefinite-linear-systems-5yjv8w0pte.pdf

On solving block-structured indefinite linear systems

For large sparse systems of linear equations iterative solution techniques are attractive. In this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner. We apply our technique to the linearized Navier--Stokes equations (the Oseen equations).

An Iteration for Indefinite Systems and Its Application to the Navier--Stokes Equations

Nonlinear normal modes for the Toda Chain

This paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem. The constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints. Among this class of convergent iterative schemes are generalizations of the relaxed Jacobi, Gauss-Seidel, and symmetric Gauss-Seidel schemes.

/pdf/the-numerical-solution-of-equality-constrained-quadratic-1i4iitpc87.pdf

The numerical solution of equality constrained quadratic programming problems

: The spectral properties of Jacobi and periodic Jacobi matrices are analyzed and algorithms for the construction of Jacobi and periodic Jacobi matrices with prescribed spectra are presented. Numerical evidence demonstrates that these algorithms are of practical utility. These algorithms have been used in studies of the periodic Toda lattice, and might also be used in studies of inverse eigenvalue problems for Sturm-Louiville equations and Hill's equation. (Author)

/pdf/the-construction-of-jacobi-and-periodic-jacobi-matrices-with-2mybo6nhgv.pdf

The construction of Jacobi and periodic Jacobi matrices with prescribed spectra

: The numerical construction of a smooth surface with prescribed weighted integrals over a domain of interest, is investigated This construction is mostly relevant to the estimation of a smooth density function over geographical regions, from data aggregated over several subregions By analogy to the definition of the univariate histospline the smooth surface is defined as the solution to a certain constrained minimization problem The application of finite element methods to the numerical solution of the minimization problem is studied It is shown that any finite element procedure, convergent for a related boundary value problem can be used to construct a sequence of finite element approximations converging to the smooth surface which solves the constrained minimization problem For the case of smoothness requirement of lowest order, a specific finite element method is considered, and its convergence as the mesh size decreases is demonstrated numerically for a particular example of volume matching (Author)

Numerical Construction of Smooth Surfaces from Aggregated Data.

: This paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem. The constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints. Among this class of convergent iterative schemes are generalizations of the relaxed Jacobi, Gauss-Seidel, and symmetric Gauss-Seidel schemes. (Author)

The Numerical Solution of a Class of Constrained Minimization Problems.

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Papers

The numerical solution of equality constrained quadratic programming problems

The construction of Jacobi and periodic Jacobi matrices with prescribed spectra

Numerical Construction of Smooth Surfaces from Aggregated Data.

The Numerical Solution of a Class of Constrained Minimization Problems.