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.

/pdf/numerical-solution-of-saddle-point-problems-l583iob5c2.pdf

Numerical solution of saddle point problems

Contents: Matrix Eigenvalue Methods.- Double Bracket Isospectral Flows.- Singular Value Decomposition.- Linear Programming.- Approximation and Control.- Balanced Matrix Factorizations.- Invariant Theory and System Balancing.- Balancing via Gradient Flows.- Sensitivity Optimization.- Linear Algebra.- Dynamical Systems.- Global Analysis.

Optimization and Dynamical Systems

In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

/pdf/conjugate-gradient-methods-for-toeplitz-systems-44re7adc4m.pdf

Conjugate Gradient Methods for Toeplitz Systems

Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.

/pdf/a-fast-algorithm-for-deblurring-models-with-neumann-boundary-vk09we2ofi.pdf

A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.

/pdf/an-iss-small-gain-theorem-for-general-networks-5bzclxdsg1.pdf

An ISS small gain theorem for general networks

A numerically stable method, that uses the structure of the matrices, for computing the eigenvalues and eigenvectors of orthogonal matrices is described. The orthogonality of the matrices reduces the eigenproblem to three singular value problems for bidiagonal matrices, whose elements are explicitly known functions of the Schur parameters.

On the eigenproblem for orthogonal matrices

The Gohberg–Semencul formula expresses the inverse of a Toeplitz matrix as the difference of products of lower triangular and upper triangular Toeplitz matrices. In this paper the idea of cyclic displacement structure is used to show that the upper triangular matrices in this formula can be replaced by circulant matrices. The resulting computational savings afforded by this modified formula is discussed.

/pdf/a-variant-of-the-gohberg-semencul-formula-involving-1ykmxa7zn5.pdf

A variant of the Gohberg-Semencul formula involving circulant matrices

This paper explores the relations between the matrix Riccati equation and the standard matrix eigenvalue methods. It is demonstrated that the mathematics of the analysis of the two objects is essentially the same; consisting of the analysis of flows on the homogeneous spaces of various Lie groups.

The geometry of matrix eigenvalue methods

We study an algorithm for the numerical solution of algebraic matrix Riccati equa- tions that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subspace of the associated Hamiltonian matrix. We describe the details of this method and compare it with other numerical methods for the solution of the algebraic Riccati equation.

https://etna.mcs.kent.edu/vol.1.1993/pp33-48.dir/abstr33-48.pdf

A multishift algorithm for the numerical solution of algebraic riccati equations

Numerical experience with a superfast real Toeplitz solver

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