Saddle Point Systems Research Articles

A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems

We consider a mixed finite element method based on simplicial triangulations for a three-field formulation of linear elasticity. The three-field formulation is based on three unknowns: displacement, stress and strain. In order to obtain an efficient discretization scheme, we use a pair of finite element bases forming a biorthogonal system for the strain and stress. The biorthogonality relation allows us to statically condense out the strain and stress from the saddle-point system leading to a symmetric and positive-definite system. The strain and stress can be recovered in a post-processing step simply by inverting a diagonal matrix. Moreover, we show a uniform convergence of the finite element approximation in the incompressible limit. Numerical experiments are presented to support the theoretical results.

MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS

In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator

A distributive Gauss---Seidel relaxation based on the least squares commutator is devised for the saddle-point systems arising from the discretized Stokes equations. Based on that, an efficient mul...

Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems

SUMMARY A generalized skew-Hermitian triangular splitting iteration method is presented for solving non-Hermitian linear systems with strong skew-Hermitian parts. We study the convergence of the generalized skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew-Hermitian triangular splitting. Then the generalized skew-Hermitian triangular splitting iteration method is applied to non-Hermitian positive semidefinite saddle-point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew-Hermitian triangular splitting iteration methods are effective for solving non-Hermitian saddle-point linear systems with strong skew-Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the instationary Stokes equations that describe instationary, incompressible flows for moderate viscosities. After a mixed finite element discretization we get a differential-algebraic system of differential index two [J. Weickert, Navier-Stokes Equations as a Differential-Algebraic System, Preprint SFB393/96-08, Department of Mathematics, Chemnitz University of Technology, Chemnitz, Germany, 1996]. To reduce this index, we follow the analytic ideas of [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] coupled with the projection idea of [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. Avoiding this explicit projection leads to solving a series of large-scale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b...

Stochastic Optimal Robin Boundary Control Problems of Advection-Dominated Elliptic Equations

In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.

Analysis of Shape Optimization for Magnetic Microswimmers

We analyze an infinite dimensional, geometrically constrained shape optimization problem for magnetically driven microswimmers (locomotors) in three-dimensional (3-D) Stokes flow and give a well-posed descent scheme for computing optimal shapes. The problem is inspired by recent experimental work in this area. We show the existence of a minimizer of the optimization problem using analytical tools for elastic rods that respect the excluded volume constraint. We derive a variational gradient descent method for computing optimal locomotor shapes using the tools of shape differential calculus. The descent direction is obtained by solving a saddle-point system, which we prove is well-posed. We also introduce a finite element approximation of the gradient descent method and prove its stability. We present numerical results illustrating our method and the effect that finite aspect ratio and external cargo can have on the optimal shape. The 3-D Stokes equations are solved by a boundary integral method.

Robust Iterative Solution of a Class of Time‐Dependent Optimal Control Problems

AbstractThe fast iterative solution of optimal control problems, and in particular PDE‐constrained optimization problems, has become an active area of research in applied mathematics and numerical analysis. In this paper, we consider the solution of a class of time‐dependent PDE‐constrained optimization problems, specifically the distributed control of the heat equation. We develop a strategy to approximate the (1, 1)‐block and Schur complement of the saddle point system that results from solving this problem, and therefore derive a block diagonal preconditioner to be used within the MINRES algorithm. We present numerical results to demonstrate that this approach yields a robust solver with respect to step‐size and regularization parameter. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Preconditioned iteration for saddle-point systems with bound constraints arising in contact problems

We introduce a new algorithm, which we shall call Mixed Iteration, or by the acronym Mix-It, for the solution of saddle point problems arising from minimization problems under equality and/or inequality constraints. Such constrained problems are found in many applications and we shall be specially interested in contact problems in solid mechanics. The algorithm is however general and can be applied to other situations. We present a real saddle point algorithm in the sense that the primal variables and Lagrange multipliers unknowns are solved simultaneously. The effectiveness of the proposed method for large-scale simulations is improved by a block preconditioner and an adequate treatment of inequality constraints. The block preconditioner involves the solution of two subsystems associated respectively with the primal and the dual variables. We present some numerical results to illustrate the performances of the proposed method.

A Constrained Ordering for Solving the Equality Constrained State Estimation

This paper applies a simple constrained ordering for the solution of the equality constrained state estimation problem. By low-rank perturbations in the semidefinite (1,1) block of the coefficient matrix, while maintaining sparsity, a saddle point matrix is formed. The vectors used for generating the perturbations are rows of the matrix associated with the equality constraints that represent the zero injections. The proposed algorithm make use of the Bridson's ordering constraint for saddle-point systems, which is sufficient to guarantee the existence of a signed Cholesky factorization for the perturbed indefinite coefficient matrix, with separate symbolic and numerical phases. The need for numerical pivoting during factorization is avoided, with clear benefits for performance. Two alternative implementations are provided, either modifying a fill-reducing ordering algorithm to incorporate this constraint or modifying an existing fill-reducing ordering to respect the constraint. The proposed method is compared with existing methods in terms of computational time and convergence robustness. The IEEE 300-bus and the FRCC 3949-bus systems are used as test beds for this study.

The parallel solution of dense saddle-point linear systems arising in stochastic programming

We present a novel approach for solving dense saddle-point linear systems in a distributed-memory environment. This work is motivated by an application in stochastic optimization problems with recourse, but the proposed approach can be used for a large family of dense saddle-point systems, in particular, for those arising in convex programming. Although stochastic optimization problems have many important applications, they can present serious computational difficulties. In particular, sample average approximation (SAA) problems with a large number of samples are often too big to solve on a single shared-memory system. Recent work has developed interior-point methods and specialized linear algebra to solve these problems in parallel, using a scenario-based decomposition that distributes the data, and work across computational nodes. Even for sparse SAA problems, the decomposition produces a dense and possibly very large saddle-point linear system that must be solved repeatedly. We developed a specialized parallel factorization procedure for these systems, together with a streamlined method for assembling the distributed dense matrix. Strong scaling tests indicate over 90% efficiency on 1024 cores on a stochastic unit commitment problem with 57 million variables. Stochastic unit commitment problems with up to 189 million variables are solved efficiently on up to 2048 cores.

Generalized Block Triangular Preconditioner for Saddle Point Systems

In this paper, based on the preconditioner for saddle point systems in Benzi and Liu [M. Benzi, J. Liu, Block preconditioning for saddle point systems with indefinite (1,1) block, International Journal of Computer Mathematics, 84(8) (2007) 1117-1129], we present the generalized block preconditioner for generalized saddle point systems, which uses some more parameters and is a generalization of Benzi and Liu's block preconditioner. Moreover, spectral properties of our generalized block triangular preconditioner are given.

A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices

In this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular (1,1) blocks. Results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given, and an optimal augmentation block preconditioner of the set is derived. These results extend previous ones in the literature. Numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported.

Combination preconditioning of saddle point systems for positive definiteness

SUMMARYAmongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyżanowski (Numerical Linear Algebra with Applications 2011; 18:123–140) identified a two‐parameter family of preconditioners in this context and Stoll and Wathen (SIAM Journal on Matrix Analysis and Applications 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self‐adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self‐adjoint with respect to a nonstandard inner product always allow a MINRES‐type method (‐PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product, a more efficient CG‐like method (‐PCG) can be reliably used. We establish eigenvalue expressions for Krzyżanowski preconditioners and show that for a specific choice of parameters, although the Krzyżanowski preconditioned saddle point matrix is self‐adjoint with respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only ‐PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, ‐PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems. Copyright © 2012 John Wiley & Sons, Ltd.

Preconditioning for Allen–Cahn variational inequalities with non-local constraints

The solution of Allen–Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal–dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach.

An Iterative Two-Grid Method of A Finite Element PML Approximation for the Two Dimensional Maxwell Problem

AbstractIn this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer(PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of H(grad) variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete H(curl)-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.

Improving the Solution of Least Squares Support Vector Machines with Application to a Blast Furnace System

The solution of least squares support vector machines (LS‐SVMs) is characterized by a specific linear system, that is, a saddle point system. Approaches for its numerical solutions such as conjugate methods Sykens and Vandewalle (1999) and null space methods Chu et al. (2005) have been proposed. To speed up the solution of LS‐SVM, this paper employs the minimal residual (MINRES) method to solve the above saddle point system directly. Theoretical analysis indicates that the MINRES method is more efficient than the conjugate gradient method and the null space method for solving the saddle point system. Experiments on benchmark data sets show that compared with mainstream algorithms for LS‐SVM, the proposed approach significantly reduces the training time and keeps comparable accuracy. To heel, the LS‐SVM based on MINRES method is used to track a practical problem originated from blast furnace iron‐making process: changing trend prediction of silicon content in hot metal. The MINRES method‐based LS‐SVM can effectively perform feature reduction and model selection simultaneously, so it is a practical tool for the silicon trend prediction task.

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

We consider time-dependent parabolic problems coupled across a common interface which we formulate using a Lagrange multiplier construction and solve by applying a monolithic solution technique. We derive an adjoint-based a posteriori error representation for a quantity of interest given by a linear functional of the solution. We establish the accuracy of our error representation formula through numerical experimentation and investigate the effect of error in the adjoint solution. Crucially, the error representation affords a distinction between temporal and spatial errors and can be used as a basis for a blockwise time-space refinement strategy. Numerical tests illustrate the efficacy of the refinement strategy by capturing the distinctive behavior of a localized traveling wave solution. The saddle point systems considered here are equivalent to those arising in the mortar finite element technique for parabolic problems.

Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems

In this article, we motivate, derive, and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems which arise in PDE-constrained optimization problems. We consider the distributed control problem involving the heat equation and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are effective for a wide range of regularization parameter values, as well as mesh sizes and time-steps.

Structured backward errors for generalized saddle point systems

In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its (2,2) block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms.