@techreport{NA-08/17,
  abstract = "We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the method is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This, makes it a lot simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.",
  author = "Kamel Nafa and Andrew J Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "October",
  number = "NA-08/17",
  title = "Local projection finite element stabilization for the generalized Stokes problem",
  year = "2008",
}


    
      @techreport{NA-08/14,
  abstract = "It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. Hence a semi-iterative method, which requires eigenvalue bounds and computes an explicit polynomial, must, for just a little less computational work, give an inferior result. In this manuscript we identify a specific situation in the context of preconditioning when the Chebyshev semi-iterative method is the method of choice since it has properties which make it superior to the Conjugate Gradient method.",
  author = "Andrew J. Wathen, Tyrone Rees",
  institution = "Oxford University Computing Laboratory",
  month = "October",
  number = "NA-08/14",
  title = "Chebyshev Semi-iteration in Preconditioning",
  year = "2008",
}


    
      @techreport{NA-08/08,
  abstract = "Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from partial differential equations and in the area of Optimization such problems are ubiquitous. In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples.",
  author = "H. Sue Dollar and Nicholas I. M. Gould and Martin Stoll and Andrew J. Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-08/08",
  title = "A Bramble-Pasciak-like method with applications in optimization",
  year = "2008",
}


    
      @techreport{NA-08/10,
  abstract = "Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other partial differential equations.",
  author = "Tyrone Rees and H. Sue Dollar and Andrew J. Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-08/10",
  title = "Optimal solvers for PDE-Constrained Optimization",
  year = "2008",
}


    
      @techreport{NA-07/11,
  abstract = "It is widely appreciated that the iterative solution of linear systems of equations with large sparse matrices is much easier when the matrix is symmetric. It is equally advantageous to employ symmetric iterative methods when a nonsymmetric matrix is self-adjoint in a non-standard inner product. Here, general conditions for such self-adjointness are considered. In particular, a number of known examples for saddle point systems are surveyed and combined to make new combination preconditioners which are self-adjoint in di erent inner products.",
  author = "Martin Stoll and Andy Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "April",
  number = "NA-07/11",
  title = "Combination preconditioning and self-adjointness in non-standard inner products with application to saddle point problems",
  year = "2007",
}


    
      @techreport{NA-07/13,
  abstract = "The Bramble-Pasciak Conjugate Gradient method is a well known tool to solve linear systems in saddle point form. A drawback of this method in order to ensure applicability of Conjugate Gradients is the need for scaling the preconditioner which typically involves the solution of an eigenvalue problem. Here, we introduce a modified preconditioner and inner product which without scaling enable the use of a MINRES variant and can be used for the simplified Lanczos process. Furthermore, the modified preconditioner and inner product can be combined with the original Bramble-Pasciak setup to give new preconditioners and inner products. We undermine the new methods by showing numerical experiments for Stokes problems.",
  author = "Martin Stoll and Andy Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-07/13",
  title = "The Bramble-Pasciak preconditioner for saddle point problems",
  year = "2007",
}


    
      @techreport{NA-07/22,
  abstract = "The simultaneous solution of&nbsp;<em>Ax=b</em>&nbsp;and&nbsp;<em>A<sup>T</sup>y=g</em>&nbsp;is required in a number of situations. Darmofal and Lu have proposed a method based on the Quasi-Minimal residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and show how its performance can be improved when using the Generalized LSQR method. We further show how preconditioners can be introduced to enhance the speed of convergence and discuss different preconditioners that can be used. The scattering amplitude&nbsp;<em>g<sup>T</sup>x</em>, a widely used quantity in signal processing for example, has a close connection to the above problem since&nbsp;<em>x</em>&nbsp;represents the solution of the forward problem and&nbsp;<em>g</em>&nbsp;is the right hand side of the adjoint system. We show how this quantity can be efficiently approximated using Gauss quadrature and introduce a Block-Lanczos process that approximates the scattering amplitude and which can also be used with preconditioners.",
  author = "Gene H. Golub and Martin Stoll and Andy Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "November",
  number = "NA-07/22",
  title = "Approximation of the scattering amplitude",
  year = "2007",
}