Title of Dissertation: ITERATIVE METHODS FOR THE STOCHASTIC DIFFUSION PROBLEM Darran G. Furnival, Doctor of Philosophy, 2008 Dissertation directed by: Professor Howard Elman Department of Computer Science It is the purpose of this thesis to develop iterative methods for solving the linear systems that arise from application of the stochastic finite element method to steady-state stochastic diff...

In this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter is included. The first algorithm reduces the saddle point system to a symmetric p...

Topology optimization is a powerful tool for global and multiscale design of structures, microstructures, and materials. The computational bottleneck of topology optimization is the solution of a large number of extremely ill-conditioned linear systems arising in the finite element analysis. Adaptive mesh refinement (AMR) is one efficient way to reduce the computational cost. We propose a new A...

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large threedimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce th...

In this paper, we consider the generalized saddle point linear system of equations which is obtained from discretizing the Euler Lagrange equations associated with image debulrring problem. This system is ill-conditioned and is of huge size. Moreover, the (2,2) block of the coefficient matrix of this system contains summation of two terms. One of these terms is a product of a Toepelitz matrix w...

We consider a stationary Stokes interface problem. In the discretization the interface is not aligned with the triangulation. For the discretization we use the P1 extended finite element space (P1-XFEM) for the pressure and the standard conforming P2 finite element space for the velocity. Since this pair is not necessarily LBB stable, a consistent stabilization term, known from the literature, ...

One of the most powerful tools for solving large and sparse systems of linear algebraic equations is a class of iterative methods called Krylov subspace methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. The use of the Krylov subspaces in iterati...

Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE constrained optimization, an important class of these problems. Block diagonal and block triangular...

We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MINRES, SYMMLQ, GMRES, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Gi...

Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyżanowski (Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditi...