This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.

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The simulation of electronic circuits involves the numerical solution of very largescale, sparse, in general nonlinear, systems of differential-algebraic equations. Often, the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller statespace dimension. In this paper, we describe the use of Krylov-subspace methods for generating such reduced-order models of linear subcircuits. Particular emphasis is on reduced-order modeling techniques that preserve the passivity of linear RLC subcircuits. Key words: Lanczos algorithm; Arnoldi process; Linear dynamical system; VLSI interconnect; Transfer function; Pad'e approximation; Stability; Passivity; Positive real function 1 Introduction Today's integrated electronic circuits are extremely complex, with up to tens of millions of devices. Prototyping of such circuits is no longer possible, and instead, computational methods are used to simulate and ...

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

The aim of this paper is to construct Levenberg-Marquardt level set methods for inverse obstacle problems, and to discuss their numerical realization. Based on a recently developed framework for the construction of level set methods, we can define LevenbergMarquardt level set methods in a general way by varying the function space used for the normal velocity. In the typical case of a PDE-constraint, the iterative method yields an indefinite linear system to be solved in each iteration step, which can be reduced to a positive definite problem for the normal velocity. We discuss the structure of this systems and possibilities for its iterative solution.

In this paper, a strategy is proposed for alternative computations of the residual vectors in Krylov subspace methods, which improves the agreement of the computed residuals and the true residuals to the level of O(u)kAkkxk. Building on earlier ideas on residual replacement and on insights in the finite precision behaviour of the Krylov subspace methods, computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals, and they are used to monitor the deviation of the two residuals and hence to select residual replacement steps, so that the recurrence relations for the computed residuals, which control the convergence of the method, are perturbed within safe bounds. Numerical examples are presented to demonstrate the effectiveness of this new residual replacement scheme. 1 Introduction Krylov subspace iterative methods for solving a large linear system Ax = b typically consist of iterations that recursively update appr...

this paper we report on our continuing investigations on eigensolvers and their preconditioning. In contrast to [7] we conduct our experiments on a domain that approximates the new cavity design for the 590 MeV ring cyclotron at the Paul Scherrer Institute (PSI) in Villigen, Switzerland, see Fig. 1 on the right-hand side. The outline of the paper is as follows. In section II we will give the mathematical formulation of the problem. In section III the Nedelec finite element approach for tetrahedral meshes is looked at from a linear algebra point of view. In sections IV to VI we detail how 2 FIG. 1: Cross section of two di#erent designs for accelerating cavities. On the left the original design that is presently in operation; on the right the future design. The plane of these two cross sections is perpendicular to the accelerator mid plane and tangential to the orbit of the accelerated particles. Both cavities extend prismatically in the radial direction over approximately 3.3 meters. we solve the large eigenvalue problems and in particular solve the systems of equations that are introduced by the spectral transformation. In section VII we compare the performance of several variants in solving the model problem. II. STATEMENT OF THE PROBLEM Without changing the basic structure of the problem one can assume that the metallic surfaces are perfectly conducting and that the inside volume of the cavity# is all in vacuum. The electromagnetic field in the cavity is described by the Maxwell equations [1]. After separation of time and space and after elimination of the magnetic field intensity the di#erential equations

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We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph coloring problems for partial matrix estimation. Numerical experiments utilizing one of these formulations demonstrate the viability of this approach.

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