M.B. Van Gijzen

Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian

Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to provide an optimal value for the shift, and to explain the meshdependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.

Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theory

This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ζ. The admittance t = 1/ζ is the classical homotopy parameter. In particular, we study the spectrum when t → ∞. We show that in the limit part of the eigenvalues remain bounded and converge to the so-called kernel points. We also show that there exist the so-called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| → ∞ enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright

A polynomial preconditioner for the GMRES algorithm

The major drawback of GMRES is that the storage demands and the number of operations per iteration increase with the number of iterations. It is important to avoid that so many iterations are needed that the work per iteration and the storage requirements become unacceptably high. This paper describes a polynomial preconditioner with which this can be achieved efficiently. The polynomial preconditioner is constructed so that it has a minimization property in an area of the complex plane. A suitable area, and hence the preconditioning polynomial, can be obtained from eigenvalue estimates. The polynomial preconditioner is very simple and easy to implement.

An analysis of element-by-element preconditioners for nonsymmetric problems

Abstract A group of preconditioners for systems of linear equations arising from finite element discretization, which is of increasing interest, is the class of element-by-element preconditioners. These preconditioners allow the solution of a system without the need to assemble the global stiffness matrix. The success of a preconditioner depends on how much it reduces the condition number of the global stiffness matrix. In this paper a bound on the condition number of the preconditioned stiffness matrix is derived in terms of norms of the element stiffness matrices, the element preconditioning matrices and the flexibility matrix. Ideas are proposed how to improve the element-by-element preconditioner.

On the use of rigid body modes in the deflated preconditioned conjugate gradient method

Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper considers the deflated preconditioned conjugate gradient method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.

Shallow-water acoustic communication with high bit rate BPSK signals

BPSK signals have been defined for transmission through a shallow-water acoustic communication channel. The signals were accompanied by two displaced carriers to facilitate carrier recovery. To correct for the adverse effects of time spreading, a pseudo-random learning sequence was transmitted ahead of the communication signal. The signal processing consists of a shift to baseband guided by the displaced carriers, a least-squares equalizer tuned to the received learning signal, coherent addition of reconstructed baseband signals (corresponding to selected channels of the receiving hydrophone array), and bit restoration with a decision-directed equalizer. In this manner, bit rates up to 4 kbit/s are successfully dealt with in moving-point-to-fixed-point communication. The main risk appears to be the fading of a displaced carrier.BPSK signals have been defined for transmission through a shallow-water acoustic communication channel. The signals were accompanied by two displaced carriers to facilitate carrier recovery. To correct for the adverse effects of time spreading, a pseudo-random learning sequence was transmitted ahead of the communication signal. The signal processing consists of a shift to baseband guided by the displaced carriers, a least-squares equalizer tuned to the received learning signal, coherent addition of reconstructed baseband signals (corresponding to selected channels of the receiving hydrophone array), and bit restoration with a decision-directed equalizer. In this manner, bit rates up to 4 kbit/s are successfully dealt with in moving-point-to-fixed-point communication. The main risk appears to be the fading of a displaced carrier.

An MSSS-preconditioned Matrix Equation Approach for the Time-Harmonic Elastic Wave Equation at Multiple Frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

Solving Large Sparse Linear Systems Efficiently on Grid Computers Using an Asynchronous Iterative Method as a Preconditioner

This paper describes an efficient iterative algorithm for solving large sparse linear systems on Grid computers. The algorithm is a combination of a synchronous flexible outer iterative method and a coarse-grain asynchronous inner iterative method as a preconditioner. The preconditioning iteration is performed on heterogeneous computing hardware. We present experimental results on a heterogeneous computing grid of a complete implementation using GridSolve as middleware for a 3D convection–diffusion problem.

Conjugate gradient-like solution algorithms for the Mixed Finite Element approximation of the Biharmonic Equation, applied to plate bending problems

Abstract Discretization of the Biharmonic Equation with the Mixed Finite Element Method yields an indefinite linear system of equations with a special structure. In this paper two variants of the Conjugate Gradient method are formulated that are suited for solving such systems. They both require the solution of a system of linear equations in every iteration. Different strategies for doing this are examined. An Incomplete Choleski decomposition is used as a preconditioner. Both iterative methods and the preconditioner are chosen so that optimal use can be made of the special block structure of the global system of equations.

A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

This work presents a new algorithm to compute eigenpairs of large unsymmetric matrices. Using the Induced Dimension Reduction method (IDR( s )), which was originally proposed for solving systems of linear equations, we obtain a Hessenberg decomposition, from which we approximate the eigenvalues and eigenvectors of a matrix. This decomposition has two main advantages. First, IDR( s ) is a short-recurrence method, which is attractive for large scale computations. Second, the IDR( s ) polynomial used to create this Hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues. Additionally, we incorporate the implicitly restarting technique proposed by D.C. Sorensen, in order to approximate specific portions of the spectrum and improve the convergence.