Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

Interior penalty method for the indefinite time-harmonic Maxwell equations

Summary In this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L²-norm. In particular, the error in the energy norm is shown to converge with the optimal order (h^min{s,}) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L²-error is shown to converge with the optimal order (h⁺¹). The theoretical results are confirmed in a series of numerical experiments.Supported by the EPSRC (Grant GR/R76615).Supported by the Swiss National Science Foundation under project 21-068126.02.Supported in part by the Natural Sciences and Engineering Council of Canada.     

A finite element method for approximating the time-harmonic Maxwell equations

Summary In this paper we study the use of Nédélec's curl conforming finite elements to approximate the time-harmonic Maxwell equations on a bounded domain. The analysis is complicated by the fact that the bilinear form is not coercive, and the principle part has a very large null-space. This difficulty is circumvented by using a discrete Helmholtz decomposition of the error vector. Numerical results are presented that compare two different linear elements.Research supported in part by grants from AFOSR and NSF     

BTTB preconditioners for BTTB least squares problems

In this paper, we consider solving the least squares problem min_x‖b-Tx‖₂ by using preconditioned conjugate gradient (PCG) methods, where T is a large rectangular matrix which consists of several square block-Toeplitz-Toeplitz-block (BTTB) matrices and b is a column vector. We propose a BTTB preconditioner to speed up the PCG method and prove that the BTTB preconditioner is a good preconditioner. We then discuss the construction of the BTTB preconditioner. Numerical examples, including image restoration problems, are given to illustrate the efficiency of our BTTB preconditioner. Numerical results show that our BTTB preconditioner is more efficient than the well-known Level-1 and Level-2 circulant preconditioners.     

Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

We present numerical results concerning the solution of the time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.     

Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems

We study the solutions of Toeplitz systemsA _n x=b by the preconditioned conjugate gradient method. Then ×n matrixA _n is of the forma ₀ I+H _n wherea ₀ is a real number,I is the identity matrix andH _n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC _n and the skew-circulant matrixS _n whereA _n =1/2(C _n +S _n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC ^–1 _n A_n andS ^–1 _n A _n . For Toeplitz matricesA _n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC _n andS _n and prove that the singular values ofC ^–1 _n A _n andS ^–1 _n A _n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.     

Hartley preconditioners for Toeplitz systems generated by positive continuous functions

In this paper, we consider the solution ofn-by-n symmetric positive definite Toeplitz systemsT _n x=b by the preconditioned conjugate gradient (PCG) method. The preconditionerM _n is defined to be the minimizer of T _n –B _{n F} over allB _n H _n whereH _n is the Hartley algebra. We show that if the generating functionf ofT _n is a positive 2-periodic continuous even function, then the spectrum of the preconditioned systemM _n ^–1 T _n will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.     

On additive Schwarz preconditioners for sparse grid discretizations

Summary Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k ² ·2 ^k/2 ), wherek denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.     

A note on best conditioned preconditioners

We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition number(M ^–1/2 AM ^–1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM _Q = {Q* _N Q | _n is ann-by-n complex diagonal matrix} andM _Q ⁺ ={Q* _n Q | _n is ann-by-n positive definite diagonal matrix}. The preconditionerM _b that minimizes(M ^–1/2 AM ^–1/2 ) overM _Q ⁺ is called the best conditioned preconditioner for the matrixA overM _Q ⁺ . We prove that ifQAQ* has Young's Property A, thenM _b is nothing new but the minimizer of M –A _F overM _Q . Here · _F denotes the Frobenius norm. Some applications are also given here.     

Circulant integral operators as preconditioners for Wiener-Hopf equations

In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {C }. Letk(t) denote the kernel function of the Wiener-Hopf equation and be its Fourier transform. We prove that for sufficiently large if {C } is uniformly bounded on the real lineR and the convolution product of the Fourier transform ofC with uniformly onR, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.Research supported in part by HKRGC grant no. 221600070.     

Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes

Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional, order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio. In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions. Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002 The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council     

Numerical experiments with two approximate inverse preconditioners

We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out on a Cray C98 vector processor. This work was partially supported by the GA AS CR under grant 2030706 and by the grant GA CR 205/96/0921.     

Optimal preconditioners of a given sparsity pattern

We consider the problem of finding the symmetric positive definite preconditionerM of a given form- e.g., having nonzero elements only in specified positions — which minimizes the ratio of the largest to smallest eigenvalue ofM ^–1 A, for a given symmetric positive definitive matrixA. We show how this problem can be expressed as one of minimizing a convex function and how an optimization code can be used to solve the problem numerically. Results are presented showing optimal preconditioners of various sparsity patterns and comparing these to preconditioners that have been proposed in the literature. Several conjectures are made, based on results from the optimization code.This work was supported by the Advanced Research Projects Agency of the Department of Defense under contract F49620-87-C-0065 and by the Applied Mathematical Sciences Program of the U.S. Department of Energy under contract DE-AC02-76ER03077.     

Parallel preconditioners for large scale partial difference equation systems

We propose a new preconditioner DASP (discrete approximate spectral preconditioner), based on the existing well-known preconditioners and our computational experience. Parallel preconditioning strategies for large scale partial difference equation systems arising from partial differential equations are investigated. Numerical results are given to show the efficiency and effectiveness of the new preconditioners for both model problems and real applications in petroleum reservoir simulation.     

A numerical study of optimized sparse preconditioners

Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n ^1/4) andO(n ^1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.Supported by The Norwegian Research Council for Science and the Humanities (NAVF) under grants no. 413.90/002 and 412.93/005.Supported by The Royal Norwegian Council for Scientific and Industrial Research (NTNF) through program no. STP.28402: Toolkits in industrial mathematics.     

Wavelet sparse approximate inverse preconditioners

We show how to use wavelet compression ideas to improve the performance of approximate inverse preconditioners. Our main idea is to first transform the inverse of the coefficient matrix into a wavelet basis, before applying standard approximate inverse techniques. In this process, smoothness in the entries ofA ⁻¹ are converted into small wavelet coefficients, thus allowing a more efficient approximate inverse approximation. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverses. Supported by grants from ONR: ONR-N00014-92-J-1890, and the Army Research Office: DAAL-03-91-C-0047 (Univ. of Tenn. subcontract ORA4466.04 Amendment 1 and 2). The first and the third author also acknowledge support from RIACS/NASA Ames NAS 2-96027 and the Alfred P. Sloan Foundation as Doctoral Dissertation Fellows, respectively. the work was supported by the Natural Sciences and Engineering Research Council of Canada, the Information Technology Research Centre (which is funded by the Province of Ontario), and RIACS/NASA Ames NAS 2-96027.     

A discontinuous Galerkin method on refined meshes for the two-dimensional time-harmonic Maxwell equations in composite materials

In this paper, a discontinuous Galerkin method for the two-dimensional time-harmonic Maxwell equations in composite materials is presented. The divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized. Due to an appropriate mesh refinement near exterior and interior corners, the singular behaviour of the electromagnetic field is taken into account. Optimal error estimates in a discrete energy norm and in the L²$L^2$-norm are proved in the case where the exact solution is singular.     

A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical study on incomplete orthogonal factorization preconditioners

We design, analyse and test a class of incomplete orthogonal factorization preconditioners constructed from Givens rotations, incorporating some dropping strategies and updating tricks, for the solution of large sparse systems of linear equations. Comprehensive accounts about how the preconditioners are coded, what storage is required and how the computation is executed for a given accuracy are presented. A number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate Krylov subspace iteration methods such as GMRES and BiCGSTAB.     

The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case

Summary. In [10,14], circulant-type preconditioners have been proposed for ill-conditioned Hermitian Toeplitz systems that are generated by nonnegative continuous functions with a zero of even order. The proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero. Therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems. In [10,14], it was claimed that this result can be the case where the generating functions have multiple zeros. The main aim of this paper is to give a complete convergence proof of the method in [10,14] for this class of generating functions. Received October 19, 1999 / Revised version received May 2, 2001 / Published online October 17, 2001