Manish Malhotra

A BLOCK QMR ALGORITHM FOR NON-HERMITIAN LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES

Abstract Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ in their right-hand sides. Instead of applying an iterative method to each of these systems individually, it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose a block version of Freund and Nachtigals quasi-minimal residual (QMR) method for the iterative solution of non-Hermitian linear systems. The block QMR method uses a novel Lanczos-type process for multiple starting vectors, which was recently developed by Aliaga, Boley, Freund, and Hernandez, to compute suitable basis vectors for the underlying block Krylov subspaces. We describe the basic block QMR method, and also give important implementation details. In particular, we show how to incorporate deflation to drop converged linear systems, and to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences. Numerical results are reported that illustrate typical features of the block QMR method.

Recent developments in finite element methods for structural acoustics

SummaryThe study of structural acoustics involves modeling acoustic radiation and scattering, primarily in exterior regions, coupled with elastic and structural wave propagation. This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics. The cost-effectiveness of finite element methods is composed of several ingredients. Boundary-value problems in unbounded domains are inappropriate for direct discretization. Employing DtN methodology yields an equivalent problem that is suitable for finite element analysis by posing impedance relations at an artificial exterior boundary. Vell-posedness of the resulting continuous formulations is discussed, leading to simple guidelines for practical implementation and verifying that DtN boundary conditions provide a suitable basis for computation.Approximation by Galerkin finite element methods results in spurious dispersion, degrading with reduced wave resolution. Accuracy is improved by Galerkin/least-squares and related technologies on the basis of detailed examinations of discrete errors in simplified settings, relaxing wave-resolution requirements. This methodology is applied to time-harmonic problems of acoustics and coupled problems of structural acoustics. Space-time finite methods based on time-discontinuous Galerkin/least-squares are derived for transient problems of structural acoustics. Numerical results validate the superior performance of Galerkin/least-squares finite elements for problems of structural acoustics.A comparative study of the cost of computation demonstrates that Galerkon/least-squares finite element methods are economically competitive with boundary element methods, the prevailing numerical approach to exterior problems of acoustics. Efficient iterative methods are derived for solving the large-scale matrix problems that arise in structural acoustics computation of realistics configuration at high wavenumbers. An a posteriori error estimator and adaptive strategy are developed for time-harmonic acoustic problems and the role of adaptivity in reducing the cost of computation is addressed.

On the implementation of the Dirichlet-to-Neumann radiation condition for iterative solution of the Helmholtz equation

The Helmholtz equation posed on an unbounded domain with the Sommerfeld condition prescribed at infinity is considered. The unbounded domain is eliminated by imposing a Dirichlet-to-Neumann (DtN) map or a modified DtN map on a truncating surface and the resulting bounded domain problem is modeled using the finite element method. The resulting system of linear equations is then solved using a Krylov subspace iterative method. New, efficient algorithms to compute matrix-vector products that are based on the structure of the DtN and the modified DtN map are presented. Connections between the DtN map and the discrete Fourier transform in two dimensions and discrete spherical transform in three dimensions are established, and are utilized to develop fast implementations of matrix-vector product algorithms. Also, an SSOR-type preconditioner that is based on a local radiation condition is considered for the modified DtN formulation. An efficient implementation is proposed by extending Eisenstats trick for the standard SSOR preconditioner. Finally, numerical examples which illustrate the efficacy of the proposed algorithms are presented.

Iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi-minimal residual algorithm

Abstract Finite-element discretizations of time-harmonic acoustic wave problems in exterior domains result in large sparse systems of linear equations with complex symmetric coefficient matrices. In many situations, these matrix problems need to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. Recently, Freund and Malhotra have proposed a block quasi-minimal residual (BL-QMR) algorithm [13] for the iterative solution of non-Hermitian linear systems with multiple right-hand sides. The BL-QMR algorithm is a block Krylov-subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences. In this paper, we describe a J -symmetric variant of the BL-QMR algorithm that introduces important simplifications for the case when the coefficient matrix is symmetric with respect to a bilinear form induced by a certain matrix J . In particular, the J -symmetric variant includes the complex symmetric form of BL-QMR as a special case. We identify suitable preconditioners for the BL-QMR algorithm applied to multiple radiation and scattering problems. Our numerical tests with the preconditioned BL-QMR algorithm for such multiple linear systems show that, instead of solving each of the linear systems individually, it is significantly more efficient to employ the block version of the iterative method. Moreover, the numerical results clearly illustrate the importance of deflation and its effect on iterative convergence.

Parallel preconditioning based on h-hierarchical finite elements with application to acoustics

In this paper, we consider the hierarchical basis preconditioning approach which is closely related to the h-version of the hierarchical finite element method. In general, finite element formulations that employ hierarchical shape functions yield better conditioned matrix problems than those based on the Lagrange nodal basis. These matrix problems are also better-suited to a faster rate of convergence with Krylov-subspace iterative methods. We consider a purely algebraic approach to describe projections between the nodal and the h-hierarchical bases functions, which are then used to construct the preconditioning operator. Implementation details of the preconditioner are provided for solving finite element problems on unstructured grids. We employ the preconditioning approach in the iterative solution of linear systems that arise from finite element discretization of the exterior Helmholtz problem. Numerical results are presented to examine convergence rates for practical discretizations in acoustics, and to illustrate the computational performance of the preconditioning algorithm on serial and data-parallel computers.

A Krylov subspace projection method for simultaneous solution of Helmholtz problems at multiple frequencies

Abstract A Krylov subspace projection method which provides simultaneous solutions of the Helmholtz equation at multiple frequencies in one solution step is presented. The projector is obtained with an unsymmetric block Lanczos algorithm applied to a transfer function derived from a finite element discretization. This approach is equivalent to a matrix-valued Pade approximation of the transfer function. The proposed method is an extension of the formulation presented in [J. Comput. Acoust. 8 (2000) 223] to unsymmetric systems and allows the treatment of a much wider range of practical problems, including near-field and fluid–structure interaction computations.

Shape sensitivity calculations for exterior acoustics problems

The purpose of this paper is to present a method to calculate the derivative of a functional that depends on the shape of an object. This functional depends on the solution of a linear acoustic problem posed in an unbounded domain. We rewrite this problem in terms of another one posed in a bounded domain using the Dirichlet‐to‐Neumann (DtN) map or the modified DtN map. Using a classical method in shape sensitivity analysis, called the adjoint method, we are able to calculate the derivative of the functional using the solution of an auxiliary problem. This method is particularly efficient because the cost of calculating the derivatives is independent of the number of parameters used to approximate the shape of the domain. The resulting variational problems are discretized using the finite‐element method and solved using an efficient Krylov‐subspace iterative scheme. Numerical examples that illustrate the efficacy of our approach are presented.

Application of Padé via Lanczos approximations for efficient multifrequency solution of Helmholtz problems

This paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies. The exterior acoustics boundary value problem is approximated using the finite element method in combination with the Dirichlet-to-Neumann (DtN) map. The restriction domain problem is formally posed in transfer function form based on the finite element solution. In order to obtain the solution over a range of frequencies, a matrix-valued Padé approximation of the transfer function is employed, using a two-sided block Lanczos algorithm. This approach provides a stable and efficient representation of the Padé approximation. In order to apply the algorithm, it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition. This is illustrated for the case of the DtN boundary condition, but there is no restriction on the approach which can also be applied to other radiation boundary conditions. Numerical tests confirm that the approach offers significant computational speed-up.

Implementing highly accurate non-reflecting boundary conditions for large scale problems in structural acoustics

Time-harmonic problems of acoustic radiation and scattering in unbounded domains are governed by the Helmholtz equation in the fluid domain, coupled to the structural equations for elastic bodies in contact with the fluid. In addition to the boundary conditions applied to the structure and the fluid-structure interface conditions, a radiation condition is applied at infinity to render the solution to the problem unique. When modeling such problems with domain based methods such as the finite element method, the finite element mesh is truncated at a finite distance from the structure and either an artificial boundary condition that replicates the infinite domain is applied, or special elements such as infinite elements are used.

An efficient algorithm for the simultaneous solution of exterior acoustic problems over a frequency band

A method for solving time‐harmonic exterior acoustics problems in a frequency band over selected regions of the computational domain is presented. The partial fields of interest include surfaces enclosing the sound source, as well as distinct points in the near‐field of the source. The discretization of the boundary‐value problem is based on finite elements. Replacing the infinite domain problem with an equivalent formulation in a bounded domain leads to the incorporation of a Dirichlet‐to‐Neumann (DtN) map, which accounts for the analytical asymptotic behavior of the solution. By interpreting the discretized DtN operator as a low‐rank update of the system matrix, a standard shifted form of the inverse of this matrix, which is incorporated in the expression for the partial fields, is obtained. Due to the complex frequency dependence of this inverse, a direct evaluation is prohibitive. Hence, a matrix‐valued Pade approximation of the inverse operator is employed via a two‐sided block Lanczos algorithm, whi...