Kurt Otto

Iterative Solution of the Helmholtz Equation by a Second-Order Method

The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a two-dimensional curvilinear duct. The problem is discretized with a second-order accurate finite difference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. We construct a preconditioner that is based on fast transforms and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. The fast transform preconditioner is compared with a symmetric successive over-relaxation (SSOR) preconditioner, and also with band Gaussian elimination. For the preconditioned iterative methods, the gains in storage requirement are large compared with band Gaussian elimination. Regarding the arithmetic complexity, the fast transform preconditioner yields a significant gain, whereas the SSOR preconditioner performs worse than band Gaussian elimination.

Iterative solution methods and preconditioners for block-tridiagonal systems of equations

Systems of equations arising from implicit time discretizations and finite difference space discretizations of systems of partial differential equations in two space dimensions are considered. The ...

Analysis of Preconditioners for Hyperbolic Partial Differential Equations

Preconditioners for conjugate gradient (CG)-like iterative methods are analyzed. The systems of equations arise from discretizations of time-dependent hyperbolic partial differential equations (PDEs) in two space dimensions. Incomplete LU (ILU) and block ILU preconditioners are considered for a model problem subject to periodic boundary conditions. The spectra of the preconditioned coefficient matrices are determined by Fourier analysis. It is found that the condition number is not improved by the ILU preconditioners. Furthermore, the number of distinct eigenvalues is not decreased. A semicirculant preconditioner applied to a problem, subject to Dirichlet boundary conditions at the inflow boundaries, is also examined. Analytical formulas for the eigenvalues and the eigenvectors are derived. For this purpose a special analysis, based on spectral decomposition and residue theory, is developed. When the grid ratio in space is less than one, the spectrum asymptotically becomes two finite curve segments, which are independent of the number of gridpoints. For the restarted generalized minimal residual (GMRES) iteration, a slight reduction of the grid ratio from one substantially improves the convergence rate. This is also predicted by an asymptotic analysis.

Semicirculant preconditioners for first-order partial differential equations

This paper considers solving time-independent systems of first-order partial differential equations (PDEs) in two space dimensions using a conjugate gradient (CG)-like iterative method. The systems of equations are preconditioned using semicirculant preconditioners. Analytical formulas for the eigenvalues and the eigenvectors are derived for a scalar model problem with constant coefficients. The main problems in constructing and analyzing the numerical methods are caused by the numerical boundary conditions required at the outflow boundaries. It is proved that, when the grid ratio is less than one, the spectrum asymptotically becomes two finite curve segments that are independent of the number of gridpoints. The same type of result for a time-dependent problem has previously been established. For the restarted generalized minimal residual (GMRES) iteration, a slight reduction of the grid ratio from one substantially improves the convergence rate. This is also predicted by an asymptotic analysis of the eig...

Object-oriented construction of parallel PDE solvers

An object-oriented approach is taken to the problem of formulating portable, easy-to-modify PDE solvers for realistic problems in three space dimensions. The resulting software library, Cogito, contains tools for writing programs to be executed on MIMD computers with distributed memory. Difference methods on composite, structured grids are supported. Most of the Cogito classes have been implemented in Fortran 77, in such a way that the object-oriented design is visible. With respect to parallel performance, these tools yield code that is comparable to parallel solvers written in plain Fortran 77. The resulting programs can be executed without modifications on a large number of multicomputer platforms, and also on serial computers.

A framework for polynomial preconditioners based on fast transforms I: Theory

Optimal and superoptimal approximations of a complex square matrix by polynomials in a normal basis matrix are considered. If the unitary transform associated with the eigenvectors of the basis matrix is computable using a fast algorithm, the approximations may be utilized for constructing preconditioners. Theorems describing how the parameters of the approximations could be efficiently computed are given, and for special cases earlier results by other authors are recovered. Also, optimal and superoptimal approximations for block matrices are determined, and the same type of theorems as for the point case are proved.

A framework for polynomial preconditioners based on fast transforms II : PDE applications

The solution of systems of equations arising from systems of time-dependent partial differential equations (PDEs) is considered. Primarily, first-order PDEs are studied, but second-order derivatives are also accounted for. The discretization is performed using a general finite difference stencil in space and an implicit method in time. The systems of equations are solved by a preconditioned Krylov subspace method. The preconditioners exploit optimal and superoptimal approximations by low-degree polynomials in a normal basis matrix, associated with a fast trigonometric transform. Numerical experiments for high-order accurate discretizations are presented. The results show that preconditioners based on fast transforms yield efficient solution algorithms, even for large quotients between the time and space steps. Utilizing a spatial grid ratio less than one, the arithmetic work per grid point is bounded by a constant as the number of grid points increases.

Analysis of semi-Toeplitz preconditioners for first-order PDEs

A semi-Toeplitz preconditioner for nonsymmetric, nondiagonally dominant systems of equations is studied. The preconditioner solve is based on a fast modified sine transform. As a model problem we study a system of equations arising from an implicit time discretization of a scalar hyperbolic partial differential equation (PDE). Analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system are derived. The convergence of a minimal residual iteration is shown to depend only on the spatial grid ratio and not on the number of unknowns.

Object-oriented software tools for the construction of preconditioners

In recent years, there has been considerable progress concerning preconditioned iterative methods for large and sparse systems of equations arising from the discretization of differential equations. Such methods are particularly attractive in the context of high-performance (parallel) computers. However, the implementation of a preconditioner is a nontrivial task. The focus of the present contribution is on a set of object-oriented software tools that support the construction of a family of preconditioners based on fast transforms. By combining objects of different classes, it is possible to conveniently construct any preconditioner within this family.

Semicirculant Solvers and Boundary Corrections for First-Order Partial Differential Equations

Fast solvers for systems of partial differential equations (PDEs) in two space dimensions are considered. The solvers are used as direct solution methods or as preconditioners for a conjugate gradient (CG)-like iteration. Mainly first-order PDEs are considered, but second-order terms may be included. Employing a semicirculant framework, PDEs with constant coefficients in one space direction and arbitrary boundary conditions are considered. A factorization of the inverse of the difference approximation matrix is described. This factorization is exploited to derive a direct solver, where the complexity for the first right-hand side is