Iterative Solution of Dense Linear Systems

Integral equation methods have been used with great success in electromagnetic scattering calculations and in other problems involving unbounded computational domains. Their application is in many cases limited by the storage requirements of dense matrices and also by the rapidly increasing computational time. However, the use of iterative solvers and special methods for computing the matrix-vector products can greatly reduce both the CPU and memory requirements. In this work, the application of iterative solvers for three problems involving integral equations has been studied. For both the volume and the surface integral equation formulations of electromagnetic scattering the complex symmetric version of QMR is working eeciently. For the volume integral equation iterative solvers converge quickly even without preconditioning. We show how the eigenvalues of the coeecient matrix and the spectrum of the corresponding integral operator are related for a spherical scatterer. The matrix-vector products can be computed with a 3-dimensional FFT. For the surface integral equation of electromagnetic scattering we have applied the sparse approximate inverse preconditioner. For this problem, the fast multipole method can be employed for the calculation of the matrix-vector product. A third application is a surface integral formulation for an electrostatics problem, where the iterative solvers converge very quickly without preconditioning.