Convergence properties of linear equation solvers applied to iterative
spectral domain integral equation methods
Integral equation techniques for the simulation of (M)MIC-structures
have become very popular in the last years since they offer a high versatility
combined with a minimum of discretization effort. On the other hand the
computational and storage complexity increases with O(N^2) or even O(N^3)
using conventional implementations of the Method of Moments (MoM).In order
to reduce this increasing computational effort we currently develop several
strategies for a combination within an overall simulation framework. Up
to a certain number of unknowns, the efficient generation of the explicit
system matrix with the subsequent solution of the linear system of equations
remains the best choice for the analysis of circuits of moderate size.
A promising method for larger structures is the iterative MoM based on
a fast implicit spectral domain matrix-vector product evaluation as part
of any iterative linear equation solver. In contrast to Conjugate Gradient
type FFT-iterative methods, our method is not restricted on uniform discretization
strategies, but it still suffers from the poor convergence properties of
the iterative solvers in context with strongly nonuniform discretized structures.In
this context we present the convergence behavior of different solvers applied
to structures with a strictly uniform discretization up to antenna subarrays
with a strongly nonuniform discretization. Furthermore we have applied
matrix scaling techniques and a complex matrix mapping to equivalent real
valued equation systems and diagonal preconditioning. Unfortunately all
measures applied so far lead to a nonsatisfactory convergence in context
with structures modeled with a strongly nonuniform discretization and we
cannot observe a significant advantage of any solver. The most stable solution
process we get so far with the Transpose Free Quasi Minimal Residual (TFQMR)-method,
but the number of necessary matrix-vector products remains nearly the same
as with other Krylov subspace methods like the Standard Conjugate Gradient
or Conjugate Residual methods.
* University of Wuppertal, Chair of Electromagnetic Theory, Fuhlrottstr.
10, 42097 Wuppertal, Germany .
|