We are interested in solving scattering problems with strong trapping using the Combined Field Integral Equation (CFIE) and the Generalized Minimal Residual method (GMRes).
In this talk, we show a new understanding of how the number of GMRes iterations depends on frequency in this situation.
The non-normal nature of CFIE makes GMRes the iterative method of choice for solving linear systems stemming from its discretisation.
GMRes has the advantage of being able to solve any non-singular linear system, in particular non-normal. But the convergence analysis becomes less straightforward in this case, because it requires more information than just the spectrum. Bounds for GMRes applied to non-normal matrices can be derived using condition number of eigenvalues, numerical range or pseudo-spectrum 12.
But in the case of trapping, an additional difficulty comes from the solution operator whose norm grows exponentially through a sequence of frequencies tending to infinity, with the density of these ``bad’’ frequencies increasing as the frequency increases. In this case, the spectrum of the associated matrix has the form of a cluster with outliers near the origin. Following 3 where matrices with similar spectra are studied, we provide a new analysis of the GMRes convergence taking into account this particular distribution.
Mark Embree, How descriptive are GMRES convergence bounds? BIT Numerical Mathematics, 1996. ↩︎
Jörg Liesen and Petr Tichy, Convergence analysis of Krylov subspace methods. GAMM-Mitteilungen, 2004. ↩︎
S. L. Campbell and I. C. F. Ipsen and C. T. Kelley and C. D. Meyer, GMRES and the minimal polynomial. Oxford University Computing Laboratory, 1999. ↩︎