國立清華大學 數學系

In several important applications - e.g. seismic exploration or earthquake prediction - one seeks to infer unknown material properties of the earth's subsurface by sending seismic waves down and measuring the scattered field which comes back. In the process of solving the inverse problem (so-called "full-waveform inversion") one needs to iteratively solve the forward scattering problem, each time using an improved guess of the unknown material properties. In practice each step is done by solving the appropriate wave equation using explicit time stepping. However in many applications the relevant signals are band-limited and it would be more efficient to solve in the frequency domain (the Helmholtz equation), except for the fact that the construction of optimal solvers for the high frequency Helmholtz equation is highly problematic. The development of fast Helmholtz solvers at high frequency is of great current interest and requires a combination of linear algebra (iterative methods for non-normal complex linear systems) and PDE theory (stability theory for highly oscillatory non-self-adjoint PDEs). In the talk I'll describe progress in this area with concentration on the use of domain decomposition methods in a non-standard setting. Most of the theory concerns the constant wave-speed case at high frequency. I will also describe at the end of the talk some recent work on stability theory for the variable wave speed case which shows that the full theory of solvers in the latter case is likely to be both rich and challenging. References: M.J. Gander, I.G. Graham, E.A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik, 2015. I.G. Graham, E.A. Spence and E. Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems using absorption. Math Comp., to appear, 2016.