会议摘要（Abstract）

This week-long research workshop will bring together the researchers working at the forefront of numerical methods for frequency-domain wave equations and the corresponding applications. The workshop will have the following three themes.

Theme A: New discretization methods for frequency-domain wave problems and their analysis.

Theme B: Fast solvers for linear systems arising from frequency-domain wave problems - formulation, implementation, and analysis.

Theme C: Applications of the fast solvers, including inverse problems, geophysical imaging, and electromagnetics.

Theme B is the central theme of the workshop, but a key component of the programme will be synergetic interactions of Theme B with Themes A and C.

The workshop aims at 

1. To disseminate recent advances in each of Themes A, B, and C to the collective audience. 

2. To exploit recent advances in Theme A in the development of fast solvers in Theme B.

3. To explore how the demands of the applications in Theme C can inform the development of fast solvers in Theme B.

4. To initiate new research projects via the interactions described above. 

举办意义(Description of the aim)

The efficient computation of wave propagation and scattering problems at high frequency on modern multiprocessor computers (i.e., the goal of Theme B) has long been considered one of the ‘hard problems’ in numerical PDEs/scientific computing and is also of huge importance in applications - e.g., in seismic imaging and electromagnetics.

These wave problems are difficult to solve numerically because (a) the solutions are often highly oscillatory and very fine discretisations are needed to resolve them, leading to large system matrices; (b) the large system matrices are also highly indefinite and non-normal, meaning that standard iterative methods are not reliable; (c) the propagative nature of these problems means that the presence of a source at any point can have considerable effect far away, thus inhibiting the performance of parallel algorithms which rely on localization techniques. Each of these essential difficulties gets worse as the frequency increases.

Despite the difficulties above, in the last decade there has been huge progress in the development of fast solvers for wave problems (i.e., Theme B) informed by progress in Helmholtz theory and discretization methods (Theme A). There are many groups active in this area, including the organizers and the invitees of this workshop. Many successful algorithms are not yet rigorously understood, and instead are justified empirically by computational experiments and partial theory, combined with intuitive physical arguments.

We hope that the following questions will be answered during the week. 

1. Given that new methods are proposed, tested and implemented on a much faster timescale than that on which their rigorous analysis can proceed, what is the appropriate role of the numerical analyst in supporting the development and application of new methods? Conversely, what is the role of the practitioner in instigating and supporting theoretical investigations?

2. Spectral coarse spaces have proved effective in solvers for high frequency wave problems in practice (Themes B,C), but a full understanding of their properties is missing. Is it possible to make progress by harnessing the growing knowledge of non-polynomial discretization techniques for wave problems (Theme A)?　

3. Much progress has been made in the last 20 years on multiscale methods for coercive elliptic problems, and in recent years this has been partly extended to obtain new results for Helmholtz problems (Theme A). There is thus great potential for synergetic interaction between Themes A and B.

4. Perfectly-matched layers (PML) are used in some of the state-of-the-art fast solvers (Theme B), and recently there has been considerable progress in understanding their properties, but their rigorous understanding in the wide context of general geometries and variable coefficients is generally an open question. Is it possible to making progress on this question by using some new tools in semiclassical analysis?

5. Practitioners in the geophysics community (Theme C) use optimized finite-difference methods for solving the wave equation, whereas much of the rigorous theory for fast solvers is set in a finite element context. What tools are needed to ensure that emerging methods are discretisation agnostic and readily available for use in applications?