right panel of [Fig. \[fig:test-waves\]]{} illustrates a more interesting, and more complicated, case. Suppose that the Trefftz training set has been generated for the original inhomogeneous scatterer – same as in the left panel. However, the unknown “dashed arrow” solution *may involve additional objects* – such as S1, S2, S3 – in the computational domain. Obviously, under this complication, little can be inferred about the unknown solution from the Trefftz set *in the whole domain*, especially in the regions around the additional scatterers. One may hope, however, that the field *within a given small subdomain* $\Omega_h$ *inside the original scatterer* can still be approximated accurately as a superposition of the known Trefftz waves. This setup is the central issue of Sections \[sec:Auxiliary-basis\], \[sec:Random-matrices\], and is inspired by our numerical experiments with pseudorandom structures of [Section \[sec:FLAME-slab\]]{}, as well as by our earlier work on multiparticle problems [@Dai-Webb11].

The overall motivation for the paper is to highlight applications of Trefftz functions to problems involving complex, inhomogeneous media. Much of mathematical analysis so far has revolved around the homogeneous case (that is, equations with constant coefficients), where cylindrical, spherical or plane waves serve as Trefftz functions for the Helmholtz equation, while harmonic 