respective total fields inside an inhomogeneous scatterer. These total fields, by definition, form a Trefftz set. This “training set” can be precomputed and then used to approximate the field induced by another wave, indicated with a dashed arrow and a question mark. For visual clarity, only three Trefftz waves are sketched, and only the incident components. Left: one inhomogeneous object is present. Right: additional scatterers (such as S1, S2, S3) may be present in the case of an unknown field (dashed arrow) whose approximation is sought within a given small subdomain $\Omega_h$.](test_waves.png "fig:"){width="0.4\linewidth"} 0.2in ![Several incident waves (schematically indicated with solid arrows) give rise to the respective total fields inside an inhomogeneous scatterer. These total fields, by definition, form a Trefftz set. This “training set” can be precomputed and then used to approximate the field induced by another wave, indicated with a dashed arrow and a question mark. For visual clarity, only three Trefftz waves are sketched, and only the incident components. Left: one inhomogeneous object is present. Right: additional scatterers (such as S1, S2, S3) may be present in the case of an unknown field (dashed arrow) whose approximation is sought within a given small subdomain $\Omega_h$.](test_waves_complex.png "fig:"){width="0.4\linewidth"}

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