Data-informed uncertainty quantification for wave scattering by heterogeneous media

We present an efficient data-driven offline/online Bayesian algorithm for uncertainty quantification (UQ) in the induced scattered field when a time-harmonic incident wave interacts with an uncertain heterogeneous medium. The incident wave of interest need not be known in advance, and the uncertainty is informed by noisy scattering data obtained from other incident waves impinging on the medium. Our UQ algorithm is accelerated by a novel stochastic reduced order model (ROM) based on the T-matrix, and the ROM is independent of both the incident wave, and other incident waves used to generate the data. This important property allows the model to be set up offline. References M. Bachmayr and A. Djurdjevac. Multilevel representations of isotropic Gaussian random fields on the sphere. IMA J. Numer. Anal. 43.4 (2023), pp. 1970–2000. doi: 10.1093/imanum/drac034 C. Borges and G. Biros. Reconstruction of a compactly supported sound profile in the presence of a random background medium. Inv. Prob. 34, 115007 (2018). doi: 10.1088/1361-6420/aadbc5 on p. C101). Y. Chen. Inv. Prob. 13 (1997), pp. 253–282. doi: 10.1088/0266-5611/13/2/005 D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory. 4th. Springer, 2019. doi: 10.1007/978-3-030-30351-8 M. Ganesh and S. C. Hawkins. A far-field based T-matrix method for two dimensional obstacle scattering. Proceedings of the 9th Biennial Engineering Mathematics and Applications Conference, EMAC-2009. Ed. by P. Howlett, M. Nelson, and A. J. Roberts. Vol. 51. ANZIAM J. 2010, pp. C215–C230. doi: 10.21914/anziamj.v51i0.2581 M. Ganesh and S. C. Hawkins. A numerically stable T-matrix method for acoustic scattering by nonspherical particles with large aspect ratios and size parameters. J. Acoust. Soc. Am. 151 (2022), pp. 1978–1988. doi: 10.1121/10.0009679 M. Ganesh and S. C. Hawkins. Algorithm 975: TMATROM–A T-matrix reduced order model software. ACM Trans. Math. Softw. 44, 9 (2017), pp. 1–8. doi: 10.1145/3054945 M. Ganesh, S. C. Hawkins, and R. Hiptmair. Convergence analysis with parameter estimates for a reduced basis acoustic scattering T-matrix method. IMA J. Numer. Anal. 32 (2012), pp. 1348–1374. doi: 10.1093/imanum/drr041 M. Ganesh, S. C. Hawkins, A. M. Tartakovsky, and R. Tipireddy. A stochastic domain decomposition and post-processing algorithm for epistemic uncertainty quantification. Int. J. Uncertain. Quant. 13 (2023), pp. 1–22. doi: 10.1615/Int.J.UncertaintyQuantification.2023045687 S. C. Hawkins. Algorithm 1009: MieSolver–An object-oriented Mie series software for wave scattering by cylinders. ACM Trans. Math. Softw. 46, 19 (2020), pp. 1–28. doi: 10.1145/3381537 on p. C109). S. C. Hawkins. Noisy far-field data. Published online 12th August 2023. doi: 10.5281/zenodo.8240111 T. Hohage. On the numerical solution of a three-dimensional inverse medium scattering problem. Inv. Prob. 17 (2001), pp. 1743–1763. doi: 10.1088/0266-5611/17/6/314 A. Kirsch and P. Monk. An analysis of the coupling of finite-element and Nyström methods in acoustic scattering. IMA J. Numer. Anal 14 (1994), pp. 523–544. doi: 10.1093/imanum/14.4.523 on p. C101). M. Löhndorf and J. M. Melenk. On Thin Plate Spline Interpolation. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Ed. by M. Bittencourt, N. Dumont, and J. Hesthaven. Vol. 119. Lecture Notes in Computational Science and Engineering. Springer, 2017, pp. 451–466. doi: 10.1007/978-3-319-65870-4_32 T. D. Mast. Empirical relationships between acoustic parameters in human soft tissues. Acoust. Res. Lett. Online 1 (2000), pp. 37–42. doi: 10.1121/1.1336896 L. Stals. Efficient Solution Techniques for a Finite Element Thin Plate Spline Formulation. J. Sci. Comput. 63 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x K. C. Tam. Two-dimensional inverse Born approximation in ultrasonic flaw characterization. J. Nondestruct. Eval. 5 (1985), pp. 95–106. doi: 10.1007/BF00566959 W. J. Wiscombe. Improved Mie Scattering Algorithms. Appl. Opt. 19 (1980), pp. 1505–1509. doi: 10.1364/AO.19.001505