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2022 ◽  
Vol 62 ◽  
pp. C112-C127
Mahadevan Ganesh ◽  
Stuart Collin Hawkins ◽  
Nino Kordzakhia ◽  
Stefanie Unicomb

We present an efficient Bayesian algorithm for identifying the shape of an object from noisy far field data. The data is obtained by illuminating the object with one or more incident waves. Bayes' theorem provides a framework to find a posterior distribution of the parameters that determine the shape of the scatterer. We compute the distribution using the Markov Chain Monte Carlo (MCMC) method with a Gibbs sampler. The principal novelty of this work is to replace the forward far-field-ansatz wave model (in an unbounded region) in the MCMC sampling with a neural-network-based surrogate that is hundreds of times faster to evaluate. We demonstrate the accuracy and efficiency of our algorithm by constructing the distributions, medians and confidence intervals of non-convex shapes using a Gaussian random circle prior. References Y. Chen. Inverse scattering via Heisenberg’s uncertainty principle. 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 Edition. Vol. 93. Applied Mathematical Sciences. References C112 Springer, 2019. doi: 10.1007/978-3-030-30351-8 R. DeVore, B. Hanin, and G. Petrova. Neural Network Approximation. Acta Num. 30 (2021), pp. 327–444. doi: 10.1017/S0962492921000052 M. Ganesh and S. C. Hawkins. A reduced-order-model Bayesian obstacle detection algorithm. 2018 MATRIX Annals. Ed. by J. de Gier et al. Springer, 2020, pp. 17–27. doi: 10.1007/978-3-030-38230-8_2 M. Ganesh and S. C. Hawkins. Algorithm 975: TMATROM—A T-matrix reduced order model software. ACM Trans. Math. Softw. 44.9 (2017), pp. 1–18. doi: 10.1145/3054945 M. Ganesh and S. C. Hawkins. Scattering by stochastic boundaries: hybrid low- and high-order quantification algorithms. ANZIAM J. 56 (2016), pp. C312–C338. doi: 10.21914/anziamj.v56i0.9313 M. Ganesh, S. C. Hawkins, and D. Volkov. An efficient algorithm for a class of stochastic forward and inverse Maxwell models in R3. J. Comput. Phys. 398 (2019), p. 108881. doi: 10.1016/ L. Lamberg, K. Muinonen, J. Ylönen, and K. Lumme. Spectral estimation of Gaussian random circles and spheres. J. Comput. Appl. Math. 136 (2001), pp. 109–121. doi: 10.1016/S0377-0427(00)00578-1 T. Nousiainen and G. M. McFarquhar. Light scattering by quasi-spherical ice crystals. J. Atmos. Sci. 61 (2004), pp. 2229–2248. doi: 10.1175/1520-0469(2004)061<2229:LSBQIC>2.0.CO;2 A. Palafox, M. A. Capistrán, and J. A. Christen. Point cloud-based scatterer approximation and affine invariant sampling in the inverse scattering problem. Math. Meth. Appl. Sci. 40 (2017), pp. 3393–3403. doi: 10.1002/mma.4056 M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 (2019), pp. 686–707. doi: 10.1016/ A. C. Stuart. Inverse problems: A Bayesian perspective. Acta Numer. 19 (2010), pp. 451–559. doi: 10.1017/S0962492910000061 B. Veihelmann, T. Nousiainen, M. Kahnert, and W. J. van der Zande. Light scattering by small feldspar particles simulated using the Gaussian random sphere geometry. J. Quant. Spectro. Rad. Trans. 100 (2006), pp. 393–405. doi: 10.1016/j.jqsrt.2005.11.053

Václav Kučera ◽  
Mária Lukáčová-Medvid’ová ◽  
Sebastian Noelle ◽  
Jochen Schütz

AbstractIn this paper we derive and analyse a class of linearly implicit schemes which includes the one of Feistauer and Kučera (J Comput Phys 224:208–221, 2007) as well as the class of RS-IMEX schemes (Schütz and Noelle in J Sci Comp 64:522–540, 2015; Kaiser et al. in J Sci Comput 70:1390–1407, 2017; Bispen et al. in Commun Comput Phys 16:307–347, 2014; Zakerzadeh in ESAIM Math Model Numer Anal 53:893–924, 2019). The implicit part is based on a Jacobian matrix which is evaluated at a reference state. This state can be either the solution at the old time level as in Feistauer and Kučera (2007), or a numerical approximation of the incompressible limit equations as in Zeifang et al. (Commun Comput Phys 27:292–320, 2020), or possibly another state. Subsequently, it is shown that this class of methods is asymptotically preserving under the assumption of a discrete Hilbert expansion. For a one-dimensional setting with some limitations on the reference state, the existence of a discrete Hilbert expansion is shown.

2021 ◽  
Vol 62 ◽  
pp. C16-C29
Ziad Georges Ghauch

Metamodels provide an efficient means for the approximation of response surfaces of systems, particularly for resource-intensive experiment designs. It is oftentimes the case that interest is focused on a specific region of the parameter space. We propose an efficient recipe for the local approximation of response surfaces using Polynomial Chaos techniques. For systems embedded in high-dimensional settings, a basis-adapted spectral representation is exploited locally for dimension reduction. The proposed approach comprises an initial heuristic global solution for parameter space exploration using an approximate global Polynomial Chaos metamodel, followed by a local design being refined through an active learning scheme. The problem of turbulent flow around a symmetric airfoil is considered. Statistical estimators based on the local, active, basis-adapted approach show less bias and faster convergence as compared to the estimators from a global solution. References B. J. Bichon, M. S. Eldred, L. P. Swiler, S. Mahadevan, and J. M. McFarland. Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J. 46(10):2459–2468, 2008. doi: 10.2514/1.34321. G. E. P. Box and N. R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987. V. Dubourg, B. Sudret, and F. Deheeger. Metamodel-based importance sampling for structural reliability analysis. Prob. Eng. Mech. 33:47–57, 2013. doi: 10.1016/j.probengmech.2013.02.002. R. G. Ghanem and P. D. Spanos. Stochastic finite element: A spectral approach. Dover, 1991. doi: 10.1007/978-1-4612-3094-6. Z. G. Ghauch. Leveraging adapted polynomial chaos metamodels for real-time Bayesian updating. J. Verif. Valid. Uncert. 4(4):041003, 2020. doi: 10.1115/1.4045693. Z. G. Ghauch, V. Aitharaju, W. R. Rodgers, P. Pasupuleti, A. Dereims, and R. G. Ghanem. Integrated stochastic analysis of fiber composites manufacturing using adapted polynomial chaos expansions. Compos. Part A: Appl. Sci. 118:179–193, 2019. doi: 10.1016/j.compositesa.2018.12.029. M. E. Johnson, L. M. Moore, and D. Ylvisaker. Minimax and maximin distance designs. J. Stat. Plan. Infer. 26(2):131–148, 1990. doi: 10.1016/0378-3758(90)90122-B. A. Notin, N. Gayton, J. L. Dulong, M. Lemaire, P. Villon, and H. Jaffal. RPCM: A strategy to perform reliability analysis using polynomial chaos and resampling. Euro. J. Comput. Mech. 19(8):795–830, 2010. doi: 10.3166/ejcm.19.795-830. OpenCFD. OpenFOAM User’s Guide. 2019. V. Picheny, D. Ginsbourger, O. Roustant, R. T Haftka, and N.-H. Kim. Adaptive designs of experiments for accurate approximation of target regions. J. Mech. Design. 132(7):071008, 2010. doi: 10.1115/1.4001873. C. Thimmisetty, P. Tsilifis, and R. Ghanem. Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem. AI EDAM 31(3):265–276, 2017. doi: 10.1017/S0890060417000166. R. Tipireddy and R. Ghanem. Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259:304–317, 2014. doi: 10.1016/ P. Tsilifis and R. G. Ghanem. Reduced Wiener chaos representation of random fields via basis adaptation and projection. J. Comput. Phys. 341:102–120, 2017. doi: 10.1016/ Turbulence Modeling Resource. NASA Langley Research Center. Washington, DC, 2018.

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