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Ganesh, M.; Hawkins, Stuart C.; Volkov, Darko

doi:10.1016/j.jcp.2019.108881

Cited by 8 publications

(7 citation statements)

References 54 publications

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“…Bayesian algorithms have been established for inverse problems of partial differential equations (pdes) posed on bounded domains [12], but they have not been widely applied to the inverse problems in scattering theory modeled by pdes posed on unbounded regions. Recent articles [4,7,10] (and related references) extended the Bayesian framework to a class of unbounded region wave propagation inverse models. The limited literature for this inverse problem is probably due to the enormous computational challenge; Bayesian algorithms typically require a very large number of evaluations of the forward model to sample the high-dimensional posterior distribution, and wave scattering models are notoriously expensive to evaluate.…”

Section: C114mentioning

confidence: 99%

“…Accordingly, we avoid such pinn-type variants in this article and, instead, we use our nn only as a surrogate, which we construct offline using a supervised training regime in which the nn is passed training data obtained from the bie. The amount of training data required is less dimension-sensitive than for other polynomial based surrogate models that we developed for low-dimensional parameter models [4,7].…”

Section: C114mentioning

confidence: 99%

“…In previous works for d 4 we accelerated the mcmc sampling by replacing the full forward model with a cheap-to-compute surrogate model based on an approximation of F(ω) using degree L polynomials [4,7]. In this work, motivated by the challenges of using polynomial-based surrogates for problems with high stochastic dimension, we develop a neural-network-based surrogate…”

Section: Neural-network-based Surrogate Modelmentioning

confidence: 99%

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An efficient Bayesian neural network surrogate algorithm for shape detection

Ganesh

Hawkins²,

Kordzakhia

et al. 2022

ANZIAMJ

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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/j.jcp.2019.108881 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/j.jcp.2018.10.045 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

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Section: C114mentioning

confidence: 99%

Section: C114mentioning

confidence: 99%

Section: Neural-network-based Surrogate Modelmentioning

confidence: 99%

See 1 more Smart Citation

An efficient Bayesian neural network surrogate algorithm for shape detection

Ganesh

Hawkins²,

Kordzakhia

et al. 2022

ANZIAMJ

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“…Focus has been on avoiding densemesh and topological low-frequency breakdown, on avoiding false resonances, and on providing unique solutions for wider ranges of material parameters. Among later contributions we mention [10,12,13,19,22,25,26,33].…”

Section: Introductionmentioning

confidence: 99%

Comparison of integral equations for the Maxwell transmission problem with general permittivities

Helsing

Rosén

2021

Adv Comput Math

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Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.

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“…In addition, the decomposition framework provides an analytical integral representation of the far-field using the scattered field, and hence our high-order FEM-BEM model provides relatively accurate approximations of the far-field arising from the heterogeneous model. For inverse wave models, accurate modeling of the far-field plays a crucial role in the identification of unknown wave propagation configuration properties from far-field measurements [2,13,23].…”

Section: Introductionmentioning

confidence: 99%

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

Domínguez

Ganesh

Sayas

2020

Journal of Computational Physics

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A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model.The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to setup a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.

show abstract

An efficient algorithm for a class of stochastic forward and inverse Maxwell models inR3

Cited by 8 publications

References 54 publications

An efficient Bayesian neural network surrogate algorithm for shape detection

An efficient Bayesian neural network surrogate algorithm for shape detection

Comparison of integral equations for the Maxwell transmission problem with general permittivities

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

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