SGN: Sparse Gauss-Newton for Accelerated Sensitivity Analysis

We present a sparse Gauss-Newton solver for accelerated sensitivity analysis with applications to a wide range of equilibrium-constrained optimization problems. Dense Gauss-Newton solvers have shown promising convergence rates for inverse problems, but the cost of assembling and factorizing the associated matrices has so far been a major stumbling block. In this work, we show how the dense Gauss-Newton Hessian can be transformed into an equivalent sparse matrix that can be assembled and factorized much more efficiently. This leads to drastically reduced computation times for many inverse problems, which we demonstrate on a diverse set of examples. We furthermore show links between sensitivity analysis and nonlinear programming approaches based on Lagrange multipliers and prove equivalence under specific assumptions that apply for our problem setting.

Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. 10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

An efficient Bayesian neural network surrogate algorithm for shape detection

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