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

Circumcentered reflections method for wavelet feasibility problems

We introduce a two-stage global-then-local search method for solving feasibility problems. The approach pairs the advantageous global tendency of the Douglas–Rachford method to find a basin of attraction for a fixed point, together with the local tendency of the circumcentered reflections method to perform faster within such a basin. We experimentally demonstrate the success of the method for solving nonconvex problems in the context of wavelet construction formulated as a feasibility problem. References F. J. Aragón Artacho, R. Campoy, and M. K. Tam. The Douglas–Rachford algorithm for convex and nonconvex feasibility problems. Math. Meth. Oper. Res. 91 (2020), pp. 201–240. doi: 10.1007/s00186-019-00691-9 R. Behling, J. Y. Bello Cruz, and L.-R. Santos. Circumcentering the Douglas–Rachford method. Numer. Algor. 78.3 (2018), pp. 759–776. doi: 10.1007/s11075-017-0399-5 R. Behling, J. Y. Bello-Cruz, and L.-R. Santos. On the linear convergence of the circumcentered-reflection method. Oper. Res. Lett. 46.2 (2018), pp. 159–162. issn: 0167-6377. doi: 10.1016/j.orl.2017.11.018 J. M. Borwein, S. B. Lindstrom, B. Sims, A. Schneider, and M. P. Skerritt. Dynamics of the Douglas–Rachford method for ellipses and p-spheres. Set-Val. Var. Anal. 26 (2018), pp. 385–403. doi: 10.1007/s11228-017-0457-0 J. M. Borwein and B. Sims. The Douglas–Rachford algorithm in the absence of convexity. Fixed-point algorithms for inverse problems in science and engineering. Springer, 2011, pp. 93–109. doi: 10.1007/978-1-4419-9569-8_6 I. Daubechies. Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41.7 (1988), pp. 909–996. doi: 10.1002/cpa.3160410705 N. D. Dizon, J. A. Hogan, and J. D. Lakey. Optimization in the construction of nearly cardinal and nearly symmetric wavelets. In: 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030889 N. D. Dizon, J. A. Hogan, and S. B. Lindstrom. Circumcentering reflection methods for nonconvex feasibility problems. arXiv preprint arXiv:1910.04384 (2019). url: https://arxiv.org/abs/1910.04384 D. J. Franklin. Projection algorithms for non-separable wavelets and Clifford Fourier analysis. PhD thesis. University of Newcastle, 2018. doi: 1959.13/1395028. D. J. Franklin, J. A. Hogan, and M. K. Tam. A Douglas–Rachford construction of non-separable continuous compactly supported multidimensional wavelets. arXiv preprint arXiv:2006.03302 (2020). url: https://arxiv.org/abs/2006.03302 D. J. Franklin, J. A. Hogan, and M. K. Tam. Higher-dimensional wavelets and the Douglas–Rachford algorithm. 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030823 B. P. Lamichhane, S. B. Lindstrom, and B. Sims. Application of projection algorithms to differential equations: Boundary value problems. ANZIAM J. 61.1 (2019), pp. 23–46. doi: 10.1017/S1446181118000391 S. B. Lindstrom and B. Sims. Survey: Sixty years of Douglas–Rachford. J. Aust. Math. Soc. 110 (2020), 1–38. doi: 10.1017/S1446788719000570 S. B. Lindstrom, B. Sims, and M. P. Skerritt. Computing intersections of implicitly specified plane curves. J. Nonlin. Convex Anal. 18.3 (2017), pp. 347–359. url: http://www.yokohamapublishers.jp/online2/jncav18-3 S. G. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2(R). Trans. Amer. Math. Soc. 315.1 (1989), pp. 69–87. doi: 10.1090/S0002-9947-1989-1008470-5 Y. Meyer. Wavelets and operators. Cambridge University Press, 1993. doi: 10.1017/CBO9780511623820 G. Pierra. Decomposition through formalization in a product space. Math. Program. 28 (1984), pp. 96–115. doi: 10.1007/BF02612715

A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. doi:10.1017/S1446181121000377

Bounds on the critical times for the Fisher-KPP equation

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Ideal planar fluid flow over a submerged obstacle: Review and extension

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour. doi:10.1017/S1446181121000341

Numerical solutions to a fractional diffusion equation used in modelling dye-sensitized solar cells

Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement. doi:10.1017/S1446181121000353

Do poor environmental conditions drive trachoma transmission in Burundi? A mathematical modelling study

Trachoma is an infectious disease and it is the leading cause of preventable blindness worldwide. To achieve its elimination, the World Health Organization set a goal of reducing the prevalence in endemic areas to less than 55% by 2020, utilizing the SAFE (surgery, antibiotics, facial cleanliness, environmental improvement) strategy. However, in Burundi, trachoma prevalences of greater than 55% are still reported in 11 districts and it is hypothesized that this is due to the poor implementation of the environmental improvement factor of the SAFE strategy. In this paper, a model based on an ordinary differential equation, which includes an environmental transmission component, is developed and analysed. The model is calibrated to recent field data and is used to estimate the reductions in trachoma that would have occurred if adequate environmental improvements were implemented in Burundi. Given the assumptions in the model, it is clear that environmental improvement should be considered as a key component of the SAFE strategy and, hence, it is crucial for eliminating trachoma in Burundi. doi:10.1017/S1446181121000389

Investigation of determinism-related issues in the Sobol′ low-discrepancy sequence for producing sound global sensitivity analysis indices

A computationally efficient and robust sampling scheme can support a sensitivity analysis of models to discover their behaviour through Quasi Monte Carlo approximation. This is especially useful for complex models, as often occur in environmental domains when model runtime can be prohibitive. The Sobol' sequence is one of the most used quasi-random low-discrepancy sequences as it can explore the parameter space significantly more evenly than pseudo-random sequences. The built-in determinism of the Sobol' sequence assists in achieving this attractive property. However, the Sobol' sequence tends to deteriorate in the sense that the estimated errors are distributed inconsistently across model parameters as the dimensions of a model increase. By testing multiple Sobol' sequence implementations, it is clear that the deterministic nature of the Sobol' sequence occasionally introduces relatively large errors in sensitivity indices produced by well-known global sensitivity analysis methods, and that the errors do not diminish by averaging through multiple replications. Problematic sensitivity indices may mistakenly guide modellers to make type I and II errors in trying to identify sensitive parameters, and this will potentially impact model reduction attempts based on these sensitivity measurements. This work investigates the cause of the Sobol' sequence's determinism-related issues. References I. A. Antonov and V. M. Saleev. An economic method of computing LPτ-sequences. USSR Comput. Math. Math. Phys. 19.1 (1979), pp. 252–256. doi: 10.1016/0041-5553(79)90085-5 P. Bratley and B. L. Fox. Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 14.1 (1988), pp. 88–100. doi: 10.1145/42288.214372 J. Feinberg and H. P. Langtangen. Chaospy: An open source tool for designing methods of uncertainty quantification. J. Comput. Sci. 11 (2015), pp. 46–57. doi: 10.1016/j.jocs.2015.08.008 on p. C90). S. Joe and F. Y. Kuo. Constructing Sobol sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30.5 (2008), pp. 2635–2654. doi: 10.1137/070709359 S. Joe and F. Y. Kuo. Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 29.1 (2003), pp. 49–57. doi: 10.1145/641876.641879 W. J. Morokoff and R. E. Caflisch. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15.6 (1994), pp. 1251–1279. doi: 10.1137/0915077 X. Sun, B. Croke, S. Roberts, and A. Jakeman. Comparing methods of randomizing Sobol’ sequences for improving uncertainty of metrics in variance-based global sensitivity estimation. Reliab. Eng. Sys. Safety 210 (2021), p. 107499. doi: 10.1016/j.ress.2021.107499 S. Tarantola, W. Becker, and D. Zeitz. A comparison of two sampling methods for global sensitivity analysis. Comput. Phys. Com. 183.5 (2012), pp. 1061–1072. doi: 10.1016/j.cpc.2011.12.015 S. Tezuka. Discrepancy between QMC and RQMC, II. Uniform Dist. Theory 6.1 (2011), pp. 57–64. url: https://pcwww.liv.ac.uk/~karpenk/JournalUDT/vol06/no1/5Tezuka11-1.pdf I. M. Sobol′. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7.4 (1967), pp. 86–112. doi: 10.1016/0041-5553(67)90144-9 I. M. Sobol′. Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp 1.4 (1993), pp. 407–414.

Bounds on isolated scattering number

The isolated scattering number is a parameter that measures the vulnerability of networks. This measure is bounded by formulas depending on the independence number. We present new bounds on the isolated scattering number that can be calculated in polynomial time. References Z. Chen, M. Dehmer, F. Emmert-Streib, and Y. Shi. Modern and interdisciplinary problems in network science: A translational research perspective. CRC Press, 2018. doi: 10.1201/9781351237307 P. Erdős and T. Gallai. On the minimal number of vertices representing the edges of a graph. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), pp. 181–203. url: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.210.7468 J. Harant and I. Schiermeyer. On the independence number of a graph in terms of order and size. Discrete Math. 232.1–3 (2001), pp. 131–138. doi: 10.1016/S0012-365X(00)00298-3 E. Korach, T. Nguyen, and B. Peis. Subgraph characterization of red/blue-split graph and Kőnig Egerváry graphs. Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms. ACM, New York, 2006, pp. 842–850. doi: 10.1145/1109557.1109650 F. Li, Q. Ye, and Y. Sun. Proceedings of the 2016 Joint Conference of ANZIAM and Zhejiang Provincial Applied Mathematics Association, ANZPAMS-2016. Ed. by P. Broadbridge, M. Nelson, D. Wang, and A. J. Roberts. Vol. 58. ANZIAM J. 2017, E81–E97. doi: 10.21914/anziamj.v58i0.10993 F. Li, Q. Ye, and X. Zhang. Isolated scattering number of split graphs and graph products. ANZIAM J. 58.3-4 (2017), pp. 350–358. doi: 10.1017/S1446181117000062 E. R. Scheinerman and D. H. Ullman. Fractional graph theory. Dover Publications, 2011. url: https://www.ams.jhu.edu/ers/wp-content/uploads/2015/12/fgt.pdf S. Y. Wang, Y. X. Yang, S. W. Lin, J. Li, and Z. M. Hu. The isolated scattering number of graphs. Acta Math. Sinica (Chin. Ser.) 54.5 (2011), pp. 861–874. url: http://www.actamath.com/EN/abstract/abstract21097.shtml M. Xiao and H. Nagamochi. Exact algorithms for maximum independent set. Inform. and Comput. 255, Part 1 (2017), pp. 126–146. doi: 10.1016/j.ic.2017.06.001

Interaction of a singular surface with a strong shock in the interstellar gas clouds

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions. doi:10.1017/S1446181121000328