Optimal design of optical analog solvers of linear systems. (English) Zbl 07451772

Summary: In this paper, given a linear system of equations \(\mathbf{A}\, \mathbf{x}= \mathbf{b} \), we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlying partial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.


65-XX Numerical analysis
35C20 Asymptotic expansions of solutions to PDEs
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
Full Text: DOI arXiv


[1] Ammari, H.; Imeri, K.; Nigam, N., Optimization of Steklov-Neumann eigenvalues, J. Compt. Phys., 406, 109211 (2020) · Zbl 1453.35056
[2] Ammari, H.; Bruno, O.; Imeri, K.; Nigam, N., Wave enhancement through optimization of boundary conditions, SIAM J. Sci. Comput., 42, 1, B207-B224 (2020) · Zbl 1430.35171
[3] Ammari, H.; Imeri, K., A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of helmholtz resonators, Wave Motion, 97, 102614 (2020) · Zbl 07328366
[4] Ammari, H.; Imeri, K.; Wei, W., A mathematical framework for tunable metasurfaces. Part I, Asymptot. Anal., 114, 3-4, 129-179 (2019) · Zbl 1443.35153
[5] Ammari, H.; Imeri, K.; Wei, W., A mathematical framework for tunable metasurfaces. Part II, Asymptot. Anal., 114, 3-4, 181-209 (2019) · Zbl 1442.35443
[6] Krutitskii, PA, The neumann problem for the 2-d helmholtz equation in a domain, bounded by closed and open curves, Int. J. Math. Math. Sci. (1998) · Zbl 0910.35040
[7] Lan, J.; Yifeng, L.; Yue, X.; Xiaozhou, L., Manipulation of acoustic wavefront by gradient metasurface based on helmholtz resonators, Scie. Rep., 7, 1, 10587 (2017)
[8] Lin, D.; Fan, P.; Hasman, E.; Brongersma, ML, Dielectric gradient metasurface optical elements, Science, 345, 6194, 298-302 (2014)
[9] Mohammadi, NE; Brian, E.; Nader, E., Inverse-designed metastructures that solve equations, Science, 363, 6433, 1333-1338 (2019) · Zbl 1431.78002
[10] Muskhelishvili, NI, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (2013), New York: Dover Books on Mathematics, Dover Publications, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.