Maxwell's equations are universal for sparse linear algebra
Abstract
Optical and wave-based analog computing has re-emerged as a route to accelerating linear algebra, with metamaterials and photonic circuits shown to invert matrices and solve integral equations. However, it has remained unclear which problems such passive structures can represent, and at what cost, once encoding, precision, conditioning and readout are accounted for. Here, we show that every polynomially conditioned sparse linear system can be compiled into a three-dimensional dielectric -- a cavity filled with lossless, isotropic material of bounded permittivity -- whose time-harmonic response encodes its solution. For m nonzero coefficients, the volume of the structure and the cost of writing it down and reading out the answer scale as Õ(m3/2), coefficients require only O(log m) bits, and every field that approximately satisfies Maxwell's equations decodes to an approximate solution. We prove the construction for the continuum equations and give a computer-assisted version for the Yee discretization underlying finite-difference solvers. Read in reverse, it implies that electromagnetic simulation is, up to this overhead, as hard as general sparse linear algebra: a sufficiently fast general-purpose Maxwell solver would yield a faster sparse linear solver.
Related
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BibTeX
@misc{augenstein2026maxwells,
title = {{Maxwell's equations are universal for sparse linear algebra}},
author = {Yannick Augenstein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38913},
eprint = {2610.38913},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38913},
}
RIS
TY - UNPB
TI - Maxwell's equations are universal for sparse linear algebra
AU - Yannick Augenstein
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38913
UR - https://arxiv.org/abs/2610.38913
AB - Optical and wave-based analog computing has re-emerged as a route to accelerating linear algebra, with metamaterials and photonic circuits shown to invert matrices and solve integral equations. However, it has remained unclear which problems such passive structures can represent, and at what cost, once encoding, precision, conditioning and readout are accounted for. Here, we show that every polynomially conditioned sparse linear system can be compiled into a three-dimensional dielectric -- a cavity filled with lossless, isotropic material of bounded permittivity -- whose time-harmonic response encodes its solution. For m nonzero coefficients, the volume of the structure and the cost of writing it down and reading out the answer scale as Õ(m^3/2), coefficients require only O(log m) bits, and every field that approximately satisfies Maxwell's equations decodes to an approximate solution. We prove the construction for the continuum equations and give a computer-assisted version for the Yee discretization underlying finite-difference solvers. Read in reverse, it implies that electromagnetic simulation is, up to this overhead, as hard as general sparse linear algebra: a sufficiently fast general-purpose Maxwell solver would yield a faster sparse linear solver.
ER -