We theoretically prove that the minimal rank of the interaction between two separated geometry blocks in an integral-equation based analysis of general three-dimensional objects, for a prescribed error bound, scales linearly with the electric size of the block diameter. We thus prove the existence of the error-bounded low-rank representation of both surface and volume based integral operators for electrodynamic analysis, irrespective of electric size and scatterer shape. The theoretical analysis developed in this work permits an analytical study of the minimal rank for a prescribed accuracy, for arbitrarily shaped objects with arbitrary electric sizes. Numerical experiments have verified its validity. This work provides a theoretical proof on why the low-rank matrix algebra can be employed to accelerate the computation of large-scale electrodynamic problems. The rank studied in this paper is based on a singular value decomposition based minimal rank approximation of integral operators, which does not rely on the separation of observation and source coordinates. Methods that do not generate a minimal rank approximation for a prescribed accuracy can result in a rank that scales with electric size at a much higher rate.
Rank, Integral Operators, Electrodynamic Analysis, Three Dimensional, Theoretical Analysis
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