We propose an analytical approach to study the rank of an integral operator, which is valid for an arbitrarily shaped object with an arbitrary electric size. With this analytical approach, we theoretically prove that for a prescribed error bound, the minimal rank of the interaction between two separated geometry blocks in an integral-equation operator, asymptotically, is a constant for 1-D distributions of source and observation points; grows very slowly with electric size as square root of the logarithm for 2-D distributions; and scales linearly with the electric size of the block diameter for 3-D problems. We thus prove the existence of an errorbounded low-rank representation of both surface- and volume-based integral operators for electromagnetic analysis, irrespective of electric size and object shape. Numerical experiments have validated the proposed analytical approach and its resultant findings on the rank of integral operators. This work provides a theoretical basis for employing and further developing low-rank matrix algebra for accelerating the computation of electrically large problems.
Rank, Integral Operators, Electrodynamic Analysis, One-, Two-, and Three-Dimensional Analysis, Theoretical Analysis
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