To facilitate the broadband modeling of integrated electronic and photonic systems from static to electrodynamic frequencies, we propose an analytical approach to study the rank of the integral operator for electromagnetic analysis, 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 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 distributions. We thus prove the existence of an error-bounded lowrank representation of both surface-and volume-based integral operators for electromagnetic analysis, irrespective of electric size and object shape. Numerical experiments validated the proposed analytical approach and the resultant findings on the rank of integral operators. This paper provides a theoretical basis for employing and further developing low-rank matrix algebra for accelerating the integral-equation-based electromagnetic analysis from static to electrodynamic frequencies.Index Terms-3-D, broadband analysis, electrodynamic analysis, integral operators, rank, theoretical analysis.

State-of-the-art integral equation based solvers rely on techniques that can perform a dense matrix-vector multiplication in linear complexity. We introduce  2 matrix as a mathematical framework to enable a highly efficient computation of dense matrices. Under this mathematical framework, as yet, no linear complexity has been established for matrix inversion. In this work, we developed a matrix inverse of linear complexity to directly solve the dense system of linear equations for the capacitance extraction involving arbitrary geometry and non-uniform materials.We theoretically proved the existence of the  2 matrix representation of the inverse of the dense system matrix, and revealed the relationship between the block cluster tree of the original matrix and that of its inverse. We analyzed the complexity and the accuracy of the proposed inverse, and proved its linear complexity as well as controlled accuracy. The proposed inverse-based direct solver has demonstrated clear advantages over state-of-the-art capacitance solvers such as FastCap and HiCap: with fast CPU time and modest memory consumption, and without sacrificing accuracy. It successfully inverts a dense matrix that involves more than one million unknowns associated with a large-scale, on-chip, 3-D interconnect embedded in inhomogeneous materials with fast CPU time and less than 5 GB memory.

State-of-the-art integral-equation-based solvers rely on techniques that can perform a matrix-vector multiplication in O(N) complexity. In this work, a fast inverse of linear complexity was developed to solve a dense system of linear equations directly for the capacitance extraction of any arbitrary shaped 3D structure. The proposed direct solver has demonstrated clear advantages over state-of-the-art solvers such as FastCap and HiCap; with fast CPU time and modest memory consumption, and without sacrificing accuracy. It successfully inverts a dense matrix that involves more than one million unknowns associated with a largescale on-chip 3D interconnect embedded in inhomogeneous materials. Moreover, we have successfully applied the proposed solver to full-wave extraction.

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Abstract-A fast LU factorization of linear complexity is developed to directly solve a dense system of linear equations for the capacitance extraction of any arbitrary shaped 3-D structure embedded in inhomogeneous materials. In addition, a higher-order scheme is developed to achieve any higher-order accuracy for the proposed fast solver without sacrificing its linear computational complexity. The proposed solver successfully factorizes dense matrices that involve more than one million unknowns in fast CPU run time and modest memory consumption. Comparisons with state-of-the-art integral-equation-based capacitance solvers have demonstrated its clear advantages. In addition to capacitance extraction, the proposed LU solver has been successfully applied to large-scale full-wave extraction.Index Terms-Direct solver, fast solver, integral-equation-based methods, interconnect extraction, LU factorization.