Wenwen Chai

A direct integral-equation solver of linear complexity for large-scale 3D capacitance and impedance extraction

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 large-scale on-chip 3D interconnect embedded in inhomogeneous materials. Moreover, we have successfully applied the proposed solver to full-wave extraction.

Dense Matrix Inversion of Linear Complexity for Integral-Equation-Based Large-Scale 3-D Capacitance Extraction

State-of-the-art integral-equation-based solvers rely on techniques that can perform a dense matrix-vector multiplication in linear complexity. We introduce the H2 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 3-D capacitance extraction involving arbitrary geometry and nonuniform materials. We theoretically proved the existence of the H2 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.

Fast

In this paper, we propose a fast <i>H</i>-matrix-based direct solution with a significantly reduced computational cost for an integral-equation-based capacitance extraction of large-scale 3-D interconnects in multiple dielectrics. We reduce the computational cost of an <i>H</i>-matrix-based computation by simultaneously optimizing the <i>H</i>-matrix partition to minimize the number of matrix blocks and minimizing the rank of each matrix block based on a prescribed accuracy. With the proposed cost-reduction method, we develop a fast LU-based direct solver. This solver possesses a complexity of <i>kC</i>_sp<i>O</i> (<i>N</i><b>log</b><i>N</i>) in storage, a complexity of <i>k</i>²<i>C</i>_sp²<i>O</i>(<i>N</i>log²N) in LU factorization, and a complexity of <i>kC</i>_sp<i>O</i>(<i>N</i>logN) in LU solution, where <i>k</i> is the maximal rank, <i>C</i>_sp is a constant dependent on matrix partition, and the constant <i>kC</i>_sp is minimized based on accuracy by the proposed cost-reduction method. The proposed solver successfully factorizes dense matrices that involve millions of unknowns in fast CPU time and modest memory consumption, and with the prescribed accuracy satisfied. As an algebraic method, the underlying fast technique is kernel independent.

{\cal H}

Integral-equation-based methods generally lead to dense systems of linear equations. The resulting matrices, although dense, can be thought of as ldquodata-sparserdquo, i.e., they can be specified by few parameters. This can be accomplished by remodeling the problem subject to underlying hierarchical dependencies such that all interactions can be constructed from a reduced set of parameters. There exists a general mathematical framework called the ldquoHierarchical (H) Matrixrdquo framework [1-2], which enables a highly compact representation and efficient numerical computation of the dense matrices resulted from integral-equation-based methods. Storage requirements and matrix-vector multiplications using H-matrices have been shown to be of complexity O(N log N). Authors in [3] later introduced H2-matrices, which are a specialized subclass of hierarchical matrices. It was shown that the storage requirements and matrix-vector products are of complexity O(N). H-matrix-based techniques have been applied to solve electrostatic and magneto-static problems [4-6]. Questions that remain open are: Can recent advances in hierarchical matrix framework reduce the computation of electrodynamic problems? Can the matrices encountered in full-wave electromagnetics-based analysis be approximated as hierarchical matrices? In this work, we demonstrate the feasibility of H-matrix-based techniques in integral-equation-based solutions of electrodynamic problems.

-Matrix-Based Direct Integral Equation Solver With Reduced Computational Cost for Large-Scale Interconnect Extraction

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 low-rank 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.

An H-matrix-based method for reducing the complexity of integral-equation-based solutions of electromagnetic problems

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.

Theoretical Study on the Rank of Integral Operators for Broadband Electromagnetic Modeling From Static to Electrodynamic Frequencies

The Integral equation (IE) based computational electromagnetic methods generally lead to a dense system of linear equations, the solution of which could be very expensive. Recently, fast solvers [1–3] such as FMM-based methods, fast low-rank compression methods, FFT-based methods, and H2-matrix based methods have been developed, which dramatically reduce the memory and CPU time of iterative IE solvers for electrodynamic problems. Fast direct solvers have also been developed. LU factorization of O(N2) time complexity and O(N1.5) memory complexity was reported [4]. Compared to iterative solvers, direct solvers have advantages when the number of iterations or the number of right hand sides is large.

An LU Decomposition Based Direct Integral Equation Solver of Linear Complexity and Higher-Order Accuracy for Large-Scale Interconnect Extraction

We develop a new rank-minimized H2-matrix-based representation of the dense system matrix arising from an integral-equation (IE)-based analysis of large-scale 3-D interconnects. Different from the H2-representation generated by the existing interpolation-based method, the new H2-representation minimizes the rank in nested cluster bases and all off-diagonal blocks at all tree levels based on accuracy. The construction algorithm of the new H2-representation is applicable to both real- and complex-valued dense matrices generated from scalar and/or vector-based IE formulations. It has a linear complexity, and hence, the computational overhead is small. The proposed new H2-representation can be employed to accelerate both iterative and direct solutions of the IE-based dense systems of equations. To demonstrate its effectiveness, we develop a linear-complexity preconditioned iterative solver as well as a linear-complexity direct solver for the capacitance extraction of arbitrarily shaped 3-D interconnects in multiple dielectrics. The proposed linear-complexity solvers are shown to outperform state-of-the-art H2-based linear-complexity solvers in both CPU time and memory consumption. A dense matrix resulting from the capacitance extraction of a 3-D interconnect having 3.71 million unknowns and 576 conductors is inverted in fast CPU time (1.6 h), modest memory consumption (4.4 GB), and with prescribed accuracy satisfied on a single core running at 3 GHz.

A complexity-reduced H-matrix based direct integral equation solver with prescribed accuracy for large-scale electrodynamic analysis

We develop a linear-complexity direct matrix solution for the surface integral equation (IE)-based impedance extraction of arbitrarily shaped 3-D nonideal conductors embedded in a dielectric material. A direct inverse of a highly irregular system matrix composed of both dense and sparse matrix blocks is obtained in O(N) complexity with N being the matrix size. It outperforms state-of-the-art impedance solvers, be they direct solvers or iterative solvers, with fast central processing unit (CPU) time, modest memory consumption, and without sacrificing accuracy, for both small and large number of unknowns. The inverse of a 2.68-million-unknown matrix arising from the extraction of a large-scale 3-D interconnect having 128 buses, which is a matrix solution for 2.68 million right-hand sides, was obtained in less than 1.5 GB memory and 1.3 h on a single CPU running at 3 GHz.

Linear-Complexity Direct and Iterative Integral Equation Solvers Accelerated by a New Rank-Minimized

The low-frequency breakdown problem in electric field integral equation (EFIE) is well recognized and has been extensively studied. However, existing approaches have not rigorously solved the problem yet since they rely on low-frequency approximations. In this work, we present a rigorous method to fundamentally eliminate the problem. In this method, the original frequency dependent problem is rigorously transformed to a generalized eigenvalue problem, from the solution of which the frequency dependence of the EFIE solution can be analytically derived. The rigor of the proposed method has been validated at frequencies as low as DC. As the first rigorous solution to EFIE at low frequencies, the proposed method can be used to benchmark the accuracy of existing low-frequency EFIE-based solvers, quantitatively answer critical design questions such as at which frequency full-wave effects become important, etc.