Fast CPU Time Research Articles

Linear-Complexity Direct and Iterative Integral Equation Solvers Accelerated by a New Rank-Minimized ${\cal H}^{2}$-Representation for Large-Scale 3-D Interconnect Extraction

We develop a new rank-minimized H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -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 H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -representation generated by the existing interpolation-based method, the new H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -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 H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -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 H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -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 H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -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.

Fast ${\cal H}$-Matrix-Based Direct Integral Equation Solver With Reduced Computational Cost for Large-Scale Interconnect Extraction

In this paper, we propose a fast H-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 H-matrix-based computation by simultaneously optimizing the H-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 kCspO (NlogN) in storage, a complexity of k2Csp2O(Nlog2N) in LU factorization, and a complexity of kCspO(NlogN) in LU solution, where k is the maximal rank, Csp is a constant dependent on matrix partition, and the constant kCsp 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.

A Quadratic Eigenvalue Solver of Linear Complexity for 3-D Electromagnetics-Based Analysis of Large-Scale Integrated Circuits

It is of critical importance to efficiently and accurately predict global resonances of a 3-D integrated circuit system that involves arbitrarily shaped lossy conductors and inhomogeneous materials. A quadratic eigenvalue solver of linear complexity and electromagnetic accuracy is developed in this paper to fulfill this task. Without sacrificing accuracy, the proposed eigenvalue solver has shown a clear advantage over state-of-the-art eigenvalue solvers in fast CPU time. It successfully solves a quadratic eigenvalue problem of over 2.5 million unknowns associated with a large-scale 3-D on-chip circuit embedded in inhomogeneous materials in 40 min on a single 3 GHz 8222SE AMD Opteron processor.

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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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.

TM Plane-Wave Scattering From Finite Rectangular Grooves in a Conducting Plane Using Overlapping T-Block Method

Transverse magnetic plane-wave scattering from finite rectangular grooves in a conducting plane is systematically analyzed with the overlapping T-block method. Multiple rectangular grooves are divided into several overlapping T-blocks to obtain the fast CPU time, simple applicability, and wide versatility. The field representations within T-blocks are expressed using the Green's function relation and mode-matching method. The scattered fields are obtained in simple closed forms including a fast-convergent integral.