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41
FAST COMPUTATION OF FOURIER INTEGRAL OPERATORS
, 2007
"... We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operat ..."
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Cited by 38 (10 self)
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We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operator (FIO) of the form ∫ e2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) isan amplitude, and Φ(x, ξ) is a phase function, which is typically as large as ξ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size N by N would require O(N 4) operations. This paper develops a new numerical algorithm which requires O(N 2.5 log N) operations and as low as O ( √ N) in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e2πiΦ(x,ξ) a(x, ξ) into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.
Algorithms in Fastimp: a fast and wideband impedance extraction program for complicated 3D geometries
 ACM/IEEE Design Automation Conference
, 2003
"... Abstract—In this paper, we describe the algorithms used in FastImp, a program for accurate analysis of wideband electromagnetic effects in very complicated geometries of conductors. The program is based on a recently developed surface integral formulation and a precorrected fast Fourier transform ( ..."
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Cited by 34 (15 self)
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Abstract—In this paper, we describe the algorithms used in FastImp, a program for accurate analysis of wideband electromagnetic effects in very complicated geometries of conductors. The program is based on a recently developed surface integral formulation and a precorrected fast Fourier transform (FFT) accelerated iterative method, but includes a new piecewise quadrature panel integration scheme, a new scaling and preconditioning technique as well as a generalized grid interpolation and projection strategy. Computational results are given on a variety of integrated circuit interconnect structures to demonstrate that FastImp is robust and can accurately analyze very complicated geometries of conductors. Index Terms—Fast integral equation solver, panel integration, parasitic extraction, preconditioner, surface integral formulation, wideband analysis. I.
Fast Methods for Extraction and Sparsification of Substrate Coupling
 In Proc. 37th Design Automation Conference
, 2000
"... The sudden increase in systemsonachip designs has renewed interest in techniques for analyzing and eliminating substrate coupling problems. Previous work on the substrate coupling analysis has focused primarily on faster techniques for extracting coupling resistances, but has offered little help ..."
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Cited by 13 (5 self)
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The sudden increase in systemsonachip designs has renewed interest in techniques for analyzing and eliminating substrate coupling problems. Previous work on the substrate coupling analysis has focused primarily on faster techniques for extracting coupling resistances, but has offered little help for reducing the resulting network whose number of resistors grows quadratically with the number of contacts. In this paper we show that an approach inspired by wavelets can be used in two ways. First, the wavelet method can be used to accurately sparsify the dense contact conductance matrix. In addition, we show that the method can be used to compute the sparse representation directly. Computational results are presented that show that for a problems with a few thousand contacts, the method can be almost ten times faster at constructing the matrix.
Simulation approaches for strongly coupled interconnect systems
 In International Conference on Computer AidedDesign
, 2001
"... Shrinking feature sizes and increasing speeds of operation make interconnectrelated effects very relevant for current circuit verification methodologies. Reliable and accurate system verification requires the full analysis of circuits together with the environment that surrounds them, including the ..."
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Cited by 12 (1 self)
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Shrinking feature sizes and increasing speeds of operation make interconnectrelated effects very relevant for current circuit verification methodologies. Reliable and accurate system verification requires the full analysis of circuits together with the environment that surrounds them, including the common substrate, the packaging structures, and perhaps even board information. In this paper we discuss circuitlevel simulation algorithms that enable the analysis of the impact of strongly coupled interconnect structures on nonlinear circuit operation, so as to allow reliable and accurate system verification. 1
FFTSVD: A fast multiscale boundaryelement method solver suitable for biomems and biomolecule simulation
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 2006
"... Abstract—This paper presents a fast boundaryelement method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and biomicroelectromechanical systems (bioMEMS) design. The algorithm, FFTSVD, is Green’sfunctionindependent for lowfrequency kernels an ..."
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Cited by 12 (4 self)
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Abstract—This paper presents a fast boundaryelement method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and biomicroelectromechanical systems (bioMEMS) design. The algorithm, FFTSVD, is Green’sfunctionindependent for lowfrequency kernels and efficient for inhomogeneous problems. FFTSVD is a multiscale algorithm that decomposes the problem domain using an octree and uses sampling to calculate lowrank approximations to dominant source distributions and responses. Longrange interactions at each length scale are computed using the FFT. Computational results illustrate that the FFTSVD algorithm performs better than precorrectedFFT (pFFT)style algorithms or the multipolestyle algorithms in FastCap. Index Terms—BioMEMS, biomolecule, boundary element, electrostatic, fast solver, FFTSVD. I.
Parasitic extraction: Current state of the art and future trends
 PROC. OF THE IEEE
, 2001
"... With the increase in circuit performance (higher speeds) and density (smaller feature size) in deep submicrometer (DSM) designs, interconnect parasitic effects are increasingly becoming more important. This paper first surveys the state of the art in parasitic extraction for resistance, capacitance, ..."
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Cited by 10 (1 self)
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With the increase in circuit performance (higher speeds) and density (smaller feature size) in deep submicrometer (DSM) designs, interconnect parasitic effects are increasingly becoming more important. This paper first surveys the state of the art in parasitic extraction for resistance, capacitance, and inductance. The paper then covers other related issues such as interconnect modeling, model order reduction, delay calculation, and signal integrity issues such as crosstalk. Some future trends on parasitic extraction, model reduction and interconnect modeling are discussed and a fairly complete list of references is given.
Analysis of eddycurrent losses over conductive substrates with applications to monolithic inductors and transformers
 IEEE Transactions on Microwave Theory and Techniques
, 2001
"... Abstract—In this paper, a closedform integral representation for the eddycurrent losses over a conductive substrate is presented. The results are applicable to monolithic inductors and transformers, especially when such structures are realized over an epitaxial CMOS substrate. The technique is ver ..."
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Cited by 9 (0 self)
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Abstract—In this paper, a closedform integral representation for the eddycurrent losses over a conductive substrate is presented. The results are applicable to monolithic inductors and transformers, especially when such structures are realized over an epitaxial CMOS substrate. The technique is verified against measured results from 100 MHz to 14 GHz for spiral inductors. Index Terms—CMOS substrate losses, eddy currents, monolithic inductors, monolithic transformers, spiral inductors, spiral transformers. I.
Efficient Statistical Capacitance Variability Modeling with Orthogonal Principle Factor Analysis
"... Abstract — Due to the everincreasing complexity of VLSI designs and IC process technologies, the mismatch between a circuit fabricated on the wafer and the one designed in the layout tool grows ever larger. Therefore, characterizing and modeling process variations of interconnect geometry has becom ..."
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Cited by 8 (1 self)
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Abstract — Due to the everincreasing complexity of VLSI designs and IC process technologies, the mismatch between a circuit fabricated on the wafer and the one designed in the layout tool grows ever larger. Therefore, characterizing and modeling process variations of interconnect geometry has become an integral part of analysis and optimization of modern VLSI designs. In this paper, we present a systematic methodology to develop a closed form capacitance model, which accurately captures the nonlinear relationship between parasitic capacitances and dominant global/local process variation parameters. The explicit capacitance representation applies the orthogonal principle factor analysis to greatly reduce the number of random variables associated with modeling conductor surface fluctuations while preserving the dominant sources of variations, and consequently the variational capacitance model can be efficiently utilized by statistical model order reduction and timing analysis tools. Experimental results demonstrate that the proposed method exhibits over 100 × speedup compared with Monte Carlo simulation while having the advantage of generating explicit variational parasitic capacitance models of high order accuracy.
A divideandconquer algorithm for 3D capacitance extraction
 IEEE Trans. CAD
, 2004
"... We present a new algorithm to improve the 3D boundary element method (BEM) for capacitance extraction. We partition large interconnect structures into small sections, set new boundary conditions using the border for each section, solve each section, and then combine the results to derive the capaci ..."
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Cited by 5 (0 self)
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We present a new algorithm to improve the 3D boundary element method (BEM) for capacitance extraction. We partition large interconnect structures into small sections, set new boundary conditions using the border for each section, solve each section, and then combine the results to derive the capacitance. The target applications are critical nets, clock trees, or packages where 3D accuracy is required. Our algorithm is a significant improvement over the traditional BEMs and their enhancements, such as the “window ” method where conductors far away are dropped, and the “shield ” method where conductors hidden behind other conductors are dropped. Experimental results show that our algorithm is a magnitude faster than the traditional BEM and the window+shield method, for medium to large structures. The error of the capacitance computed by the new algorithm is within 2 % for self capacitance and 7 % for coupling capacitance, compared with the results obtained by solving the entire system using BEM. Furthermore, our algorithms gives accurate distributed RC, where none of the previous 3D BEM algorithms and their enhancements can. 1.