Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation
|
|
- Alexina Sanders
- 5 years ago
- Views:
Transcription
1 Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation Tuck Chan EE Department UCLA, CA Fang Gong EE Department UCLA, CA ABSTRACT The increasing process variation and spatial correlation between process parameters makes capacitance extraction of interconnects an important yet challenging problem in modern VLSI designs. In this presentation, we will first introduce basic capacitance extraction flow follow by advance model considering process variation (E.g., orthogonal polynomial method). After that, approaches that solve the resulted augmented system will be presented, such as spectral methods based on polynomial chaos, including Galerkin and collocation methods. Experiment results will also be presented to compare the performance between these methods as well as Monte Carlo method. Finally, some potential research topics within this area will be discussed. Categories and Subject Descriptors B.7.2 [Integrated Circuits]: Design Aids simulation, verification General Terms Design, Algorithms Keywords Process variations, capacitance, parasitic extraction, random variable reduction, principle factor analysis, orthogonal polynomial 1. INTRODUCTION Along with the scaling of CMOS technology, process-induced variability is increasing and has huge impacts on the circuit performance in sub-100nm VLSI technologies. Such process variation has to be considered throughout VLSI design flows to ensure robust circuit design as well as good yield rate. In order to consider the impacts of process variations on interconnects, an efficient stochastic RLC extraction method is required. In this report, we surveyed several approaches on statistical extraction of capacitance considering process variations proposed in [4, 5, 7, 6, 2, 1] under different variational models. Method in [4], uses analytical formulae to consider the variations in capacitance extraction and it has only first-order accuracy. The FastSies program considers the rough surface effects of the interconnect conductors [7]. It assumes only Gaussian distributions and has high computation costs. In [5], a method combining hierarchical extraction and principle factor analysis is proposed. The capacitance extraction is done based on second-order perturbation, which can generate quadratic variational capacitance for better accuracy. Recently, a spectral stochastic collocation based capacitance extraction method was proposed [2]. This approach is based on the Hermite orthogonal polynomial representation of the variational capacitance. It applies the collocation idea, where the capacitance extraction processes (by solving the potential coefficient matrices)are performed many times by sampling so that the coefficients of orthogonal polynomials of variational capacitance can be computed using the weighted least square method. The number of samplings is O(m2), where m is the number of variables. So if m is large, the approach will lose it efficiency compared to the Monte Carlo method. Instead of using collocation method, [1] use a different spectral stochastic method, where the Galerkin scheme is used. Galerkin-based Spectral stochastic method has been applied for statisitical interconenct modeling and on-chip power grid analysis consider process variations. The new method, called statcap, first transforms the original stochastic potential coefficient equations into a deterministic and larger one and then solves it using iterative method. It avoids the sampling process in the existing collocation-based extraction approach. As a result, the potential coefficient equations and the corresponding augmented system only need to be set up once versus many times in the collocation based sampling method. This can lead to a significant saving in CPU time. Also the augmented potential coefficient system is sparse, symmetric and low-rank, which is further exploited by an iterative solver to gain further speedup. The rest of this report is organized as follows: Section 2 reviews the capacitance extraction method in [3]. Section 3 and 4 present the details of stochastic capacitance extraction method proposed in [5] and [1], respectively. Section 5 shows the summary of experimental results and Section 6 concludes this report.
2 2. PRELIMINARY The capacitances among m conductors can be summarized by an m m capacitance matrix C, Cṽ = q, (1) where q,ṽ R m 1 are conductor charge distribution and potential vectors, respectively. The diagonal entries C ii of C are positive, representing the self-capacitance of conductor i. The non-diagonal entries C ij are negative, representing the coupling capacitance between conductors i and j. The j th column of C can be calculated by solving for the total charges on each of the conductors when the j th conductor is at unit potential and all the other conductors are at zero potential. Then the charge on conductor i, q i, is equal to C ij. This procedure is repeated m times to compute all columns of C. 2.1 BEM Capacitance Extraction Boundary element methods (BEM), also referred to as panel methods or the method of moments, have been adopted as the main approach for 3D capacitance calculation. Due to the fact that the charge is restricted to the surface of the conductors, the surfaces of m conductors with non-uniform charge distribution need to be discretized into a total of n twodimensional panels and the charge distribution on each panel is assumed to be even. Then for each panel k, an equation is written that relates the potential at the center of the k th panel to the sum of contributions to that potential from the charge distribution on all n panels and the contribution from the l th panel is determined by the potential coefficient, P kl = 1 a l Z panel 1 1 x k x da, (2) x k x l where x l and x k are the centers of the lth and kth panels. Then a system of equations can be constructed to solve for the discretized conductor surface charges Pq = v, (3) where P R n n is the potential coefficient matrix and q, v R nx1 are panel charge distribution and potential vectors. To compute the j th column of the capacitance matrix, Eq. 3 must be solved for q, given a v vector where v k = 1 if panel k is on the j th conductor, and v k = 0. Then C ij of the capacitance matrix is computed by summing all the panel charges on the ith conductor, X C ij = q k, (4) k conductor i 2.2 Hierarchical Capacitance Algorithms The main obstacle of solving q is that the coefficient matrix in Eq. 3 is very dense and direct linear system solvers, such as Gaussian elimination or Cholesky decomposition, become computationally intractable if the number of panels exceeds several hundred. Therefore, multipole accelerated and hierarchical algorithms have been proposed to address this problem. Conductor surfaces can be hierarchically, instead of uniformly, divided into smaller panels. The hierarchical panel refinement can be fully described by a multipletree, in which the root panel of each tree corresponds to a conductor surface. If the estimated potential coefficient between two panels is larger than a threshold value P, they are further divided into smaller panels. Otherwise, a link recording the potential coefficient is created between these two panels. All recorded potential coefficients compose a link matrix H R NxN, where N is the number of all panels. For any two panels i and j with no links in between, the corresponding entry in H is zero, otherwise, the recorded potential coefficient evaluated by using Eq. 2 is filled into H ij. For each panel in the hierarchical data structure, its charge is equal to the summation of charges on its two child panels. Therefore, one can choose a specific set of panels, called basis panels, such that all panel charges can be uniquely represented as linear combinations of charges on those panels. The coefficient matrix of those linear combinations is called the structure matrix J R Nxn, where n is the number of leaf panels. For a particular multiple-tree structure, there are many possible bases and each of them has its own structure matrix J and potential coefficient matrix P, which can be expressed in terms of H and J as charges on the ith conductor, P = J T HJ. (5) Then in the new linear system, the q vector in Eq. 3 will represent charges on those basis panels instead of leaf panels. It has been discovered in [3] that the potential coefficient matrix in Eq. 5 is dense when leaf panels are chosen as the basis. On the contrary, if all root panels and left hand side panels are chosen as the basis, it is provable that the P matrix related to this basis is sparse and contains O(n) non-zeros. Therefore, equations in Eq. 3 can be efficiently solved by preconditioned Krylov subspace solvers in linear time. 3. STOCHASTIC CAPACITANCE EXTRAC- TION USING PFA In this section, the details of solving statistical system equation, modeling of surface fluctuation due to process variation and PFA method are described. 3.1 Variational Capacitance Approximation Assume for now that process variations induce some perturbations in the nominal potential coefficient P kl between panel k and panel l in Eq. 2, and the variational potential coefficient P can be represented in terms of the nominal value P and normal random variables δ = [δ 1 δ 2... δ k ] T as P = P + X J T H i Jδ i + X J T H ij Jδ iδ j, i i,j {z } P where P i = J T H i J and P ij = J T H ij J. P is the potential coefficient matrix without considering the process variations, and P, which is the summation of the second and third terms in Eq. 6, represents the variational part of (6)
3 In [5], the spatial variation between these panels are modeled by Gaussian correlation function: Γ ij = e x i x j 2 /η2, (11) Figure 1: Conductor with position perturbations on leaf panels where e is Euler constant and η is user specified correlation length. x i and x j are the centers of leaf panels i and j, respectively. Then the correlation matrix can be written as P. Let q denote the variational charge distribution vector, q is expressed in a quadratic form, such that Γ( ñ) = (Γ ij) n n (12) q = q + X q i δ i + X q ij δ iδ j, i i,j {z } q where q, q i, q ijr n 1. From Eq. 7, it is clear that the quadratic expressions of self and coupling capacitances can be easily obtained by using Eq. 4. From Eq. 6 and Eq. 7, the variational linear system can then represented as (7) (P + P)(q + q) = v. (8) Substituting the normal equation in Eq. 3 into Eq. 8 and applying T aylor expansion (up to second order terms), q can be expressed as q = Aq + A 2 q, A = P 1 P. Let the quadratic form representation of the first term on the right hand side of Eq. 9, q 1, to be q 1 = X i q i 1δ i + X ij q ij 1 δ iδ j. By using Eq. 6 and P q 1 = Pq, we can get P q i 1 = P i q, P q ij 1 = P ij q (9) (10) Therefore, the quadratic expression of q 1 can be calculated by solving (k + k 2 ) linear systems. Since P is sparse, each linear system in Eq. 10 can be efficiently solved by preconditioned iterative methods with O(n) complexity. So the total complexity of solving q 1 is O((k 2 + k)n). 3.2 Process Variation Modeling After the hierarchical panel discretization process, the positions of those most delicate panels, leaf panels, may be varying due to process variations. The surface fluctuation of a conductor can be described as a statistical perturbation on each nominal leaf panel smooth surface along its normal direction as shown in Fig. 1. The leaf panel position variation is modeled as a random variable vector ñ, where the random perturbation on the i th leaf is denote as ñ i and the variance-covariance matrix Σ of ñ can be written as Σ( ñ) = (Γ ijσ iσ j) n n (13) where σ is the variance for a particular leaf panel. 3.3 Random Variable Reduction Since only a portion of the random variables have significant effects on the conductor surface fluction, principle factor analysis is used to reduce number of random variables in [5]. The orthogonal principle factor analysis (OPFA), also referred to as principle component model, assumes that µ(d) = 0, Σ(d) = I. After applying PFA on to the covariance matrix, Σ, it can be represents as Σ( ñ) = L δ. (14) where L is a loading matrix L R n k and is given by L = [λ 1e 1λ 2e 2...λ k e k ], (15) where λ and e are the eigen pair of Σ( ñ). It should be noted that the number of random variable has been reduced from n to k using this method. In [5], it is shown that about 90% of the total random variables can be compressed/discarded with minor error in accuracy. 4. ORTHOGONAL POLYNOMIAL METHOD In this section, we will briefly review orthogonal polynomial based stochastic analysis methods that introduced in [1]. 4.1 Hermite Polynomial Basis First, we can consider a random variable ξ(θ) which is expressed as a function of θ that is the random event. Hermite Polynomial utilizes a series of orthogonal polynomials (with respect to the Gaussian distribution) to facilitate stochastic analysis. So, these polynomials are used as orthogonal basis to decompose a random process. With the Hermite Polynomials, if given a random variable v(t, ξ) with variation, where ξ = [ξ 1,..., ξ n] denotes a vector of orthonormal Gaussian random variables with zero mean,
4 the random variable can be approximated by a truncated Hermite PC expansion as follows: v(t,ξ) = PX a k Hk n (ξ) (16) k=0 where n is the number of independent random variables, H n k (ξ) are n-dimensional Hermite polynomials, and a k are the deterministic coefficients. The Hermite polynomials are orthogonal with respect to Gaussian weighted expectation, which can be expressed with following: < H i(ξ),h j(ξ) >=< H 2 i (ξ) > δ ij (17) where δ ij is the Kronecker delta and <, > denotes an inner product. Thus, in order to determine the orthogonal expansion coefficients a k, the orthogonality can be used to evaluate by the projection operation onto the Hermite Polynomial basis: a k (t) = < v(t, ξ), H k(ξ) > H 2 k (ξ) (18) 4.2 Extraction With Hermite Polynomials With Hermite Polynomials, we can present the capacitance extraction with them considering the process variation in this section. or CG. As such, the capacitance extraction under process variation is to fix the orthogonal expansion of q as: q(ξ) = q 0 + KX q ih i(ξ) (23) Once the Hermite Polynomial Expansion of q(ξ) is known, the mean and variance of q(ξ) can be evaluated trivially. Given an example, for one random variable, the mean and variance are calculated as: i=1 E(q(ξ)) = q 0 (24) V ar(q(ξ)) = q q 2 2 (25) 5. EXPERIMENT In this section, we compare the results of both PFA and OP based methods against the Monte Carlo method. The two approaches are tested on 2x2 bus test case with 352 panels, which is shown in Fig.2. Also, the standard deviation is set as 10% of the wire width and the η, the correlation length, as 200of the wire width Process Variation Modeling For each panel, we assume there is a perturbation n i along its normal direction, which can describe the geometric variation on conductor surface. Also, the potential coefficient matrix P can be approximated with its integral kernel, when panels are far away (their distance is much larger than the panel area). We can have the following approximation: P ij G(x i, x j), i j (19) G(x i, x j) = 1 x i x j (20) Therefore, the approximation for P ij can be obtained: nx P ij(ξ) = p k ξ k (21) k=1 As we have mentioned, the coefficients p k will be evaluated with orthogonal property in Equa.(18) OP based Stochastic Capacitance Extraction The nominal system Pq = v can be converted to P(ξ)q(ξ) = v, and potential coefficient matrix: P(ξ) = P 0 + P 1 = P 0 + PX p 1iH i(ξ). (22) In this way, the perturbated system P(ξ)q(ξ) = v will be solved with Iterative Linear System Solver, such as GMRES i=1 Figure 2: 2x2 Bus Test Example. 5.1 PFA based method First, for the 2x2 bus crossing problem, probability density functions (PDF) obtained from the canonical linear model and the quadratic model are shown in Fig.3 and compared with that from Monte Carlo simulation. It is illustrated that there is a significant accuracy improvement by using the second order quadratic model instead of the canonical model. In Table 1, the run times of Monte Carlo method and the quadratic model with 10 dominant factors for 2x2 bus crossing benchmarks are compared. It is clear that the quadratic model exhibits over 100X speedup compared with Monte Carlo simulation. Statistical distribution-related parameters, such as mean value, standard deviation, and skewness are normally within 3% errors. Combined with the results from previous experiments, We can safely conclude that, currently, the second order approximation is accurate enough for variational parasitic capacitance modeling. In the second experiment, the PDFs of the second order quadratic models with different number of dominant factors
5 Figure 3: First and second order capacitance models and their comparisons with Monte Carlo method for the bus 2x2 benchmark (δ = 20%). Table 1: Simulation Comparison between MC and PFA method. Method Time Mean µ std variation δ skewness η MC QualMod Speedup/Err 186.7X 3.7% 2.2% 3.2 are compared in Fig. 4. In this test, PFA with only ten factors is very close to the result PDF from Monte Carlo simulation, so that ninety percent random variable reduction has been achieved by PFA. And in this case, the error compared with Monte Carlo is less than 3%. Furthermore, as the number of factors increases, the PDFs from the quadratic models quickly converge to those from Monte Carlo simulation. 5.2 OP based method First, we compare the CPU times of the two methods. The results are summarized in Table 1, where MC(1000) means that 1000 runs used in MC method.it can be seen that OP method (268 second) is much faster than Monte Carlo method ( second), which can deliver about three orders of magnitude speedup over Monte Carlo method. Next, we perform the accuracy comparison. For 2x2 bus case, we carry out 6000 times runs for Monte Carlo simulation. The results are summarized in Table CONCLUSION In this paper, we survey some papers about stochastic capacitance extraction considering process variation. Especially, we explain the main flows of Principle Factor Analysis and Table 2: Capacitance mean values for the 2x2 bus. 2x2 Bus, Panel num=352, δ = 0.1, η = 2 MC OP C C C C C C C Figure 4: Second order parasitic capacitance modeling with different number of factors and the comparison with Monte Carlo method for bus 2x2 benchmark. orthogonal polynomial methods, which involve variational geometric modeling, perturbated system building and solution. Other related methods base on these approaches to improve the accuracy or speedup. Experiments show that both methods are many orders of magnitude faster than the Monte Carlo method. 7. REFERENCES [1] J. Cui, G. Chen, R. Shen, S. Tan, W. Yu, and J. Tong. Variational capacitance modeling using orthogonal polynomial method. In in Proc. 18th ACM Great lake s symposium on VLSI., pages ACM, [2] H.Zhu, X.Zeng, W.Cai, J.Xue, and D.Zhou. A sparse grid based spectral stochastic collocation method for variations-aware capacitance extraction of interconnects under nanometer process technology. In in Proc. European Design and Test Conf. (DATE), pages IEEE, [3] R. Jiang, Y.-H. Chang, and C.-P. Chen. Iccap: A linear time sparse transformation and reordering algorithm for 3d ben capacitance extraction. In in Proc. Design Automation Conf. (DAC), pages IEEE, [4] A. Labun. Rapid method to account for process variation in full-chip capacitance extraction. IEEE Trans. on Computer-Aided Design of Inregrated Circuits and Systems, 23: , June [5] R.jiang, W.Fu, J. Wang, V.lin, and C.-P. Chen. Efficient statistical capacitance variability modeling with orthogonal principle factor analysis. In in Proc. Int. Conf. on Computer Aided Design (ICCAD), pages IEEE, [6] Y.Zhou, Z.Li, Y.Tian, W.Shi, and F.Liu. A new methodology for interconnect parasitics extraction considering photo-lithography effects. In in Proc. Asia South pacific design Automation Conf. (ASPDAC), pages IEEE, January [7] Z.Zhu and J.White. Fastsies: a fast stochastic integral equation solver for modeling the rough surface effect. In in Proc. Int. Conf. on Computer Aided Design (ICCAD), pages IEEE, 2005.
Variational Capacitance Modeling Using Orthogonal Polynomial Method
Variational Capacitance Modeling Using Orthogonal Polynomial Method Jian Cui, Gengsheng Chen, Ruijing Shen, Sheldon X.-D. Tan, Wenjian Yu and Jiarong Tong Department of Electrical Engineering, University
More information1556 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 18, NO. 11, NOVEMBER 2010
1556 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL 18, NO 11, NOVEMBER 2010 Variational Capacitance Extraction and Modeling Based on Orthogonal Polynomial Method Ruijing Shen, Student
More informationPiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation
PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation Fang Gong UCLA EE Department Los Angeles, CA 90095 gongfang@ucla.edu Hao Yu Berkeley Design Automation
More informationAS IC designs are approaching processes below 45 nm,
IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL 20, NO 9, SEPTEMBER 2012 1729 A Parallel and Incremental Extraction of Variational Capacitance With Stochastic Geometric Moments Fang
More informationVariation-Aware Interconnect Extraction using Statistical Moment Preserving Model Order Reduction
Variation-Aware Interconnect Extraction using Statistical Moment Preserving Model Order Reduction The MIT Faculty has made this article openly available. Please share how this access benefits you. Your
More information1420 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2005
14 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 4, NO. 9, SEPTEMBER 5 Sparse Transformations and Preconditioners for 3-D Capacitance Extraction Shu Yan, Student Member,
More informationChapter 2 Basic Field-Solver Techniques for RC Extraction
Chapter 2 Basic Field-Solver Techniques for RC Extraction Because 3-D numerical methods accurately model the realistic geometry, they possess the highest precision. The field solver based on 3-D numerical
More informationEstimating functional uncertainty using polynomial chaos and adjoint equations
0. Estimating functional uncertainty using polynomial chaos and adjoint equations February 24, 2011 1 Florida State University, Tallahassee, Florida, Usa 2 Moscow Institute of Physics and Technology, Moscow,
More informationAccounting for Variability and Uncertainty in Signal and Power Integrity Modeling
Accounting for Variability and Uncertainty in Signal and Power Integrity Modeling Andreas Cangellaris& Prasad Sumant Electromagnetics Lab, ECE Department University of Illinois Acknowledgments Dr. Hong
More informationProcess-variation-aware electromagnetic-semiconductor coupled simulation
Title Process-variation-aware electromagnetic-semiconductor coupled simulation Author(s) Xu, Y; Chen, Q; Jiang, L; Wong, N Citation The 2011 IEEE International Symposium on Circuits and Systems (ISCAS),
More informationUncertainty analysis of large-scale systems using domain decomposition
Center for Turbulence Research Annual Research Briefs 2007 143 Uncertainty analysis of large-scale systems using domain decomposition By D. Ghosh, C. Farhat AND P. Avery 1. Motivation and objectives A
More informationA Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models
A Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models Sarvesh Bhardwaj, Praveen Ghanta, Sarma Vrudhula Department of Electrical Engineering, Department of Computer Science
More informationEfficient Solvers for Stochastic Finite Element Saddle Point Problems
Efficient Solvers for Stochastic Finite Element Saddle Point Problems Catherine E. Powell c.powell@manchester.ac.uk School of Mathematics University of Manchester, UK Efficient Solvers for Stochastic Finite
More informationStatistical Performance Analysis and Modeling Techniques for Nanometer VLSI Designs
Statistical Performance Analysis and Modeling Techniques for Nanometer VLSI Designs Ruijing Shen Sheldon X.-D. Tan Hao Yu Statistical Performance Analysis and Modeling Techniques for Nanometer VLSI Designs
More informationAn adaptive fast multipole boundary element method for the Helmholtz equation
An adaptive fast multipole boundary element method for the Helmholtz equation Vincenzo Mallardo 1, Claudio Alessandri 1, Ferri M.H. Aliabadi 2 1 Department of Architecture, University of Ferrara, Italy
More informationA Solenoidal Basis Method For Efficient Inductance Extraction Λ
A Solenoidal Basis Method For Efficient Inductance Extraction Λ Hemant Mahawar Department of Computer Science Texas A&M University College Station, TX 77843 mahawarh@cs.tamu.edu Vivek Sarin Department
More informationA reduced-order stochastic finite element analysis for structures with uncertainties
A reduced-order stochastic finite element analysis for structures with uncertainties Ji Yang 1, Béatrice Faverjon 1,2, Herwig Peters 1, icole Kessissoglou 1 1 School of Mechanical and Manufacturing Engineering,
More informationFast Statistical Analysis of RC Nets Subject to Manufacturing Variabilities
Fast Statistical Analysis of RC Nets Subject to Manufacturing Variabilities Yu Bi, Kees-Jan van der Kolk, Jorge Fernández Villena,Luís Miguel Silveira, Nick van der Meijs CAS, TU Delft. Delft, The Netherlands.
More informationUNCERTAINTY ASSESSMENT USING STOCHASTIC REDUCED BASIS METHOD FOR FLOW IN POROUS MEDIA
UNCERTAINTY ASSESSMENT USING STOCHASTIC REDUCED BASIS METHOD FOR FLOW IN POROUS MEDIA A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT
More informationPolynomial chaos expansions for structural reliability analysis
DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for structural reliability analysis B. Sudret & S. Marelli Incl.
More informationReluctance/Inductance Matrix under
Generating Stable and Sparse Reluctance/Inductance Matrix under Insufficient Conditions Y. Tanji, Kagawa University, Japan T. Watanabe, The University it of Shizuoka, Japan H. Asai, Shizuoka University,
More informationPARALLEL COMPUTATION OF 3D WAVE PROPAGATION BY SPECTRAL STOCHASTIC FINITE ELEMENT METHOD
13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No. 569 PARALLEL COMPUTATION OF 3D WAVE PROPAGATION BY SPECTRAL STOCHASTIC FINITE ELEMENT METHOD Riki Honda
More informationVector Potential Equivalent Circuit Based on PEEC Inversion
43.2 Vector Potential Equivalent Circuit Based on PEEC Inversion Hao Yu EE Department, UCLA Los Angeles, CA 90095 Lei He EE Department, UCLA Los Angeles, CA 90095 ABSTRACT The geometry-integration based
More informationHierarchical Parallel Solution of Stochastic Systems
Hierarchical Parallel Solution of Stochastic Systems Second M.I.T. Conference on Computational Fluid and Solid Mechanics Contents: Simple Model of Stochastic Flow Stochastic Galerkin Scheme Resulting Equations
More informationFast Buffer Insertion Considering Process Variation
Fast Buffer Insertion Considering Process Variation Jinjun Xiong, Lei He EE Department University of California, Los Angeles Sponsors: NSF, UC MICRO, Actel, Mindspeed Agenda Introduction and motivation
More informationVariation-aware Modeling of Integrated Capacitors based on Floating Random Walk Extraction
Variation-aware Modeling of Integrated Capacitors based on Floating Random Walk Extraction Paolo Maffezzoni, Senior Member, IEEE, Zheng Zhang, Member, IEEE, Salvatore Levantino, Member, IEEE, and Luca
More informationNumerical Characterization of Multi-Dielectric Green s Function for 3-D Capacitance Extraction with Floating Random Walk Algorithm
Numerical Characterization of Multi-Dielectric Green s Function for 3-D Capacitance Extraction with Floating Random Walk Algorithm Hao Zhuang 1, 2, Wenjian Yu 1 *, Gang Hu 1, Zuochang Ye 3 1 Department
More informationTHE boundary-element, or method-of-moments [1], technique
18 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL 47, NO 1, JANUARY 1999 Capacitance Extraction of 3-D Conductor Systems in Dielectric Media with High-Permittivity Ratios Johannes Tausch and
More informationStochastic Collocation with Non-Gaussian Correlated Process Variations: Theory, Algorithms and Applications
Stochastic Collocation with Non-Gaussian Correlated Process Variations: Theory, Algorithms and Applications Chunfeng Cui, and Zheng Zhang, Member, IEEE Abstract Stochastic spectral methods have achieved
More informationApplications of Randomized Methods for Decomposing and Simulating from Large Covariance Matrices
Applications of Randomized Methods for Decomposing and Simulating from Large Covariance Matrices Vahid Dehdari and Clayton V. Deutsch Geostatistical modeling involves many variables and many locations.
More informationKarhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques
Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,
More informationSolving the steady state diffusion equation with uncertainty Final Presentation
Solving the steady state diffusion equation with uncertainty Final Presentation Virginia Forstall vhfors@gmail.com Advisor: Howard Elman elman@cs.umd.edu Department of Computer Science May 6, 2012 Problem
More informationJ.TAUSCH AND J.WHITE 1. Capacitance Extraction of 3-D Conductor Systems in. Dielectric Media with high Permittivity Ratios
JTAUSCH AND JWHITE Capacitance Extraction of -D Conductor Systems in Dielectric Media with high Permittivity Ratios Johannes Tausch and Jacob White Abstract The recent development of fast algorithms for
More informationCollocation based high dimensional model representation for stochastic partial differential equations
Collocation based high dimensional model representation for stochastic partial differential equations S Adhikari 1 1 Swansea University, UK ECCM 2010: IV European Conference on Computational Mechanics,
More informationAdaptive Collocation with Kernel Density Estimation
Examples of with Kernel Density Estimation Howard C. Elman Department of Computer Science University of Maryland at College Park Christopher W. Miller Applied Mathematics and Scientific Computing Program
More informationA Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations
An Extended Abstract submitted for the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada January 26 Preferred Session Topic: Uncertainty quantification and stochastic methods for CFD A Non-Intrusive
More informationUncertainty Quantification in MEMS
Uncertainty Quantification in MEMS N. Agarwal and N. R. Aluru Department of Mechanical Science and Engineering for Advanced Science and Technology Introduction Capacitive RF MEMS switch Comb drive Various
More informationPolynomial Chaos and Karhunen-Loeve Expansion
Polynomial Chaos and Karhunen-Loeve Expansion 1) Random Variables Consider a system that is modeled by R = M(x, t, X) where X is a random variable. We are interested in determining the probability of the
More informationA hybrid reordered Arnoldi method to accelerate PageRank computations
A hybrid reordered Arnoldi method to accelerate PageRank computations Danielle Parker Final Presentation Background Modeling the Web The Web The Graph (A) Ranks of Web pages v = v 1... Dominant Eigenvector
More informationSolving Corrupted Quadratic Equations, Provably
Solving Corrupted Quadratic Equations, Provably Yuejie Chi London Workshop on Sparse Signal Processing September 206 Acknowledgement Joint work with Yuanxin Li (OSU), Huishuai Zhuang (Syracuse) and Yingbin
More informationHyperbolic Polynomial Chaos Expansion (HPCE) and its Application to Statistical Analysis of Nonlinear Circuits
Hyperbolic Polynomial Chaos Expansion HPCE and its Application to Statistical Analysis of Nonlinear Circuits Majid Ahadi, Aditi Krishna Prasad, Sourajeet Roy High Speed System Simulations Laboratory Department
More informationSecond-Order Balanced Truncation for Passive Order Reduction of RLCK Circuits
IEEE RANSACIONS ON CIRCUIS AND SYSEMS II, VOL XX, NO. XX, MONH X Second-Order Balanced runcation for Passive Order Reduction of RLCK Circuits Boyuan Yan, Student Member, IEEE, Sheldon X.-D. an, Senior
More informationZhigang Hao, Sheldon X.-D. Tan, E. Tlelo-Cuautle, Jacob Relles, Chao Hu, Wenjian Yu, Yici Cai & Guoyong Shi
Statistical extraction and modeling of inductance considering spatial correlation Zhigang Hao, Sheldon X.-D. Tan, E. Tlelo-Cuautle, Jacob Relles, Chao Hu, Wenjian Yu, Yici Cai & Guoyong Shi Analog Integrated
More informationFast Numerical Methods for Stochastic Computations
Fast AreviewbyDongbinXiu May 16 th,2013 Outline Motivation 1 Motivation 2 3 4 5 Example: Burgers Equation Let us consider the Burger s equation: u t + uu x = νu xx, x [ 1, 1] u( 1) =1 u(1) = 1 Example:
More informationStochastic structural dynamic analysis with random damping parameters
Stochastic structural dynamic analysis with random damping parameters K. Sepahvand 1, F. Saati Khosroshahi, C. A. Geweth and S. Marburg Chair of Vibroacoustics of Vehicles and Machines Department of Mechanical
More informationPerformance Evaluation of Generalized Polynomial Chaos
Performance Evaluation of Generalized Polynomial Chaos Dongbin Xiu, Didier Lucor, C.-H. Su, and George Em Karniadakis 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA, gk@dam.brown.edu
More informationImproving the Robustness of a Surface Integral Formulation for Wideband Impendance Extraction of 3D Structures
Improving the Robustness of a Surface Integral Formulation for Wideband Impendance Extraction of 3D Structures Zhenhai Zhu, Jingfang Huang, Ben Song, Jacob White Department of Electrical Engineering and
More information3-D Inductance and Resistance Extraction for Interconnects
3-D Inductance and Resistance Extraction for Interconnects Shuzhou Fang, Liu Yang and Zeyi Wang Dept. of Computer Science & Technology Tsinghua University, Beijing 100084, China Aug. 20, 2002 Content PEEC
More informationA Capacitance Solver for Incremental Variation-Aware Extraction
A Capacitance Solver for Incremental Variation-Aware Extraction Tarek A. El-Moselhy Research Lab in Electronics Massachusetts Institute of Technology tmoselhy@mit.edu Ibrahim M. Elfadel Systems & Technology
More informationPolynomial chaos expansions for sensitivity analysis
c DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for sensitivity analysis B. Sudret Chair of Risk, Safety & Uncertainty
More informationA Vector-Space Approach for Stochastic Finite Element Analysis
A Vector-Space Approach for Stochastic Finite Element Analysis S Adhikari 1 1 Swansea University, UK CST2010: Valencia, Spain Adhikari (Swansea) Vector-Space Approach for SFEM 14-17 September, 2010 1 /
More informationSolving the Stochastic Steady-State Diffusion Problem Using Multigrid
Solving the Stochastic Steady-State Diffusion Problem Using Multigrid Tengfei Su Applied Mathematics and Scientific Computing Advisor: Howard Elman Department of Computer Science Sept. 29, 2015 Tengfei
More informationA Novel Approach for Solving the Power Flow Equations
Vol.1, Issue.2, pp-364-370 ISSN: 2249-6645 A Novel Approach for Solving the Power Flow Equations Anwesh Chowdary 1, Dr. G. MadhusudhanaRao 2 1 Dept.of.EEE, KL University-Guntur, Andhra Pradesh, INDIA 2
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM Repetition of Previous Lecture Sparse grids in Uncertainty Quantification
More informationStabilization and Acceleration of Algebraic Multigrid Method
Stabilization and Acceleration of Algebraic Multigrid Method Recursive Projection Algorithm A. Jemcov J.P. Maruszewski Fluent Inc. October 24, 2006 Outline 1 Need for Algorithm Stabilization and Acceleration
More informationA Polynomial Chaos Approach to Robust Multiobjective Optimization
A Polynomial Chaos Approach to Robust Multiobjective Optimization Silvia Poles 1, Alberto Lovison 2 1 EnginSoft S.p.A., Optimization Consulting Via Giambellino, 7 35129 Padova, Italy s.poles@enginsoft.it
More informationStochastic dominant singular vectors method for variation-aware extraction
Stochastic dominant singular vectors method for variation-aware extraction The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation
More informationDinesh Kumar, Mehrdad Raisee and Chris Lacor
Dinesh Kumar, Mehrdad Raisee and Chris Lacor Fluid Mechanics and Thermodynamics Research Group Vrije Universiteit Brussel, BELGIUM dkumar@vub.ac.be; m_raisee@yahoo.com; chris.lacor@vub.ac.be October, 2014
More informationFast Simulation of VLSI Interconnects
Fast Simulation of VLSI Interconnects Jitesh Jain, Cheng-Kok Koh, and Venkataramanan Balakrishnan School of Electrical and Computer Engineering Purdue University, West Lafayette, IN 4797-1285 {jjain,chengkok,ragu}@ecn.purdue.edu
More informationStatistical Modeling and Analysis of Chip-Level Leakage Power by Spectral Stochastic Method
Statistical Modeling and Analysis of Chip-Level Leakage Power by Spectral Stochastic Method Ruijing Shen,NingMi, Sheldon X.-D. Tan,YiciCai and Xianlong Hong Department of Electrical Engineering, University
More informationRegularized Discriminant Analysis and Reduced-Rank LDA
Regularized Discriminant Analysis and Reduced-Rank LDA Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Regularized Discriminant Analysis A compromise between LDA and
More informationEfficient Incremental Analysis of On-Chip Power Grid via Sparse Approximation
Efficient Incremental Analysis of On-Chip Power Grid via Sparse Approximation Pei Sun and Xin Li ECE Department, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 1513 {peis, xinli}@ece.cmu.edu
More informationSimulating with uncertainty : the rough surface scattering problem
Simulating with uncertainty : the rough surface scattering problem Uday Khankhoje Assistant Professor, Electrical Engineering Indian Institute of Technology Madras Uday Khankhoje (EE, IITM) Simulating
More informationNon-Linear Statistical Static Timing Analysis for Non-Gaussian Variation Sources
Non-Linear Statistical Static Timing Analysis for Non-Gaussian Variation Sources Lerong Cheng 1, Jinjun Xiong 2, and Prof. Lei He 1 1 EE Department, UCLA *2 IBM Research Center Address comments to lhe@ee.ucla.edu
More informationStochastic Integral Equation Solver for Efficient Variation-Aware Interconnect Extraction
Stochastic Integral Equation Solver for Efficient Variation-Aware Interconnect Extraction ABSTRACT Tarek El-Moselhy Computational Prototyping Group Research Laboratory in Electronics Massachusetts Institute
More informationDominant Feature Vectors Based Audio Similarity Measure
Dominant Feature Vectors Based Audio Similarity Measure Jing Gu 1, Lie Lu 2, Rui Cai 3, Hong-Jiang Zhang 2, and Jian Yang 1 1 Dept. of Electronic Engineering, Tsinghua Univ., Beijing, 100084, China 2 Microsoft
More informationReview of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits. and Domenico Spina
electronics Review Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits Arun Kaintura *, Tom Dhaene ID and Domenico Spina IDLab, Department of Information
More informationBuffered Clock Tree Sizing for Skew Minimization under Power and Thermal Budgets
Buffered Clock Tree Sizing for Skew Minimization under Power and Thermal Budgets Krit Athikulwongse, Xin Zhao, and Sung Kyu Lim School of Electrical and Computer Engineering Georgia Institute of Technology
More informationUncertainty Quantification and Validation Using RAVEN. A. Alfonsi, C. Rabiti. Risk-Informed Safety Margin Characterization. https://lwrs.inl.
Risk-Informed Safety Margin Characterization Uncertainty Quantification and Validation Using RAVEN https://lwrs.inl.gov A. Alfonsi, C. Rabiti North Carolina State University, Raleigh 06/28/2017 Assumptions
More informationFast On-Chip Inductance Simulation Using a Precorrected-FFT Method
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 22, NO. 1, JANUARY 2003 49 Fast On-Chip Inductance Simulation Using a Precorrected-FFT Method Haitian Hu, Member, IEEE,
More informationWHEN studying distributed simulations of power systems,
1096 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 21, NO 3, AUGUST 2006 A Jacobian-Free Newton-GMRES(m) Method with Adaptive Preconditioner and Its Application for Power Flow Calculations Ying Chen and Chen
More informationEE731 Lecture Notes: Matrix Computations for Signal Processing
EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten
More informationEfficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration
Efficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration Shan Zeng, Wenjian Yu, Jin Shi, Xianlong Hong Dept. Computer Science & Technology, Tsinghua University, Beijing
More informationHowever, reliability analysis is not limited to calculation of the probability of failure.
Probabilistic Analysis probabilistic analysis methods, including the first and second-order reliability methods, Monte Carlo simulation, Importance sampling, Latin Hypercube sampling, and stochastic expansions
More informationNumerical methods for the discretization of random fields by means of the Karhunen Loève expansion
Numerical methods for the discretization of random fields by means of the Karhunen Loève expansion Wolfgang Betz, Iason Papaioannou, Daniel Straub Engineering Risk Analysis Group, Technische Universität
More informationA Fast N-Body Solver for the Poisson(-Boltzmann) Equation
A Fast N-Body Solver for the Poisson(-Boltzmann) Equation Robert D. Skeel Departments of Computer Science (and Mathematics) Purdue University http://bionum.cs.purdue.edu/2008december.pdf 1 Thesis Poisson(-Boltzmann)
More informationEfficient Variability Analysis of Electromagnetic Systems via Polynomial Chaos and Model Order Reduction
IEEE TCPMT, VOL. XXX, NO. XXX, 213 1 Efficient Variability Analysis of Electromagnetic Systems via Polynomial Chaos and Model Order Reduction Domenico Spina, Francesco Ferranti, Member, IEEE, Giulio Antonini,
More informationImproved pre-characterization method for the random walk based capacitance extraction of multi-dielectric VLSI interconnects
INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS Int. J. Numer. Model. 2016; 29:21 34 Published online 8 January 2015 in Wiley Online Library (wileyonlinelibrary.com)..2042
More informationAn Accelerated Block-Parallel Newton Method via Overlapped Partitioning
An Accelerated Block-Parallel Newton Method via Overlapped Partitioning Yurong Chen Lab. of Parallel Computing, Institute of Software, CAS (http://www.rdcps.ac.cn/~ychen/english.htm) Summary. This paper
More informationAN INDEPENDENT LOOPS SEARCH ALGORITHM FOR SOLVING INDUCTIVE PEEC LARGE PROBLEMS
Progress In Electromagnetics Research M, Vol. 23, 53 63, 2012 AN INDEPENDENT LOOPS SEARCH ALGORITHM FOR SOLVING INDUCTIVE PEEC LARGE PROBLEMS T.-S. Nguyen *, J.-M. Guichon, O. Chadebec, G. Meunier, and
More informationFast Multipole Methods for The Laplace Equation. Outline
Fast Multipole Methods for The Laplace Equation Ramani Duraiswami Nail Gumerov Outline 3D Laplace equation and Coulomb potentials Multipole and local expansions Special functions Legendre polynomials Associated
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More informationUncertainty Quantification in Computational Science
DTU 2010 - Lecture I Uncertainty Quantification in Computational Science Jan S Hesthaven Brown University Jan.Hesthaven@Brown.edu Objective of lectures The main objective of these lectures are To offer
More informationWavelet-Based Passivity Preserving Model Order Reduction for Wideband Interconnect Characterization
Wavelet-Based Passivity Preserving Model Order Reduction for Wideband Interconnect Characterization Mehboob Alam, Arthur Nieuwoudt, and Yehia Massoud Rice Automated Nanoscale Design Group Rice University,
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large
More informationPreface to the Second Edition. Preface to the First Edition
n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................
More informationAn Efficient Graph Sparsification Approach to Scalable Harmonic Balance (HB) Analysis of Strongly Nonlinear RF Circuits
Design Automation Group An Efficient Graph Sparsification Approach to Scalable Harmonic Balance (HB) Analysis of Strongly Nonlinear RF Circuits Authors : Lengfei Han (Speaker) Xueqian Zhao Dr. Zhuo Feng
More informationA THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS
A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS Victor S. Ryaben'kii Semyon V. Tsynkov Chapman &. Hall/CRC Taylor & Francis Group Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor
More informationPrincipal Component Analysis
Machine Learning Michaelmas 2017 James Worrell Principal Component Analysis 1 Introduction 1.1 Goals of PCA Principal components analysis (PCA) is a dimensionality reduction technique that can be used
More informationDynamic System Identification using HDMR-Bayesian Technique
Dynamic System Identification using HDMR-Bayesian Technique *Shereena O A 1) and Dr. B N Rao 2) 1), 2) Department of Civil Engineering, IIT Madras, Chennai 600036, Tamil Nadu, India 1) ce14d020@smail.iitm.ac.in
More informationHIGH-PERFORMANCE circuits consume a considerable
1166 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL 17, NO 11, NOVEMBER 1998 A Matrix Synthesis Approach to Thermal Placement Chris C N Chu D F Wong Abstract In this
More informationTODAY S application-specific integrated circuit (ASIC)
1618 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 25, NO. 9, SEPTEMBER 2006 Dynamic-Range Estimation Bin Wu, Student Member, IEEE, Jianwen Zhu, Member, IEEE, and
More informationFast Brownian Dynamics for Colloidal Suspensions
Fast Brownian Dynamics for Colloidal Suspensions Aleksandar Donev, CIMS and collaborators: Florencio Balboa CIMS) Andrew Fiore and James Swan MIT) Courant Institute, New York University Modeling Complex
More informationFast multipole boundary element method for the analysis of plates with many holes
Arch. Mech., 59, 4 5, pp. 385 401, Warszawa 2007 Fast multipole boundary element method for the analysis of plates with many holes J. PTASZNY, P. FEDELIŃSKI Department of Strength of Materials and Computational
More informationSampling and low-rank tensor approximation of the response surface
Sampling and low-rank tensor approximation of the response surface tifica Alexander Litvinenko 1,2 (joint work with Hermann G. Matthies 3 ) 1 Group of Raul Tempone, SRI UQ, and 2 Group of David Keyes,
More informationAccounting for Non-linear Dependence Using Function Driven Component Analysis
Accounting for Non-linear Dependence Using Function Driven Component Analysis Lerong Cheng Puneet Gupta Lei He Department of Electrical Engineering University of California, Los Angeles Los Angeles, CA
More informationPAPER A Fast Delay Computation for the Hybrid Structured Clock Network
1964 PAPER A Fast Delay Computation for the Hybrid Structured Clock Network Yi ZOU a), Student Member, Yici CAI b), Qiang ZHOU c), Xianlong HONG d), and Sheldon X.-D. TAN e), Nonmembers SUMMARY This paper
More informationEFFICIENT NUMERICAL METHODS FOR CAPACITANCE EXTRACTION BASED ON BOUNDARY ELEMENT METHOD. A Dissertation SHU YAN
EFFICIENT NUMERICAL METHODS FOR CAPACITANCE EXTRACTION BASED ON BOUNDARY ELEMENT METHOD A Dissertation by SHU YAN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment
More informationOverlay Aware Interconnect and Timing Variation Modeling for Double Patterning Technology
Overlay Aware Interconnect and Timing Variation Modeling for Double Patterning Technology Jae-Seok Yang, David Z. Pan Dept. of ECE, The University of Texas at Austin, Austin, Tx 78712 jsyang@cerc.utexas.edu,
More information