A Solenoidal Basis Method For Efficient Inductance Extraction H emant Mahawar Vivek Sarin Weiping Shi Texas A&M University College Station, TX.

Slides:

Advertisements

Similar presentations

05/11/2005 Carnegie Mellon School of Computer Science Aladdin Lamps 05 Combinatorial and algebraic tools for multigrid Yiannis Koutis Computer Science.

Advertisements

Zhen Lu CPACT University of Newcastle MDC Technology Reduced Hessian Sequential Quadratic Programming(SQP)

Modulation Spectrum Factorization for Robust Speech Recognition Wen-Yi Chu 1, Jeih-weih Hung 2 and Berlin Chen 1 Presenter : 張庭豪.

Applied Linear Algebra - in honor of Hans SchneiderMay 25, 2010 A Look-Back Technique of Restart for the GMRES(m) Method Akira IMAKURA † Tomohiro SOGABE.

Solving Linear Systems (Numerical Recipes, Chap 2)

Rayan Alsemmeri Amseena Mansoor. LINEAR SYSTEMS Jacobi method is used to solve linear systems of the form Ax=b, where A is the square and invertible.

Numerical Parallel Algorithms for Large-Scale Nanoelectronics Simulations using NESSIE Eric Polizzi, Ahmed Sameh Department of Computer Sciences, Purdue.

Iterative Methods and QR Factorization Lecture 5 Alessandra Nardi Thanks to Prof. Jacob White, Suvranu De, Deepak Ramaswamy, Michal Rewienski, and Karen.

Modern iterative methods For basic iterative methods, converge linearly Modern iterative methods, converge faster –Krylov subspace method Steepest descent.

" Characterizing the Relationship between ILU-type Preconditioners and the Storage Hierarchy" " Characterizing the Relationship between ILU-type Preconditioners.

Volkan Cevher, Marco F. Duarte, and Richard G. Baraniuk European Signal Processing Conference 2008.

Computational Solutions of Helmholtz Equation Yau Shu Wong Department of Mathematical & Statistical Sciences University of Alberta Edmonton, Alberta, Canada.

Chapter 5 Orthogonality

Weiping Shi Department of Computer Science University of North Texas HiCap: A Fast Hierarchical Algorithm for 3D Capacitance Extraction.

Matrices. Special Matrices Matrix Addition and Subtraction Example.

A Bayesian Formulation For 3d Articulated Upper Body Segmentation And Tracking From Dense Disparity Maps Navin Goel Dr Ara V Nefian Dr George Bebis.

Avoiding Communication in Sparse Iterative Solvers Erin Carson Nick Knight CS294, Fall 2011.

Code and Decoder Design of LDPC Codes for Gbps Systems Jeremy Thorpe Presented to: Microsoft Research

An Algebraic Multigrid Solver for Analytical Placement With Layout Based Clustering Hongyu Chen, Chung-Kuan Cheng, Andrew B. Kahng, Bo Yao, Zhengyong Zhu.

A continuous function, with zero values at the endpoints (blue), and a piecewise linear approximation (red).

CS240A: Conjugate Gradients and the Model Problem.

The Radiosity Method Donald Fong February 10, 2004.

Newton's Method for Functions of Several Variables

Subdivision Analysis via JSR We already know the z-transform formulation of schemes: To check if the scheme generates a continuous limit curve ( the scheme.

A Factored Sparse Approximate Inverse software package (FSAIPACK) for the parallel preconditioning of linear systems Massimiliano Ferronato, Carlo Janna,

Parallel Performance of Hierarchical Multipole Algorithms for Inductance Extraction Ananth Grama, Purdue University Vivek Sarin, Texas A&M University Hemant.

Surface Simplification Using Quadric Error Metrics Michael Garland Paul S. Heckbert.

Algorithms for a large sparse nonlinear eigenvalue problem Yusaku Yamamoto Dept. of Computational Science & Engineering Nagoya University.

Fast Low-Frequency Impedance Extraction using a Volumetric 3D Integral Formulation A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONE EURATOM/ENEA/CREATE.

PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation Fang Gong 1, Hao Yu 2, and Lei He 1 1 Electrical Engineering.

AN ORTHOGONAL PROJECTION

ParCFD Parallel computation of pollutant dispersion in industrial sites Julien Montagnier Marc Buffat David Guibert.

" Characterizing the Relationship between ILU-type Preconditioners and the Storage Hierarchy" " Characterizing the Relationship between ILU-type Preconditioners.

A study of two-dimensional quantum dot helium in a magnetic field Golam Faruk * and Orion Ciftja, Department of Electrical Engineering and Department of.

ECE 576 – Power System Dynamics and Stability Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign.

Scalable Symbolic Model Order Reduction Yiyu Shi*, Lei He* and C. J. Richard Shi + *Electrical Engineering Department, UCLA + Electrical Engineering Department,

The swiss-carpet preconditioner: a simple parallel preconditioner of Dirichlet-Neumann type A. Quarteroni (Lausanne and Milan) M. Sala (Lausanne) A. Valli.

On the Use of Sparse Direct Solver in a Projection Method for Generalized Eigenvalue Problems Using Numerical Integration Takamitsu Watanabe and Yusaku.

1 Complex Images k’k’ k”k” k0k0 -k0-k0 branch cut   k 0 pole C1C1 C0C0 from the Sommerfeld identity, the complex exponentials must be a function.

Molecular Simulation of Reactive Systems. _______________________________ Sagar Pandit, Hasan Aktulga, Ananth Grama Coordinated Systems Lab Purdue University.

Parallel Iterative Solvers with the Selective Blocking Preconditioning for Simulations of Fault-Zone Contact Kengo Nakajima GeoFEM/RIST, Japan. 3rd ACES.

Forward modelling The key to waveform tomography is the calculation of Green’s functions (the point source responses) Wide range of modelling methods available.

Parallel Solution of the Poisson Problem Using MPI

CS240A: Conjugate Gradients and the Model Problem.

On implicit-factorization block preconditioners Sue Dollar 1,2, Nick Gould 3, Wil Schilders 2,4 and Andy Wathen 1 1 Oxford University Computing Laboratory,

Case Study in Computational Science & Engineering - Lecture 5 1 Iterative Solution of Linear Systems Jacobi Method while not converged do { }

October 2008 Integrated Predictive Simulation System for Earthquake and Tsunami Disaster CREST/Japan Science and Technology Agency (JST)

Lecture 21 MA471 Fall 03. Recall Jacobi Smoothing We recall that the relaxed Jacobi scheme: Smooths out the highest frequency modes fastest.

Outline Introduction Research Project Findings / Results

Ch 6 Vector Spaces. Vector Space Axioms X,Y,Z elements of  and α, β elements of  Def of vector addition Def of multiplication of scalar and vector These.

Consider Preconditioning – Basic Principles Basic Idea: is to use Krylov subspace method (CG, GMRES, MINRES …) on a modified system such as The matrix.

Krylov-Subspace Methods - I Lecture 6 Alessandra Nardi Thanks to Prof. Jacob White, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy.

Monte Carlo Linear Algebra Techniques and Their Parallelization Ashok Srinivasan Computer Science Florida State University

Inductance Screening and Inductance Matrix Sparsification 1.

Fast Method of Fundamental Solution

F. Fairag, H Tawfiq and M. Al-Shahrani Department of Math & Stat Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 Preconditioning Technique.

Fast BEM Algorithms for 3D Interconnect Capacitance and Resistance Extraction Wenjian Yu EDA Lab, Dept. Computer Science & Technology, Tsinghua University.

Linear System expensive p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9]; e=[1,9]; n = length(p); % number of nodes m = size(t,2); % number of elements.

Multipole-Based Preconditioners for Sparse Linear Systems. Ananth Grama Purdue University. Supported by the National Science Foundation.

Conjugate gradient iteration One matrix-vector multiplication per iteration Two vector dot products per iteration Four n-vectors of working storage x 0.

Monte Carlo Linear Algebra Techniques and Their Parallelization Ashok Srinivasan Computer Science Florida State University

DAC, July 2006 Model Order Reduction of Linear Networks with Massive Ports via Frequency-Dependent Port Packing Peng Li and Weiping Shi Department of ECE.

The Landscape of Sparse Ax=b Solvers Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More RobustLess Storage More Robust More.

A computational loop k k Integration Newton Iteration

Finite Element Method To be added later 9/18/2018 ELEN 689.

Meros: Software for Block Preconditioning the Navier-Stokes Equations

Supported by the National Science Foundation.

Conjugate Gradient Method

Inductance Screening and Inductance Matrix Sparsification

A computational loop k k Integration Newton Iteration

Presentation transcript:

A Solenoidal Basis Method For Efficient Inductance Extraction H emant Mahawar Vivek Sarin Weiping Shi Texas A&M University College Station, TX

Introduction

Background Inductance between current carrying filaments Kirchoff’s law enforced at each node

Background … Current density at a point Linear system for current and potential Inductance matrix Kirchoff’s Law

Linear System of Equations Characteristics  Extremely large; R, B: sparse; L: dense  Matrix-vector products with L use hierarchical approximations Solution methodology  Solved by preconditioned Krylov subspace methods  Robust and effective preconditioners are critical Developing good preconditioners is a challenge because system is never computed explicitly!

First Key Idea Current Components  Fixed current satisfying external condition I d (left)  Linear combination of cell currents (right)

Solenoidal Basis Method Linear system Solenoidal basis  Basis for current that satisfies Kirchoff’s law  Solenoidal basis matrix P:  Current obeying Kirchoff’s law: Reduced system  Solve via preconditioned Krylov subspace method

Local Solenoidal Basis Cell current k consists of unit current assigned to the four filaments of the kth cell There are four nonzeros in the kth column of P: 1, 1, -1, -1

Second Key Idea Observe: where Approximate reduced system Approximate by

Preconditioning Preconditioning involves multiplication with

Hierarchical Approximations Components of system matrix and preconditioner are dense and large Hierarchical approximations used to compute matrix-vector products with both L and  Used for fast decaying Greens functions, such as 1/r (r : distance from origin)  Reduced accuracy at lower cost Examples  Fast Multipole Method: O(n)  Barnes-Hut: O(nlogn)

FASTHENRY Uses mesh currents to generate a reduced system Approximation to reduced system computed by sparsification of inductance matrix Preconditioner derived from Sparsification strategies  DIAG: self inductance of filaments only  CUBE: filaments in the same oct-tree cube of FMM hierarchy  SHELL: filaments within specified radius (expensive)

Experiments Benchmark problems  Ground plane  Wire over plane  Spiral inductor Operating frequencies: 1GHz-1THz Strategy  Uniform two-dimensional mesh  Solenoidal function method  Preconditioned GMRES for reduced system Comparison  FASTHENRY with CUBE & DIAG preconditioners

Ground Plane

Problem Sizes Mesh Potential Nodes Current Filaments Linear System Solenoidal functions 33x331,0892,1123,2011,024 65x654,2258,32012,5454, x12916,64133,02449,66516, x25766,049131,584197,63365,536

Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method 33x x x x x

Comparison … Time and Memory (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method Time (sec) Mem (MB) Time (sec) Mem (MB) Time (sec) Mem (MB) 33x x x x x

Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) x331/ x651/ x1291/ x2561/

Wire Over Ground Plane

Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method 33x x x x x

Comparison … Time and Memory (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method Time (sec) Mem (MB) Time (sec) Mem (MB) Time (sec) Mem (MB) 33x x x x x

Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) x331/ x651/ x1291/ x2571/

Spiral Inductor

Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) x331/ x651/ x1291/ x2571/

Concluding Remarks Preconditioned solenoidal method is very effective for linear systems in inductance extraction Near-optimal preconditioning assures fast convergence rates that are nearly independent of frequency and mesh width Significant improvement over FASTHENRY w.r.t. time and memory Acknowledgements: National Science Foundation