Introduction Basics Balanced Truncation for Many Ports Outlook and Acknowledgement A new Balanced Truncation Model Reduction Approach for Large Scale LTI Systems with many Ports Peter Benner and Andr´ e Schneider Chemnitz University of Technology Faculty of Mathematics 16th South-East-German Colloquium of Numerical Analysis Dresden April 30th, 2010 System Reduction for Nanoscale IC Design 1/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Basics Balanced Truncation for Many Ports Outlook and Acknowledgement SyreNe – An Overview SyreNe is the abbreviation for the research network System Reduction for Nanoscale IC Design within the program Mathematics for Innovations in Industry and Services funded by the German Federal Ministry of Education and Science (BMBF). Subproject 4 at Chemnitz UT deals with the Reduced Representation of Power Grid Models . 2/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Basics Balanced Truncation for Many Ports Outlook and Acknowledgement SyreNe Employees � � Universit¨ TU Chemnitz �� �� at Hamburg Prof. Dr. Dipl.-Math. techn. Dipl.-Math. techn. Prof. Dr. Dipl.-Math. techn. Dipl.-Math. Ulrich Peter Benner Andr´ e Schneider Thomas Mach Michael Hinze Martin Kunkel Matthes � � � � TU Braunschweig TU Braunschweig �� �� Prof. Dr. Dipl.-Math. techn. Prof. Dr. Juan Pablo Matthias Bollh¨ ofer Andr´ e Eppler Heike Faßbender Amorocho M.Sc. � � � � ITWM Kaiserslautern TU Berlin �� �� Dr. Patrick Dipl.-Math. Dr. Tatjana Dr. Andreas Lang Oliver Schmidt Stykel Steinbrecher � � 3/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Basics Balanced Truncation for Many Ports Outlook and Acknowledgement Outline Introduction 1 Introductory Example The Setting Model Order Reduction Basics 2 Balanced Truncation Solutions of Lyapunov Equations LR-ADI Balanced Truncation for Many Ports 3 Hankel Singular Values Projection Matrices Outlook and Acknowledgement 4 4/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Wafer Development – Power Grid Model Typical applications are the simulation of power grids and clock distribution networks. 5/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Wafer Development – Power Grid Model Typical applications are the simulation of power grids and clock distribution networks. Source: http://www.swamppolitics.com. 5/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Wafer Development – Power Grid Model Typical applications are the simulation of power grids and clock distribution networks. Recently, use of nano-scale chip manufacturing process (30nm-level), increasing number of (parasitic) elements ( Intel Nehalem , 2–4 kernels, 820 millionen transistors, 45 nm), and production of Cut through a multilayer board with a BGA. multi-layered ICs, Intel 12 Source: http://de.academic.ru/dic.nsf/dewiki/255301. layers. (more known?) 5/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
Introduction Introductory Example Basics The Setting Balanced Truncation for Many Ports Model Order Reduction Outlook and Acknowledgement Descriptor system Using the modified nodal analysis (MNA) in general leads to a differential-algebraic equation (DAE) of the implicit form f ( x, ˙ x, t ) = 0 , with det( ∂f x ) ≡ 0 . ∂ ˙ The DAE in semi-explicit form leads to a linear time-invariant continuous-time system, called descriptor system, C ˙ x ( t ) = − Gx ( t ) + Bu ( t ) , x (0) = x 0 , (1) y ( t ) = Lx ( t ) , with C, G ∈ R n × n , B ∈ R n × m , L T ∈ R n × p , x ∈ R n containing the internal state variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, x 0 ∈ R n the initial value and n the number of state variables, called the order of the system. 6/18 andre.schneider@mathematik.tu-chemnitz.de A. Schneider Balanced Truncation MOR for VLSI Systems with many Ports
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