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Prologue to Model Order Reduction Luca Daniel Massachusetts Institute of Technology luca@mit.edu http://onigo.mit.edu/~dluca/2006PisaMOR www.rle.mit.edu/cpg

Course Outline Numerical Simulation Quick introduction to PDE Solvers Quick introduction to ODE Solvers Model Order decrease Linear frameworks Common building hone Optimal methods as far as model exactness Efficient strategies regarding time and memory Non-Linear Systems Parameterized Model Order Reduction Linear Systems Non-Linear Systems Today

Model Order Reduction of Non-Linear Systems Introduction Reduction of pitifully nonlinear frameworks (Volterra Series) Reduction of firmly nonlinear frameworks Trajectory PieceWise Linear (TPWL) and Polynomial (PWP) with minute coordinating with Truncated Balance Realization

Jet motor channel case Inlet thickness unsettling influence d :

NonLinear gadgets in Systems-On-Chip

Example of a nonlinear transmission line demonstrate

Example of a microswitch DISCRETIZE

Reduction of NonLinear Systems NonLinear Reduction PDE Field Solvers or Circuit Simulators Non-Linear simple segments e.g. MEMs, VCO, LNA

Change of factors Projection Framework Equation Testing

Projection structure for nonlinear element frameworks in Substitute: Reduced framework: An issue: q=10 n=10 4 n=10 4 q=10 Using V T F(Ux) is excessively costly!

Nonlinear Problem is MUCH Harder In what premise would it be advisable for us to extend? No straightforward recurrence area understanding No eigenmodes No Krylov subspace How would you speak to the projection? New issue for nonlinear

How to get the change of variable network Analysis of linearized models [Ma88] Sampling of time-reproduction information + primary parts investigation [Sirovich87] Nonlinear adjusting [Scherpen93], Time-subsidiaries [Gunupudi99], …

Model Order Reduction for NonLinear Systems Representation of F (x) utilizing a polynomial (e.g. Taylor's developments, Volterra Series) [Phillips00]***** Representation of F(x) utilizing a few linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] Representation of F(x) with a few polynomials (PWP PieceWise Polynomial) [Dong03]

Introduction to Model Order Reduction III.2 – Reduction of Weakly Non Linear Systems Luca Daniel Thanks to Joel Phillips, Cadence Berkeley Labs

Representation of F(x) in light of Taylor's extensions Approximate nonreduced show: MOR for nonlinear element frameworks (J. Phillips, Y. Chen, F. Wang): Linear, quadratic lessened request models:

Expand nonlinearity in multi-dimensional polynomial arrangement Differential condition gets to be To match initial few terms in useful arrangement extension, just need initial couple of polynomial terms Each term is a - dimensional tensor, spoke to as a lattice Polynomial Approximations [note : not static models; recurrence impacts retained] where and so on

Draw from decreased space as Identity for Kronecker items Project tensors Gives diminished model Projection of polynomial terms and so forth.

Projection methodology produces lessened model in same polynomial frame Key tensor parts are compacted to lower dimensionality Kronecker shapes give advantageous general documentation How to get projection spaces V? Decreased polynomial models

Introduce variational parameter Consider parametrized framework Expand reaction in forces of Substitute into differential condition Compare and gather terms for every force of Variational Analysis

Coupled arrangement of differential conditions Each set registers reaction at one request Each set is straight in its own state variable First set is just linearized demonstrate Variational Analysis and so on

First request condition set driven by contribution of low rank Krylov subspace for projection Second request condition set driven by info Key understanding : review is very much approximated by accordingly Model Reduction

Second request condition set is driven by Generate second Krylov space utilizing Compute models by projection every condition arrange set independently, or venture unique polynomial framework utilizing UNION of number of coordinated snapshots of request i given by size of suitable Krylov subspace Model Reduction

Size of Krylov spaces becomes rapidly with practical arrangement arrange conceivably huge models inborn in polynomial depictions (e.g. Volterra arrangement) Practical for feeble nonlinearities requiring just few terms in utilitarian arrangement illustration: RF datapath solid nonlinearities incorporated into inclination conditions Many neighborhood advancements, augmentations are conceivable two-sided projectors different lessening steps Computational Considerations

- 20 - 60 - 80 0 50 100 150 200 MHz RF blender case dB

Projection + polynomial arrangement + variational examination = thorough nonlinear decrease system Compact Kronecker documentation is extremely helpful Is there an approach to diminish number of terms in higher-arrange Krylov space? Littler models speedier Diagnose when models in light of first request framework are fizzling? Take after on diminishment? Synopsis pitifully nonlinear MOR

Model Order Reduciton for NonLinear Systems Representation of F (x) utilizing a polynomial (e.g. Taylor's extensions, Volterra Series) [Phillips00]***** Representation of F(x) utilizing a few linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] Representation of F(x) with a few polynomials (PWP PieceWise Polynomial) [Dong03]

III.3 - Reduction of Strongly NonLinear Systems by means of Trajectory PieceWise Linear Method Luca Daniel Massachusetts Institute of Technology Thanks to Michał Rewie ń ski, Jacob White and Brad Bond www.rle.mit.edu/cpg

Piecewise-direct representation [M. Rewienski, J. White ICCAD01] Linearizations around x i , i=0,… , s-1 Weighted mix of the models: Project linearized models:

Background – TPWL [Reiwenski01] x 8 x 1 x 7 x 6 x 5 x 2 x 4 x 14 x 10 x a x b x 0 x 11 x 9 x 13 x 3 x 15 Model i just substantial close x i x 2 #linearizations =#samples n = 10 4 #samples = 100 10000 = LARGE Use gathering of straight models x 1

Background – TPWL: Picking Linearization Points Use preparing directions to pick linearization focuses y(t) x 2 State Space Time Domain Simulation t x 1 Linearization at current state x i

Quasi-piecewise-straight MOR – figuring weights For i=0,… ,(s-1) register: Compute For i=0,… ,(s-1) process: Normalize w i .

= Model from linearization 1 Model from linearization k Model from linearization 2 K 2 A 1 K 1 A 2 A k Background – TPWL Reduction of the Linearized Systems

= Background – TPWL Constructing the projection framework V Use minutes from EACH straight model to develop V

Background – TPWL: Weighting/Simulation [Riewinski01,Tiwary05,Dong05] Linearization 3 x 2 Use weighting capacities to join direct models amid reenactment additionally all around approximated C – inadequately approximated Well approximated Current state Linearization 2 x 1 Linearization 1

Test case of a nonlinear circuit:

Computational results – circuit illustration Input: preparing input testing input

Computational results - circuit case, SSS

DISCRETIZE Example of a microswitch

Computational results - microswitch case Input: preparing input testing input

Computational results - microswitch case

Computational results – microswitch case preparing input testing input

Simulation with the piecewise-direct model

Generate the lessened direct model Use the diminished direct framework for recreation x 2 x 1 x 0 A x 3 Fast surmised reproduction x 2 x 1

Computational results - show arrange decrease * Matlab usage

Computational results – OpAmp case OpAmp: 70 MOSFETs 13 resistors 9 capacitors 51 hubs preparing input testing input

Computational results – OpAmp case OpAmp: 70 MOSFETs 13 resistors 9 capacitors 51 hubs

Computational results – OpAmp case, SSS

Computational results – stun engendering in stream motor delta Density unsettling influence d :

Computational results – stun proliferation in 1D Burgers condition:

Questions yet to be replied: How would it be a good idea for us to pick preparing inputs? How "regularly" would it be advisable for us to register linearized models along the preparation direction? What are the mistakes of speaking to a nonlinear frameworks as a direction piecewise straight framework? What are "the best" decreased request bases for the direction PWL demonstrate?

Conclusions TPWL macromodels can be exact for various firmly nonlinear dynamical frameworks, beating diminished models in view of single-state polynomial developments. TPWL models utilize basic, cost-effective representation of framework's nonlinearity, bringing about short reenactment times. TPWL models might be extricated productively by utilizing a quick estimated test system.

Model Order Reduction of Non-Linear Systems Introduction Reduction of feebly nonlinear frameworks (Volterra Series) Reduction of firmly nonlinear frameworks (TPWL) Trajectory Pieace Wise Linear and Polynomial with minute coordinating with Truncated Balance Realization

III.4 Reduction of Strongly NonLinear Systems through Trajectory-PieceWise Linear (TPWL) + Truncated Balance Realization (TBR) Luca Daniel Massachusetts Institute of Technology Thanks to Dmitry Vasilyev, Michał Rewie ń ski, Jacob White

Krylov-subspace strategies Fast Don't ensure precision Balanced-truncation techniques Expensive (~n 3 ) Guarantee exactness Obtaining pr

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