Introduction to Model Order Reduction Luca Daniel Massachusetts Institute of Technology email@example.com http://onigo.mit.edu/~dluca/2006PisaMOR www.rle.mit.edu/cpg
Course Outline Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE Solvers Model Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of 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 weakly nonlinear systems (Volterra Series) • Reduction of strongly nonlinear systems Trajectory PieceWise Linear (TPWL) and Polynomial (PWP) • with moment matching • with Truncated Balance Realization
Jet engine inlet example • Inlet density disturbance d:
Example of a microswitch DISCRETIZE
Reduction of NonLinear Systems NonLinear Reduction PDE Field Solvers or Circuit Simulators Non-Linear analog components e.g. MEMs, VCO, LNA
Change of variables Projection Framework Equation Testing
Projection framework for nonlinear dynamic systems in • Substitute: • Reduced system: • A problem: q=10 n=104 n=104 q=10 • Using VTF(Ux) is too expensive!
Nonlinear Problem is MUCH Harder • In what basis should we project? • No simple frequency domain insight • No eigenmodes • No Krylov subspace • How do you represent the projection? • New problem for nonlinear
How to get the change of variable matrix • Analysis of linearized models [Ma88] • Sampling of time-simulation data + principal components analysis [Sirovich87] • Nonlinear balancing [Scherpen93], • Time-derivatives [Gunupudi99], …
Model Order Reduction for NonLinear Systems • Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]***** • Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] • Representation of F(x) with several 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) based on Taylor’s expansions • Approximate nonreduced model: MOR for nonlinear dynamic systems (J. Phillips, Y. Chen, F. Wang): • Linear, quadratic reduced order models:
Expand nonlinearity in multi-dimensional polynomial series Differential equation becomes To match first few terms in functional series expansion, only need first few polynomial terms Each term is a -dimensional tensor, represented as an matrix Polynomial Approximations [note : not static models; frequency effects retained] where etc.
Draw from reduced space as Identity for Kronecker products Project tensors Gives reduced model Projection of polynomial terms etc.
Projection procedure produces reduced model in same polynomial form Key tensor components are compressed to lower dimensionality Kronecker forms provide convenient general notation How to get projection spaces V? Reduced polynomial models
Introduce variational parameter Consider parametrized system Expand response in powers of Substitute into differential equation Compare and collect terms for each power of Variational Analysis
Coupled set of differential equations Each set computes response at one order Each set is linear in its own state variable First set is just linearized model Variational Analysis etc.
First order equation set driven by input of low rank Krylov subspace for projection Second order equation set driven by input Key insight : recall is well-approximated by thus Model Reduction
Second order equation set is driven by Generate second Krylov space using Compute models by projection each equation order set separately, or project original polynomial system using UNION of number of matched moments of order-i given by size of appropriate Krylov subspace Model Reduction
Size of Krylov spaces grows quickly with functional series order potentially large models intrinsic in polynomial descriptions (e.g. Volterra series) Practical for weak nonlinearities needing only few terms in functional series example: RF datapath strong nonlinearities included in bias conditions Many local optimizations, extensions are possible two-sided projectors multiple reduction steps Computational Considerations
-20 -60 -80 0 50 100 150 200 MHz RF mixer example dB
Projection + polynomial series + variational analysis = rigorous nonlinear reduction procedure Compact Kronecker notation is very useful Is there a way to reduce number of terms in higher-order Krylov space? Smaller models faster Diagnose when models based on first order system are failing? Follow-on reduction? Summary weakly nonlinear MOR
Model Order Reduciton for NonLinear Systems • Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]***** • Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01] • Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]
III.3 - Reduction of Strongly NonLinear Systems via Trajectory PieceWise Linear Method Luca Daniel Massachusetts Institute of Technology Thanks to Michał Rewieński, Jacob White and Brad Bond www.rle.mit.edu/cpg
Piecewise-linear representation [M. Rewienski, J. White ICCAD01] • Linearizations around xi , i=0,…, s-1 • Weighted combination of the models: • Project linearized models:
Background – TPWL [Reiwenski01] x8 x1 x7 x6 x5 x2 x4 x14 x10 xa xb x0 x11 x9 x13 x3 x15 Model i only valid near xi x2 #linearizations =#samplesn n = 104 #samples = 100 10010000 = LARGE Use collection of linear models x1
Background – TPWL: Picking Linearization Points Use training trajectories to pick linearization points y(t) x2 State Space Time Domain Simulation t x1 Linearization at current state xi
Quasi-piecewise-linear MOR – computing weights • For i=0,…,(s-1) compute: • Compute • For i=0,…,(s-1) compute: • Normalize wi.
= Model from linearization 1 Model from linearization k Model from linearization 2 K2 A1 K1 A2 Ak Kk Background – TPWLReduction of the Linearized Systems
= Background – TPWLConstructing the projection matrix V Use moments from EACH linear model to construct V
Background – TPWL: Weighting / Simulation[Riewinski01,Tiwary05,Dong05] Linearization 3 x2 Use weighting functions to combine linear models during simulation also well approximated C – poorly approximated Well approximated Current state Linearization 2 x1 Linearization 1
Computational results – circuit example • Input: training input testing input
DISCRETIZE Example of a microswitch
Computational results - microswitch example • Input: training input testing input
Computational results – microswitch example training input testing input
Generate the reduced linear model Use the reduced linear system for simulation x2 x1 x0 A x3 Fast approximate simulation x2 x1
Computational results - model order reduction *Matlab implementation
Computational results – OpAmp example OpAmp: • 70 MOSFETs • 13 resistors • 9 capacitors • 51 nodes training input testing input
Computational results – OpAmp example OpAmp: • 70 MOSFETs • 13 resistors • 9 capacitors • 51 nodes
Computational results – shock propagation in jet engine inlet • Density disturbance d:
Computational results – shock propagation in 1D • Burgers equation:
Questions yet to be answered: • How should we pick training inputs? • How “often” should we compute linearized models along the training trajectory? • What are the errors of representing a nonlinear systems as a trajectory piecewise linear system? • What are “the best” reduced order bases for the trajectory PWL model?