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Introduction to Model Order Reduction

Instructor: Prof. Luca Daniel

Affiliation: Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology

Duration: 15 hours

Period: June 19 - June 23, 2006

Place: Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica, Telecomunicazioni, via G. Caruso, meeting room, ground floor

Credits: 4

Contacts: Ing. F. De Bernardinis, Ing. P. Nuzzo

Description

The performance of many large engineering systems and complex components often critically depends on what the designers like to address as “second order effects”. These are typically phenomena that can be captured accurately only by computationally demanding partial differential equation solvers (e.g. Maxwell, Nevier-Stokes, or heat diffusion field solvers). Designers, however, would greatly benefit from the availability of very small models that capture the input-output behavior of complex systems with the same accuracy as field solvers. In this series of lectures we will survey several techniques to generate automatically such reduced order models preserving field solver accuracy. We will further describe techniques to generate field solver accurate parameterized reduced order models that can be instantiated for a range of values of specified design parameters, hence enabling fast design exploration and optimization. Detailed examples will be presented, drawn from a variety of engineering disciplines e.g. Electrical Engineering (interconnect networks including parasitics; fullwave electromagnetic structures; analog and digital circuits including nonlinear semiconductor devices and Micro-Electro-Mechanical Devices), Mechanical Engineering (frame modeling, heat diffusion), and Civil Engineering (structural problems).

Outline

Introduction. Motivations. Model Order Reduction problem definition.

PART I: Assembling Dynamical State Space Systems from Engineering problems

1. Sample problems from Electrical, Mechanical and Civil Engineering.
2. Basic problem formulation techniques (e.g. nodal analysis).
3. Basic Partial Differential Equation (PDE) solvers (e.g. Finite Difference, Finite Element, Boundary Element Methods).
4. Basic numerical techniques to manipulate dynamical systems models:
   1. Linear System solvers (LU decomposition, Krylov iterative methods);
   2. An efficient PDE based matrix-vector product algorithm;
   3. Non-Linear System solvers (Newton method);
   4. Time domain simulation of dynamical systems models.
5. Important dynamical systems’ properties (e.g. stability, passivity).

PART II: Model Order Reduction of Linear Dynamical Systems

1. The classical engineering approach: modal analysis (the eigenvalue method).
2. Reducing Linear Time Invariant (LTI) Dynamical State Space System models:
   1. Pade’ approximation and Asymptotic Waveform Evaluation (AWE).
   2. The Projection Framework.
   3. Proper Orthogonal Decomposition (POD), or Karhunen-Lo`eve decomposition (KLD), or principal component analysis (PCA), or singular value decomposition (SVD).
   4. Time domain Chebyshev polynomial and W. approach.
   5. Transfer Function Moment Matching (PVL).
   6. Passivity and stability preserving Moment Matching (PRIMA)
   7. Truncated Balance Realizations (TBR) and Hankel reduction.
   8. Positive Real and Bounded Real TBR to preserve passivity.
   9. TBR with approximate system Grammians.
3. Reducing LTI Distributed Systems (with frequency dependent matrices).
4. Reduction of Linear Time Varying Dynamical Systems.
5. Reduced order modeling from transfer function data or measurements:
   1. Curve fitting, least square methods;
   2. Quasi-convex optimization methods.

PART III: Model Order Reduction of Non-Linear Dynamical Systems.

1. Volterra Series based reduction method.
2. Trajectory Piece-Wise Linear (TPWL) + moment matching reduction.
3. Trajectory Piece-Wise Linear (TPWL) + balance realizations (TBR) reduction.
4. Trajectory Piece-Wise Polynomial (PWP) + moment matching reduction.

PART IV: Model Order Reduction of Parameterized Dynamical Systems.

1. Motivations and applications (library characterization, design optimization, inverse problems for automatic design synthesis).
2. Reducing linear models with linear dependency on parameters.
3. Reducing linear models with non-linear dependency on parameters.
   1. Single and Multi-Point Parameterized Moment Matching approach
   2. Parameterized Quasi-convex optimization based approach.
   3. Variational spectrally-weighted balanced truncation.
4. Reducing non-linear models with non-linear dependency on parameters.