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Practicable Simulation-Free Model Order Reduction by Nonlinear Moment Matching

Maria Cruz Varona, Raphael Gebhart, Julian Suk, Boris Lohmann

TL;DR

A practicable simulation-free model order reduction method by nonlinear moment matching is developed that relies on the solution of nonlinear systems of equations rather than on the expensive simulation of the original model or the difficult solution of a nonlinear partial differential equation.

Abstract

In this paper, a practicable simulation-free model order reduction method by nonlinear moment matching is developed. Based on the steady-state interpretation of linear moment matching, we comprehensively explain the extension of this reduction concept to nonlinear systems presented in [1], provide some new insights and propose some simplifications to achieve a feasible and numerically efficient nonlinear model reduction algorithm. This algorithm relies on the solution of nonlinear systems of equations rather than on the expensive simulation of the original model or the difficult solution of a nonlinear partial differential equation.

Practicable Simulation-Free Model Order Reduction by Nonlinear Moment Matching

TL;DR

A practicable simulation-free model order reduction method by nonlinear moment matching is developed that relies on the solution of nonlinear systems of equations rather than on the expensive simulation of the original model or the difficult solution of a nonlinear partial differential equation.

Abstract

In this paper, a practicable simulation-free model order reduction method by nonlinear moment matching is developed. Based on the steady-state interpretation of linear moment matching, we comprehensively explain the extension of this reduction concept to nonlinear systems presented in [1], provide some new insights and propose some simplifications to achieve a feasible and numerically efficient nonlinear model reduction algorithm. This algorithm relies on the solution of nonlinear systems of equations rather than on the expensive simulation of the original model or the difficult solution of a nonlinear partial differential equation.

Paper Structure

This paper contains 25 sections, 5 theorems, 32 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Moment Matching for linear systems
  3. Notion of Moments and Krylov subspaces
  4. Equivalence of Krylov subspaces and Sylvester equations
  5. Time domain interpretation of Moment Matching
  6. Moment Matching for nonlinear systems
  7. Nonlinear Petrov-Galerkin projection
  8. Notion of Nonlinear Moments
  9. Steady-State-Based Nonlinear Moment Matching
  10. Simulation-Free Nonlinear Model Reduction by Moment Matching
  11. Linear Projection
  12. Nonlinear signal generator
  13. Linear signal generator
  14. Zero signal generator
  15. Time discretization with collocation points
  16. ...and 10 more sections

Key Result

Lemma 1

The moments $\boldsymbol{m}_i(\sigma)$ of system eq:linear-FOM around shifts $\sigma \not\in \uplambda(\boldsymbol{E}^{-1} \boldsymbol{A})$ are equivalently given by where, according to eq:multimom-block-Krylov-V, $\boldsymbol{V}_i$ is calculated as or, alternatively, $\boldsymbol{V} \!=\! \left[\boldsymbol{V}_0, \ldots, \boldsymbol{V}_{r-1}\right]$ corresponds to the unique solution of the Sylv

Figures (3)

  • Figure 1: Diagram depicting the interconnection between the linear FOM/ROM and the linear signal generator to illustrate the time domain interpretation of moment matching for linear systems.
  • Figure 2: Diagram depicting the interconnection between the nonlinear FOM/ROM and the nonlinear signal generator to illustrate the time domain interpretation of moment matching for nonlinear systems.
  • Figure 3: Limit cycle behavior and outputs of the FHN model for test signal $\boldsymbol{u}(t) \!=\! \left[i_0(t), \, 1\right]^{\mathsf T}$ with $i_0(t) \!=\! 5 \cdot 10^4 \, t^3 \, \mathrm{e}^{-15 t}$ ($r_{\mathrm{defl}} \!=\! 22$)

Theorems & Definitions (7)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Remark 1
  • Theorem 2
  • Corollary 2