Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

Wenqiang Yang, Wenyuan Wu, Greg Reid

TL;DR

The paper tackles the challenge of structural analysis for polynomial and analytic differential-algebraic equations that become degenerate for certain parameter values, i.e., $S$-unamenable cases. It introduces an embedding method that regularizes a real-analytic DAE by constructing an equivalent system with a full-rank Jacobian, and a witness-point strategy to detect degeneration across all constraint components, enabling a global numerical structural analysis. By leveraging real algebraic geometry (real witness sets, distance-based optimization, and homotopy continuation) and a constant-rank embedding framework, the approach avoids expensive symbolic eliminations and extends structural analysis to DAEs with symbolic cancellation and numerical degeneration. The method is demonstrated on representative examples (transistor amplifier, nonlinear pendulum, beam bending), showing improved robustness and efficiency relative to traditional substitution/augmentation techniques, with potential impact on reliable simulation and design of complex engineering systems.

Abstract

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for polynomial systems of differential-algebraic equations, numerical methods are given to solve such cases using numerical real algebraic geometry. First, we propose an embedding method that for a given real analytic system constructs an equivalent system with a full-rank Jacobian matrix. Secondly, we introduce a witness point method, which can help to detect degeneration on all components of constraints of such systems. Thirdly, the two methods above lead to a numerical global structural analysis method for structurally unamenable differential-algebraic equations on all components of constraints.

Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

TL;DR

The paper tackles the challenge of structural analysis for polynomial and analytic differential-algebraic equations that become degenerate for certain parameter values, i.e., -unamenable cases. It introduces an embedding method that regularizes a real-analytic DAE by constructing an equivalent system with a full-rank Jacobian, and a witness-point strategy to detect degeneration across all constraint components, enabling a global numerical structural analysis. By leveraging real algebraic geometry (real witness sets, distance-based optimization, and homotopy continuation) and a constant-rank embedding framework, the approach avoids expensive symbolic eliminations and extends structural analysis to DAEs with symbolic cancellation and numerical degeneration. The method is demonstrated on representative examples (transistor amplifier, nonlinear pendulum, beam bending), showing improved robustness and efficiency relative to traditional substitution/augmentation techniques, with potential impact on reliable simulation and design of complex engineering systems.

Abstract

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for polynomial systems of differential-algebraic equations, numerical methods are given to solve such cases using numerical real algebraic geometry. First, we propose an embedding method that for a given real analytic system constructs an equivalent system with a full-rank Jacobian matrix. Secondly, we introduce a witness point method, which can help to detect degeneration on all components of constraints of such systems. Thirdly, the two methods above lead to a numerical global structural analysis method for structurally unamenable differential-algebraic equations on all components of constraints.
Paper Structure (14 sections, 11 theorems, 26 equations, 2 figures, 3 tables)

This paper contains 14 sections, 11 theorems, 26 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Problem Statement and Contributions
  4. Preliminaries
  5. Detecting Degeneration by Points
  6. The Embedding Method for S-unamenable daes
  7. Global Structural Analysis Method For Polynomial daes
  8. Examples
  9. Transistor Amplifier
  10. Non-linearly Modified Pendulum
  11. Analysis of Bending Deformation of a Beam
  12. Result Analysis
  13. Conclusions
  14. Algorithms

Key Result

Proposition 2.1

Let $\{\bm{\mathcal{J}}_i\}$ be the set of Jacobian matrices of $\{\bm{B}_i \}$. For any $0 \leq i < j \leq k_c$, $\bm{\mathcal{J}}_{i}$ is a sub-matrix of $\bm{\mathcal{J}}_{j}$. Moreover, if $\bm{\mathcal{J}}_{k_c}$ has full rank, then any $\bm{\mathcal{J}}_{i}$ also has full rank.

Figures (2)

  • Figure 1: The Superposition Deformation of Beams under Collinear Bending Moments
  • Figure 2: Numerical Solution of $3$ Examples by the Embedding Method

Theorems & Definitions (31)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 2.1: Real zero set, Component
  • Example 2.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Degree of freedom (DOF) for daes
  • Definition 2.5
  • ...and 21 more