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Scaled relative graphs for pairs of operators beyond classical monotonicity

Jan Quan, Alexander Bodard, Konstantinos Oikonomidis, Panagiotis Patrinos

TL;DR

This work generalizes the scaled relative graph (SRG) to pairs of operators, yielding a geometric tool to visualize and reason about paired monotonicity and incremental properties in nonlinear resolvents. By defining the SRG for operator pairs and establishing calculus rules and SRG-fullness, the authors connect paired monotonicity to convergence properties of warped and transformed resolvents, showing tight SRG-based bounds that imply firm nonexpansiveness or contraction. The framework is then applied to circuit theory, demonstrating that linear mappings composed with monotone mappings (including NPN transistor models) fit the framework and yield convergent, preconditioned iterations for solving circuit inclusions, even in highly nonmonotone settings. Overall, the paper provides a practical, visual approach to stability and convergence analysis for nonlinear, nonsmooth operator systems and offers a path to analyze or design stable feedback circuits using paired monotonicity concepts.

Abstract

We introduce a generalization of the scaled relative graph (SRG) to pairs of operators, enabling the visualization of their relative incremental properties. This novel SRG framework provides the geometric counterpart for the study of nonlinear resolvents based on paired monotonicity conditions. We demonstrate that these conditions apply to linear operators composed with monotone mappings, a class that notably includes NPN transistors, allowing us to compute the response of multivalued, nonsmooth and highly nonmonotone electrical circuits.

Scaled relative graphs for pairs of operators beyond classical monotonicity

TL;DR

This work generalizes the scaled relative graph (SRG) to pairs of operators, yielding a geometric tool to visualize and reason about paired monotonicity and incremental properties in nonlinear resolvents. By defining the SRG for operator pairs and establishing calculus rules and SRG-fullness, the authors connect paired monotonicity to convergence properties of warped and transformed resolvents, showing tight SRG-based bounds that imply firm nonexpansiveness or contraction. The framework is then applied to circuit theory, demonstrating that linear mappings composed with monotone mappings (including NPN transistor models) fit the framework and yield convergent, preconditioned iterations for solving circuit inclusions, even in highly nonmonotone settings. Overall, the paper provides a practical, visual approach to stability and convergence analysis for nonlinear, nonsmooth operator systems and offers a path to analyze or design stable feedback circuits using paired monotonicity concepts.

Abstract

We introduce a generalization of the scaled relative graph (SRG) to pairs of operators, enabling the visualization of their relative incremental properties. This novel SRG framework provides the geometric counterpart for the study of nonlinear resolvents based on paired monotonicity conditions. We demonstrate that these conditions apply to linear operators composed with monotone mappings, a class that notably includes NPN transistors, allowing us to compute the response of multivalued, nonsmooth and highly nonmonotone electrical circuits.

Paper Structure

This paper contains 6 sections, 9 theorems, 24 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Scaled relative graphs of operator pairs
  4. Properties of nonlinear resolvents
  5. Application to circuit theory
  6. Conclusion

Key Result

Proposition II.1

Let $A,B,C : \mathcal{H} \rightrightarrows\mathcal{H}$, and $\alpha,\beta \in \mathbb{R}\setminus\{0\}$. Then,

Figures (9)

  • Figure 1: Visualization standard SRG approach. Instead of traditional algebraic inequalities to characterize the properties of $B$, geometric manipulations transform the SRG of $A$ to find the SRG of $B$ (or a superset thereof). Then, SRG-fullness is used to conclude the properties of $B$.
  • Figure 2: Numerical SRG of $(A_{\rm lin}, B)$.
  • Figure 3: Numerical SRG of $(A_{\rm NPN}, B)$.
  • Figure 4: Geometry of proof \ref{['prop:transf_resv']}.
  • Figure 5: Geometry of proof \ref{['prop:warp_resv']}.
  • ...and 4 more figures

Theorems & Definitions (35)

  • Definition II.1: SRG of a pair of operators
  • Proposition II.1: Basic calculus
  • Proposition II.2: Additional calculus
  • Definition II.2: SRG-full operator classes of pairs
  • Proposition II.3
  • Definition II.3: Semimonotone operator pairs
  • Proposition II.4
  • Example II.1
  • Example II.2
  • Definition III.1
  • ...and 25 more