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Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms

Albani Olivieri, Gleb Pogudin, Boris Kramer

TL;DR

QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE, is introduced, an algorithm based on symbolic computation and discrete optimization that is the first computational tool to find quadratizations for PDEs to date.

Abstract

Quadratization for partial differential equations (PDEs) is a process that transforms a nonquadratic PDE into a quadratic form by introducing auxiliary variables. This symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of nonlinear and nonquadratic PDE models. This paper presents a rigorous definition of PDE quadratization, theoretical results for the PDE quadratization problem of spatially one-dimensional PDEs-including results on existence and complexity-and introduces QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.

Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms

TL;DR

QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE, is introduced, an algorithm based on symbolic computation and discrete optimization that is the first computational tool to find quadratizations for PDEs to date.

Abstract

Quadratization for partial differential equations (PDEs) is a process that transforms a nonquadratic PDE into a quadratic form by introducing auxiliary variables. This symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of nonlinear and nonquadratic PDE models. This paper presents a rigorous definition of PDE quadratization, theoretical results for the PDE quadratization problem of spatially one-dimensional PDEs-including results on existence and complexity-and introduces QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.
Paper Structure (33 sections, 4 theorems, 54 equations, 2 figures, 2 tables, 4 algorithms)

This paper contains 33 sections, 4 theorems, 54 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Results
  3. Notation and limitations
  4. Definitions
  5. Existence and NP-hardness
  6. PDE Quadratization Algorithm
  7. Proposed algorithm: QuPDE
  8. Monomial decomposition of nonquadratic terms and Module 1
  9. Verification of a quadratization and Module 2
  10. Efficient search strategies and Module 3
  11. Search strategy
  12. Branching strategy
  13. Pruning rules
  14. Polynomialization of rational PDEs and Module 4
  15. QuPDE overview
  16. ...and 18 more sections

Key Result

Theorem 1

A PDE system of the form eq:def-pde of order $h$ has a monomial quadratization of differential order$3h$.

Figures (2)

  • Figure 1: Example of a search tree of monomial decompositions represented by $\mathcal{M}_1, \dots, \mathcal{M}_7$.
  • Figure 2: Flowchart for QuPDE to illustrate the interaction between the different modules.

Theorems & Definitions (15)

  • Example 1
  • Remark 1
  • Definition 1
  • Theorem 1: Existence of a PDE quadratization
  • Proposition 1
  • Example 2: Monomial decompositions
  • Example 3: Application of Module \ref{['alg:mon-decomp']}
  • Definition 2
  • Proposition 2
  • Example 4: Verification of a quadratization
  • ...and 5 more