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Carleman Linearization of Parabolic PDEs: Well-posedness, convergence, and efficient numerical methods

Bernhard Heinzelreiter, John W. Pearson

TL;DR

This work develops a rigorous framework for the Carleman linearization of nonlinear parabolic PDEs by extending it to infinite-dimensional Hilbert spaces with quadratic nonlinearities and proving well-posedness and convergence of truncated linearizations. A lifted, tensorized representation using Kronecker sums yields a finite, linear system that can be analyzed independently of discretization, enabling a clear linearization–discretization error decomposition and discretization-free convergence results under small-nonlinearity assumptions. The authors derive detailed function-space settings, operator bounds, and a transformed-coercive formulation to establish existence, uniqueness, and stability of the truncated Carleman system, along with explicit convergence bounds that depend on truncation level $N$, time horizon $T$, and data size. They show how non-standard discretizations, notably sparse grids, can mitigate the curse of dimensionality and validate the theory through Burgers-type numerical experiments, illustrating both the benefits and limitations of the approach for PDEs. Overall, the results provide a solid theoretical foundation for applying Carleman linearization to parabolic PDEs and point to promising avenues for efficient, structure-exploiting numerical methods.

Abstract

We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman linearization under suitable assumptions on the dynamical system, which encompass common parabolic semi-linear partial differential equations such as the Navier-Stokes equations and nonlinear diffusion-advection-reaction equations. Upon discretization, we show that the total approximation error of the linearization decomposes into two independent components: the discretization error and the linearization error. This decomposition yields a convergence radius and convergence rate for the discretized linearization that are independent of the discretization. We thus justify the application of the linearization to parabolic PDE problems. Furthermore, it motivates the use of non-standard structure-exploiting numerical methods, such as sparse grids, taming the curse of dimensionality associated with the Carleman linearization. Finally, we verify the results with numerical experiments.

Carleman Linearization of Parabolic PDEs: Well-posedness, convergence, and efficient numerical methods

TL;DR

This work develops a rigorous framework for the Carleman linearization of nonlinear parabolic PDEs by extending it to infinite-dimensional Hilbert spaces with quadratic nonlinearities and proving well-posedness and convergence of truncated linearizations. A lifted, tensorized representation using Kronecker sums yields a finite, linear system that can be analyzed independently of discretization, enabling a clear linearization–discretization error decomposition and discretization-free convergence results under small-nonlinearity assumptions. The authors derive detailed function-space settings, operator bounds, and a transformed-coercive formulation to establish existence, uniqueness, and stability of the truncated Carleman system, along with explicit convergence bounds that depend on truncation level , time horizon , and data size. They show how non-standard discretizations, notably sparse grids, can mitigate the curse of dimensionality and validate the theory through Burgers-type numerical experiments, illustrating both the benefits and limitations of the approach for PDEs. Overall, the results provide a solid theoretical foundation for applying Carleman linearization to parabolic PDEs and point to promising avenues for efficient, structure-exploiting numerical methods.

Abstract

We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman linearization under suitable assumptions on the dynamical system, which encompass common parabolic semi-linear partial differential equations such as the Navier-Stokes equations and nonlinear diffusion-advection-reaction equations. Upon discretization, we show that the total approximation error of the linearization decomposes into two independent components: the discretization error and the linearization error. This decomposition yields a convergence radius and convergence rate for the discretized linearization that are independent of the discretization. We thus justify the application of the linearization to parabolic PDE problems. Furthermore, it motivates the use of non-standard structure-exploiting numerical methods, such as sparse grids, taming the curse of dimensionality associated with the Carleman linearization. Finally, we verify the results with numerical experiments.

Paper Structure

This paper contains 32 sections, 20 theorems, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Nonlinear Cauchy Problems in Hilbert Spaces
  3. Weak Formulation
  4. Linear Equation
  5. Nonlinear Equation
  6. Truncated Carleman Linearization
  7. Well-Posedness of the Truncated Carleman Linearization
  8. Function Spaces
  9. Properties of $A_k$, $B_k$, and $F_k$
  10. Lifted Linearization and Well-Posedness
  11. Convergence in the Case of Small Nonlinearities
  12. Assumptions and Implications of the Result
  13. Examples of Operators $A$ and $B$
  14. Zeroth-order derivatives
  15. Fluid flow problems
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $T < \infty$ and $A(t)$ be a family of $V$-$H$ operators fulfilling assumption:ABounded and assumption:AVHCoercive for $t \in (0, T)$ with constants $\beta$, $\gamma$, and $\lambda$ independent of $t$. Then, for every $y_0 \in H$ and $f \in L^2(0, T; V')$, the linear Cauchy problem has a unique solution $y \in W(0, T; V, V')$. This solution satisfies the estimate for all $t \in [0, T)$. More

Figures (4)

  • Figure 1: Snapshots of solutions to the linearization of the Burgers' equation $y_N^{(1)}(t)$ for different truncation levels $N$ compared to the exact solution $y_e(t)$.
  • Figure 2: Error plots of snapshots of solutions to the linearization of the Burgers' equation $(y_N^{(1)}(t) - y_e(t)) / \|y_e(t)\|_H$ for different truncation levels $N$.
  • Figure 3: Convergence of the Carleman linearization with respect to the truncation level $N$ measured by the error $\|\eta\|_{L^\infty(0, T; H)}$. Each plot indicates that the error behaves exponentially in $N$, whereas different sets of model parameters affect the error bounds in Theorem \ref{['thm:Convergence']} and Corollary \ref{['corollary:ConergenceAlphaZero']}.
  • Figure 4: Number of degrees of freedom (denoted DOFs) $\dim U_h(N)$ for different discretization methods for varying truncation levels $N$ and refinement levels $J$ (the corresponding maximum cell size of the mesh is given by $h=2^{-J}$).

Theorems & Definitions (43)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 33 more