Time: June 3, 2025, 11:00
Room: RICAM, SP2 416-1

Titel: Carleman linearization of PDEs: Well-posedness, convergence, and efficient numerical methods

Abstract: 

The Carleman linearization is a technique to embed a polynomially nonlinear dynamical system into an infinite-dimensional linear system. Since its introduction in [1], this approach has received significant attention in fields such as control theory and quantum computing. A central question is under which conditions a finite-dimensional truncation of the linearization converges and how the approximation error depends on the truncation size. In the last decade, this was answered for finite-dimensional systems [2]. When dealing with infinite-dimensional systems like partial differential equations (PDEs), linearization is typically applied to a discretized version of the original dynamical system. However, the error bounds from [2] are generally not robust with respect to the discretization.
In this talk, we explore how the concept of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. First, we show the well-posedness and convergence of the corresponding truncated Carleman linearization under appropriate assumptions on the Hilbert spaces and operators of the dynamical system, which cover common parabolic semi-linear PDEs such as the Navier-Stokes equations and nonlinear diffusion-advection-reaction equations. Additionally, we provide error estimates for discretizations through the Galerkin method. We further discuss how our theoretical findings motivate the use of nonstandard structure-exploiting numerical methods such as sparse grids, taming the curse of dimensionality of the Carleman linearization.
Finally, we verify the results with numerical experiments.
[1] Torsten Carleman. Application de la th´eorie des ´equations int´egrales lin´eaires aux syst`emes d’´equations diff´erentielles non lin´eaires. In Acta Mathematica, 1932, 59:63-87. doi: 10.1007/BF02546499. [2] Arash Amini, Cong Zheng, Qiyu Sun, and Nader Motee. Carleman linearization of nonlinear systems and its finite-section approximations. In Discrete and Continuous Dynamical Systems - B, 2025, 30(2): 577–603.doi: 10.3934/dcdsb.2024102. The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. Is e buidheann carthannais a th’ ann an Oilthigh Dh`un `Eideann, cl`araichte an Alba, `aireamh cl`araidh SC005336.