Prof. Zlatko Drmac | Residuals and mixed backward shadowing stability of the EDMD/KMD

Item: Prof. Zlatko Drmac | Residuals and mixed backward shadowing stability of the EDMD/KMD
Rating: 4
Author: PostSingularity Institute

by Prof. Zlatko Drmac

Formal & Physical Sciences Mathematics PBKCalculus and mathematical analysis PBKSNumerical analysis

🎙 Prof. Zlatko Drmac 👥 8K 📅 August 20, 2026 ⏱ 42 min 👁 4 📄 original study 🧭 2026-08-20

Available in: English (current) Français

Keywords

EDMDKMDbackward errorshadowingKoopman operator

Summary

The talk introduces a new concept called ‘mixed backward shadowing stability’ for the Extended Dynamic Mode Decomposition (EDMD) and Koopman Mode Decomposition (KMD). The speaker, a numerical analyst, addresses the issue of spurious eigenvalues and eigenvectors in EDMD approximations. He proposes a backward error analysis approach: instead of asking how accurate the computed eigenpairs are, he constructs a single perturbation of the Koopman operator that makes all selected eigenpairs exact. This perturbation, denoted ΔK, is built from the residuals of the selected pairs. The key insight is to then interpret the use of this perturbed operator as equivalent to evaluating the observable along a pseudo-trajectory of the original dynamical system. Using shadowing theory, the speaker argues that this pseudo-trajectory is shadowed by a true trajectory of the system with a slightly perturbed initial condition. This provides a rigorous justification for using EDMD/KMD predictions even when the system is sensitive to initial conditions. The talk outlines the theoretical framework, connects it to existing shadowing theory, and discusses conditions under which the shadowing holds, linking the size of the perturbation to the residuals and the quality of the dictionary.

187 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk presents a novel and valuable theoretical framework for understanding the reliability of EDMD/KMD. The argumentation is rigorous and builds on established concepts from numerical analysis (backward error) and dynamical systems (shadowing). The speaker clearly motivates the problem, introduces the concept of a composite backward error, and then extends it to a ‘mixed’ stability that connects the operator perturbation to perturbations in the initial condition. The reasoning is well-structured and demonstrates a deep understanding of both numerical linear algebra and dynamical systems theory.

Scientific Rigor, Source Quality, Title Accuracy

The presentation is scientifically rigorous, with a clear mathematical derivation of the main results. The speaker references the work of Palmer on shadowing theory, which is a standard reference in the field. The title accurately reflects the content, focusing on residuals and the new concept of mixed backward shadowing stability. The talk is part of a scientific workshop at the Isaac Newton Institute, which adds to its credibility. The description provides links to the institute and the specific seminar page, which are relevant sources.

183 words

Title / Content Match

The title accurately reflects the content, which focuses on residuals and a novel concept of mixed backward shadowing stability for EDMD/KMD.

Quality & Reliability

8/10

Presentation of new theoretical results by a recognized expert in numerical analysis, with rigorous mathematical reasoning and references to established shadowing theory. The talk is recent and technical, but the lack of published peer-reviewed details at this stage slightly limits verifiability.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Isaac Newton Institute for Mathematical Sciences — Host institution and general information.
Seminar page for OMDW01 — Event page with details about the workshop and talk.

Concurring Sources

Shadowing lemma — The talk builds on this classical result to justify the existence of a true trajectory near the computed pseudo-trajectory.

Contribution & Novelties

The talk introduces a novel concept of ‘mixed backward shadowing stability’ for EDMD/KMD, which provides a rigorous justification for using approximate eigenpairs by showing they correspond to a true trajectory of the system with a slightly perturbed initial condition. This goes beyond simple residual checks and connects the numerical analysis of the operator approximation to the dynamical systems theory of shadowing.

Pour aller plus loin :

Extended Dynamic Mode Decomposition — Provides background on the EDMD method.
Shadowing lemma — Core concept used in the talk to justify the stability of pseudo-trajectories.
Koopman operator — The operator theory underlying EDMD/KMD.
Backward error analysis — The numerical analysis principle used to interpret the residuals.

113 words

Radar Profile

The radar profile shows a highly technical and information-dense talk, with very high scores in information quantity, quality, and technical level. The reliability score is also high, reflecting the rigorous mathematical approach and the speaker's expertise. The profile is consistent with a specialized research seminar.

Reliability 8/10