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Preservation of dissipativity in dimensionality reduction arxiv.org/abs/2410.11889

Preservation of dissipativity in dimensionality reduction

Systems with predetermined Lyapunov functions play an important role in many areas of applied mathematics, physics and engineering: dynamic optimization methods (objective functions and their modifications), machine learning (loss functions), thermodynamics and kinetics (free energy and other thermodynamic potentials), adaptive control (various objective functions, stabilization quality criteria and other Lyapunov functions). Dimensionality reduction is one of the main challenges in the modern era of big data and big models. Dimensionality reduction for systems with Lyapunov functions requires it preserving dissipativity: the reduced system must also have a Lyapunov function, which is expected to be a restriction of the original Lyapunov function on the manifold of the reduced motion. An additional complexity of the problem is that the equations of motion themselves are often unknown in detail in advance and must be determined in the course of the study, while the Lyapunov function could be determined based on incomplete data. Therefore, the projection problem arises: for a given Lyapunov function, find a field of projectors such that the reduction of it any dissipative system is again a dissipative system. In this paper, we present an explicit construction of such projectors and prove their uniqueness. We have also taken the first step beyond the approximation by manifolds. This is required in many applications. For this purpose, we introduce the concept of monotone trees and find a projection of dissipative systems onto monotone trees that preserves dissipativity.

arXiv.org

Distributed MPC Formation Path Following for Acoustically Communicating Underwater Vehicles arxiv.org/abs/2410.11959

Distributed MPC Formation Path Following for Acoustically Communicating Underwater Vehicles

We propose and analyse a model predictive control (MPC) strategy tailored for networks of underwater agents tasked with maintaining formation while following a shared path and using acoustic communication channels. The strategy accommodates both time-division and frequency-division medium access schemes, and addresses the inherent challenges of lossy and broadcast communication over acoustic media. Our approach extends an existing distributed control algorithm originally assuming standard double precision in exchanged data, and designed for synchronous, bidirectional, and reliable communication. Here we introduce adaptations for handling broadcast asynchronous communication, for mitigating packet losses, and for quantising exchanged data. These modifications are general and intended to be applicable to other distributed control schemes that were developed under idealised assumptions. Our goal is thus to help facilitating deployment also of other control schemes in practical field conditions. We provide simulation results that quantify the impact of these adaptations on the performance of the original controller, along with sensitivity analyses on how performance losses are influenced by key hyperparameters. Additionally, we characterise the data rate savings vs. control performance losses that may be achieved through tuning such hyperparameters, showcasing the feasibility of implementing the proposed strategy for practical purposes using commercially available full-duplex or half-duplex modems.

arXiv.org

Algebraic and Topological Persistence arxiv.org/abs/2410.08323

Algebraic and Topological Persistence

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory. Moreover, we present the fundamental concepts of algebraic topology, including exact and short exact sequences and relative homology groups derived from quotienting with subspaces of a topological space. These tools are used to prove the Excision Theorem in algebraic topology. Subsequently, the theorem is applied to demonstrate the equivalence of simplicial and singular homology for triangulable topological spaces, i.e. those topological spaces which admit a simplicial structure. This enables a more general theory of homology to be adopted in the study of filtrations of point clouds. The chapter on homological persistence makes use of these tools throughout. We develop the theory of persistent homology, the homology of filtrations of topological spaces, and the corresponding dual concept of persistent cohomology. This work aims to provide mathematicians with a robust foundation for productive engagement with the aforementioned theories. The majority of the proofs have been rewritten to clarify the relationships between the techniques discussed. The novel aspect of this contribution is the canonical presentation of persistence theory and the associated ideas through a rigorous mathematical treatment for triangulable topological spaces and closing some gaps in the existing literature.

arXiv.org
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