From Low-Rank Retractions to Dynamical Low-Rank Approximation and Back: Exploring Manifold-Constrained ODEs

Kernekoncepter

The author explores the application of retractions in numerical integration schemes for manifold-constrained ordinary differential equations, focusing on low-rank matrix manifolds.

Resumé

The content delves into the utilization of retractions in solving optimization problems on manifolds, particularly focusing on dynamical low-rank approximation techniques. It introduces novel numerical integration schemes like AFE and PRH, highlighting their advantages through experiments on DLRA examples.
Several existing DLRA techniques are discussed in relation to retractions, emphasizing their close connection and applicability in time integration methods. The Weingarten map and second fundamental form are introduced to explain the acceleration of exact solutions in DLRA schemes.
The article also presents new integration algorithms like AFE and PRH based on retractions, showcasing their local truncation error order three. The discussion extends to Hermite interpolation methods using retractions for manifold curves with prescribed endpoints and velocities.
Overall, the content provides a comprehensive exploration of how retractions play a crucial role in designing efficient numerical integration schemes for manifold-constrained ODEs.

Statistik

For instance, any retraction allows one to define a manifold curve passing through a prescribed point with prescribed velocity.
The two methods are proven to have local truncation error of order three.
The Weingarten map can be computed as NVpΣ−1V⊤ + UΣ−1U⊤pN ∈ TY Mr.
The RH interpolant achieves O(∆t4) approximation error as ∆t → 0.
The accelerated forward Euler scheme has a local truncation error of order O(∆t3).

Citater

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From low-rank retractions to dynamical low-rank approximation and back

by Axel... kl. arxiv.org 03-11-2024

https://arxiv.org/pdf/2309.06125.pdf

Dybere Forespørgsler

How do different types of retraction impact the accuracy and stability of numerical integration schemes

Retractions play a crucial role in determining the accuracy and stability of numerical integration schemes. Different types of retractions can have varying impacts on these aspects. For example, second-order retractions are known to provide higher accuracy compared to first-order retractions. This is because second-order retractions allow for the approximation of curves with better precision, leading to reduced errors in the integration process. Additionally, second-order retractions often result in smoother manifold curves, which can contribute to improved stability by reducing oscillations or erratic behavior in the numerical solution.
On the other hand, certain types of retractions may introduce errors or instabilities if not carefully implemented. For instance, if a retraction does not preserve important geometric properties of the manifold or fails to accurately approximate tangent vectors and accelerations along the curve, it can lead to inaccuracies in the numerical solution. In such cases, the choice of retraction becomes critical in ensuring that the integration scheme maintains both accuracy and stability throughout its execution.
Overall, understanding how different types of retractions affect accuracy and stability is essential for designing effective numerical integration methods on manifolds.

What are the practical implications of utilizing second-order retractions in developing novel integration methods

Utilizing second-order retractions in developing novel integration methods has several practical implications for enhancing computational efficiency and solution quality.

Improved Accuracy: Second-order retractions enable more accurate approximations of curves on manifolds by considering higher-order terms during interpolation or extrapolation processes. This leads to reduced errors in estimating solutions at each time step, resulting in more precise numerical results.

Enhanced Stability: By incorporating acceleration information through second-order corrections provided by these retractions, new integration methods become more stable over longer simulation periods or when dealing with stiff differential equations. The additional control over acceleration helps mitigate issues related to rapid changes or high-frequency components present in some dynamical systems.

Efficient Curve Generation: Second-order retractions facilitate efficient generation of smooth manifold curves with prescribed initial conditions and velocities while maintaining curvature information along these trajectories. This capability allows for robust trajectory planning and tracking applications where accurate path following is essential.

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