Spaces of Relative Parallelism

Knebelman, M. S.

doi:10.2307/1969562

Cited by 9 publications

(9 citation statements)

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“…We will show in Section 3.1 that the curvature force (first two terms on the right-hand side of Equation 17) ensures the velocity remains tangent to the topography at all time. Thus, to model this effect, a tangent transport is applied (e.g., Knebelman, 1951…”

Section: The Shaltop Numerical Modelmentioning

confidence: 99%

“…Thus, to model this effect, a tangent transport is applied [e.g. Knebelman, 1951]. Considering the physical velocity ì V = (cu, s t u) in one cell with topography normal vectors ì n, the transported velocity ì V in a neighboring cell with normal vector ì n is computed with:…”

Section: The Shaltop Numerical Modelmentioning

confidence: 99%

“…We will show in Section 3.1 that the curvature force (first two terms on the right‐hand side of Equation 17) ensures the velocity remains tangent to the topography at all time. Thus, to model this effect, a tangent transport is applied (e.g., Knebelman, 1951). Considering the physical velocity

\vec{V} = (c boldu, {bolds}^{t} boldu)

in one cell with topography normal vectors

\vec{n}

, the transported velocity

\vec{V^{'}}

in a neighboring cell with normal vector

\vec{n^{'}}

is computed with:

\vec{V^{'}} = \vec{V} - \frac{\vec{V} \cdot \vec{n^{'}}}{1 + \vec{n} \cdot \vec{n^{'}}} ()(, \vec{n} + \vec{n^{'}})

…”

Section: Modeling Approach Using Thin‐layer Equationsmentioning

confidence: 99%

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Topography Curvature Effects in Thin‐Layer Models for Gravity‐Driven Flows Without Bed Erosion

et al. 2021

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The propagation of rapid gravity-driven flows (Iverson & Denlinger, 2001) occurring in mountainous or volcanic areas is a complex and hazardous phenomenon. A wide variety of events are associated with these flows, such as rock avalanches, debris avalanches and debris, mud or hyper-concentrated flows (Hungr et al., 2014). The understanding and estimation of their propagation processes is important for sediment fluxes quantification, for the study of landscapes dynamics. Besides, gravity-driven flows can have a significant economic impact and endanger local populations (Hungr et al., 2005;Petley, 2012;Froude & Petley, 2018). In order to mitigate these risks, it is of prior importance to estimate the runout, dynamic impact and travel time of potential gravitational flows.This can be done empirically, but physically based modeling is needed to investigate more precisely the dynamics of the flow, in particular due to the first-order role of local topography. Over the past decades, thin-layer models (also called shallow-water models) have been increasingly used by practitioners. Their main assumption is that the flow extent is much larger than its thickness, so that the kinematic unknowns are reduced to two variables: the flow thickness and its depth-averaged velocity. The dimension of the problem is thus lower, allowing for relatively fast numerical computations. The first and simplest form of thin-layer equations was given by Barré de Saint-Venant (1871) for almost flat topographies. The 1D formulation (i.e., for topographies given by a 1D graph Z = Z(X)) for any bed inclination and small curvatures was derived by Savage and Hutter (1991). This model has since been extended to real 2D topographies (i.e., given by a 2D graph Z = Z (X, Y)). Some of the software products based on thin-layer equations are currently used for hazard assessment to derive, for instance, maps of maximum flow height and velocity. Examples include RAMMS (

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Section: The Shaltop Numerical Modelmentioning

confidence: 99%

Section: The Shaltop Numerical Modelmentioning

confidence: 99%

\vec{V} = (c boldu, {bolds}^{t} boldu)

in one cell with topography normal vectors

\vec{n}

, the transported velocity

\vec{V^{'}}

in a neighboring cell with normal vector

\vec{n^{'}}

is computed with:

\vec{V^{'}} = \vec{V} - \frac{\vec{V} \cdot \vec{n^{'}}}{1 + \vec{n} \cdot \vec{n^{'}}} ()(, \vec{n} + \vec{n^{'}})

…”

Section: Modeling Approach Using Thin‐layer Equationsmentioning

confidence: 99%

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Topography Curvature Effects in Thin‐Layer Models for Gravity‐Driven Flows Without Bed Erosion

et al. 2021

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“…To estimate base covariance matrices, we examine several mean covariance estimation methods, including Euclidean mean of the state covariance estimates of each subject, Log Euclidean mean [15], and covariance computed on concatenated brain region time courses over cognitive states. For comparison with our proposed transport, we explore the concept of parallel transport, which provides the least deforming way of moving geometric objects along a curve between two points on a manifold [16,17]. To perform parallel transport, we use the Schild's ladder algorithm [18,19], which dates back to the 70's and has recently been revitalized for applications, such as longitudinal deformation analysis [20] and object tracking [17].…”

Section: Transport On Riemannian Manifold For Connectivity-based Braimentioning

confidence: 99%

“…Another way of transporting the state covariance matrices of all subjects to a common tangent space is to use parallel transport [16], which provides the least deforming way of moving geometric objects along a curve on a manifold [17]. One way of performing parallel transport is to use the Schild's ladder algorithm [18,19], as summarized in Fig.…”

Section: Parallel Transport With Schild's Laddermentioning

confidence: 99%

Transport on Riemannian Manifold for Connectivity-Based Brain Decoding

Varoquaux

Poline

et al. 2016

IEEE Trans. Med. Imaging

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Abstract-There is a recent interest in using functional magnetic resonance imaging (fMRI) for decoding more naturalistic, cognitive states, in which subjects perform various tasks in a continuous, self-directed manner. In this setting, the set of brain volumes over the entire task duration is usually taken as a single sample with connectivity estimates, such as Pearson's correlation, employed as features. Since covariance matrices live on the positive semidefinite cone, their elements are inherently inter-related. The assumption of uncorrelated features implicit in most classifier learning algorithms is thus violated. Coupled with the usual small sample sizes, the generalizability of the learned classifiers is limited, and the identification of significant brain connections from the classifier weights is nontrivial. In this paper, we present a Riemannian approach for connectivity-based brain decoding. The core idea is to project the covariance estimates onto a common tangent space to reduce the statistical dependencies between their elements. For this, we propose a matrix whitening transport, and compare it against parallel transport implemented via the Schild's ladder algorithm. To validate our classification approach, we apply it to fMRI data acquired from twenty four subjects during four continuous, selfdriven tasks. We show that our approach provides significantly higher classification accuracy than directly using Pearson's correlation and its regularized variants as features. To facilitate result interpretation, we further propose a non-parametric scheme that combines bootstrapping and permutation testing for identifying significantly discriminative brain connections from the classifier weights. Using this scheme, a number of neuroanatomically meaningful connections are detected, whereas no significant connections are found with pure permutation testing.

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NPTC-net: Narrow-Band Parallel Transport Convolutional Neural Networks on Point Clouds

Jin

Lai

et al. 2021

J Sci Comput

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Spaces of Relative Parallelism

Cited by 9 publications

References 0 publications

Topography Curvature Effects in Thin‐Layer Models for Gravity‐Driven Flows Without Bed Erosion

Topography Curvature Effects in Thin‐Layer Models for Gravity‐Driven Flows Without Bed Erosion

Transport on Riemannian Manifold for Connectivity-Based Brain Decoding

NPTC-net: Narrow-Band Parallel Transport Convolutional Neural Networks on Point Clouds

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