Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

Ma, Tianyu; Matveev, Vladimir S.; Pavlyukevich, Ilya

doi:10.1007/s12220-021-00723-z

Cited by 4 publications

(2 citation statements)

References 42 publications

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“…Jørgensen proved that on complete Riemannian manifolds that satisfy mild assumptions (which in particular hold for compact manifolds), the scaling limit of such a geodesic random walk is the Brownian motion on (M, g), see [16]. This result has recently been generalized to the setting of Finsler manifolds, see [20].…”

Section: Introductionmentioning

confidence: 99%

Efficient Random Walks on Riemannian Manifolds

Herrmann¹,

Schwarz²,

Sturm³

et al. 2022

Preprint

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According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge to the correct Brownian motion in the Skorokhod topology as long as the geodesic equation is approximated up to second order. As a result we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.

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Section: Introductionmentioning

confidence: 99%

Efficient Random Walks on Riemannian Manifolds

Herrmann¹,

Schwarz²,

Sturm³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Available books discuss applications in optics, thermodynamics, and biology [11], as well as modern physical settings, including spinor-type structures [12]. Finsler geometry and its generalizations have also been used for describing anisotropic space-time, general relativity, quantum fields, gravitation, electromagnetism, and diffusion [13][14][15][16][17][18][19]. The current work implements a continuum mechanical framework for the physical response of solid bodies.…”

Section: Introductionmentioning

confidence: 99%

Generalized Finsler Geometry and the Anisotropic Tearing of Skin

Clayton

2023

Symmetry

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A continuum mechanical theory with foundations in generalized Finsler geometry describes the complex anisotropic behavior of skin. A fiber bundle approach, encompassing total spaces with assigned linear and nonlinear connections, geometrically characterizes evolving configurations of a deformable body with the microstructure. An internal state vector is introduced on each configuration, describing subscale physics. A generalized Finsler metric depends on the position and the state vector, where the latter dependence allows for both the direction (i.e., as in Finsler geometry) and magnitude. Equilibrium equations are derived using a variational method, extending concepts of finite-strain hyperelasticity coupled to phase-field mechanics to generalized Finsler space. For application to skin tearing, state vector components represent microscopic damage processes (e.g., fiber rearrangements and ruptures) in different directions with respect to intrinsic orientations (e.g., parallel or perpendicular to Langer’s lines). Nonlinear potentials, motivated from soft-tissue mechanics and phase-field fracture theories, are assigned with orthotropic material symmetry pertinent to properties of skin. Governing equations are derived for one- and two-dimensional base manifolds. Analytical solutions capture experimental force-stretch data, toughness, and observations on evolving microstructure, in a more geometrically and physically descriptive way than prior phenomenological models.

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Efficient Random Walks on Riemannian Manifolds

Schwarz,

Herrmann,

Sturm

et al. 2023

Found Comput Math

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According to a version of Donsker’s theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge in distribution to the correct Brownian motion as long as the geodesic equation is approximated up to second order. As a result, we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.

show abstract

Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

Abstract: We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

Cited by 4 publications

References 42 publications

Efficient Random Walks on Riemannian Manifolds

Efficient Random Walks on Riemannian Manifolds

Generalized Finsler Geometry and the Anisotropic Tearing of Skin

Efficient Random Walks on Riemannian Manifolds

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