Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Hardering, Hanne; Sander, Oliver

doi:10.1007/s10208-014-9230-z

Cited by 35 publications

(73 citation statements)

References 68 publications

(72 reference statements)

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“…We emphasize that using the Riemannian metric (5) makes the powerful machinery of manifolds available for which a lot of theory and methodology has been developed. Examples include, but are not limited to, wavelet-type multiscale transforms [71][72][73], manifold-valued partial differential equations [74], and statistics on Riemannian manifolds [60,[75][76][77][78][79].…”

Section: Discussionmentioning

confidence: 99%

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

Baust

Weinmann

Wieczorek

et al. 2016

IEEE Trans. Med. Imaging

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In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.

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Section: Discussionmentioning

confidence: 99%

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

Baust

Weinmann

Wieczorek

et al. 2016

IEEE Trans. Med. Imaging

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show abstract

“…Note that for every w ∈ [0, 1], ∇u(w) : R → T u(w) M and ∇u 1 = [0,1] ∇u(w) dw. Now we can formulate a discrete version of the problem (4.1) by restricting the space of functions to V M h which is the space of all geodesic finite element functions for M associated with a regular grid on [0, 1]; see [18,38]. We refer to [38] for the definition of geodesic finite element spaces V M h .…”

Section: Denoising On a Riemannian Manifoldmentioning

confidence: 99%

ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

Grohs

Hosseini

2015

Adv Comput Math

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This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping x → ∂ ε f (x) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein-ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.

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“…Relationship to geodesic finite elements. Projection-based finite elements are closely related to the geodesic finite elements proposed in [19,33,34] and analyzed in [20,21]. Geodesic finite elements are constructed by replacing polynomial interpolation of values (c i ) i∈I…”

Section: 2mentioning

confidence: 99%

“…The proper quantity is the homogeneous norm | · | Wm ,p + | · |m W 1,mp , known, e.g., from [9]. It replaces the unwieldy smoothness descriptor used in corresponding results for geodesic finite elements [20].…”

Section: -Valued Interpolation We Now Turn To Error Bounds For Thementioning

confidence: 99%

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Projection-Based Finite Elements for Nonlinear Function Spaces

Hardering¹,

Sander²,

Sprecher³

2019

SIAM J. Numer. Anal.

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We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise onto the manifold. We show optimal interpolation error bounds with respect to Lebesgue and Sobolev norms. Additionally, we show similar bounds for the test functions, i.e., variations of discrete functions. Combining these results with a nonlinear Céa lemma, we prove optimal L 2 and H 1 discretization error bounds for harmonic maps from a planar domain into a smooth manifold. All these error bounds are also verified numerically.We investigate the discrete approximation of functions from a Euclidean domain Ω to a closed embedded submanifold M of R n , n ∈ N. Such functions are involved in a variety of partial differential equations (PDEs), from fields like liquid crystal physics [4] and micromagnetics [13]. In these applications, the manifold is M = S 2 , the unit sphere in R 3 . In Cosserat-type material models [29,30,36] the manifold is M = R 3 ×SO(3), where SO(3) is the special orthogonal group. Further examples are the investigation of harmonic maps into manifolds [6], signal processing of manifoldvalued signals [32], and the denoising of manifold-valued images [5].We are interested in functions of Sobolev smoothness. By this we mean functions from spaceswhere we denote by W k,p (Ω, R n ) the standard Sobolev space for k ∈ N and p ∈ [1, ∞]. Throughout the paper, | · | W l,p and · W l,p will denote the corresponding Sobolev semi norm and full norm of R n -valued functions, respectively.Spaces of approximating functions will be constructed by pointwise projection. Given a finite element grid of Ω, and a set of values c i ∈ M ⊂ R n at Lagrange points on Ω, we construct nonlinear finite element functions by first interpolating in R n by piecewise polynomials in R n , and then projecting pointwise onto M . This results in a finite-dimensional set of functions V h (Ω, M ) which, as it turns out, is a subspace of W 1,p for arbitrary p ≥ 1. While the approach presented here is based on Lagrangian interpolation in R n , other linear FE space can be used in principle (see [37] for an example).The idea to generalize finite elements spaces by a pointwise projection operator has already appeared several times [16,35,37]. For functions taking values in 1 arXiv:1803.06576v1 [math.NA]

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Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Cited by 35 publications

References 68 publications

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

Projection-Based Finite Elements for Nonlinear Function Spaces

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