Optimal adaptive ridgelet schemes for linear advection equations

Obermeier, Axel

doi:10.1016/j.acha.2015.06.003

Cited by 5 publications

(7 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ridgelets turned out to be particularly well-suited for discretising (1.2) - [Gro11] essentially proved (II) of the ingredients mentioned above, while [GO15] showed (I) & (III). In this sense, the present work complements and completes these previous works by aiming to show the last remaining ingredient (IV), which will then allow to achieve the desired highly efficient algorithms, see Theorem 1.2.…”

Section: • Discontinuous Galerkin Methodsmentioning

confidence: 94%

On the approximation of functions with line singularities by ridgelets

Obermeier

2019

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal -by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate.Structurally, the proof resembles [Can01], where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes [CDD01] we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

show abstract

Section: • Discontinuous Galerkin Methodsmentioning

confidence: 94%

On the approximation of functions with line singularities by ridgelets

Obermeier

2019

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…So far, some first attempts have been made in utilizing anisotropic frames for solving elliptic PDEs. In [28,31] optimal ridgelet-based solvers were developed for linear advection equations, whereas in [17,19] shearlet-based solvers for general advection equations were developed. Although these works constitute major successes in advancing anisotropic frame-based solvers, it was not possible to impose boundary conditions.…”

Section: Motivation: Adaptive Frame Methodsmentioning

confidence: 99%

Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

Petersen

Raslan

2019

Adv Comput Math

View full text Add to dashboard Cite

In this paper, we study a newly developed hybrid shearlet-wavelet system on bounded domains which yields frames for H s (Ω) for some s ∈ N, Ω ⊂ R 2 . We will derive approximation rates with respect to H s (Ω) norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-wavelet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties.

show abstract

“…Notable first steps towards anisotropic frame systems for the numerical solution of PDEs have already been taken in [31,34], where optimal adaptive ridgelet-based solvers are constructed for linear advection equations and [15,16,18], where a shearlet-based construction is used to solve general advection equations. Also related is the work [22,3], where frames of wave atoms, respectively curvelets, are used for the efficient representation and computation of wave propagators.…”

Section: Adaptive Schemes and Framesmentioning

confidence: 99%

Anisotropic multiscale systems on bounded domains

Kutyniok

Petersen

et al. 2020

Adv Comput Math

Self Cite

View full text Add to dashboard Cite

We provide a construction of multiscale systems on a bounded domain Ω ⊂ R 2 coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct boundary shearlet systems that form frames for the Sobolev spaces H s (Ω), s ∈ N ∪ {0}, with controllable frame bounds and admit optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuity. We show that the constructed systems allow to incorporate boundary conditions. Furthermore, for s ≥ 0 and f ∈ H s (Ω) we prove that weighted ℓ 2 norms of the L 2 −analysis coefficients of f are equivalent to its H s (Ω) norm. This yields in particular, that the reweighted systems are frames also for H −s (Ω). Moreover, we demonstrate numerically, that the associated L 2 −synthesis operator is also stable as a map to H s (Ω) which, in combination with the previous result, strongly indicates that these systems constitute so-called Gelfand frames for (H s (Ω), L 2 (Ω), H −s (Ω)).

show abstract

Optimal adaptive ridgelet schemes for linear advection equations

Cited by 5 publications

References 22 publications

On the approximation of functions with line singularities by ridgelets

On the approximation of functions with line singularities by ridgelets

Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

Anisotropic multiscale systems on bounded domains

Contact Info

Product

Resources

About