This paper is concerned with near-optimal approximation of a given univariate function with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Bin...
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This paper is concerned with near-optimal approximation of a given univariate function with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.
This paper is concerned with new discretization methods for the numerical treatment of elliptic partial differential equations. We derive an adaptive approximation scheme that is based on frames of quarkonial type, wh...
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This paper is concerned with new discretization methods for the numerical treatment of elliptic partial differential equations. We derive an adaptive approximation scheme that is based on frames of quarkonial type, which can be interpreted as a wavelet version of hp finite element dictionaries. These new frames are constructed from a finite set of functions via translation, dilation and multiplication by monomials. By using nonoverlapping domain decomposition ideas, we establish quarkonial frames on domains that can be decomposed into the union of parametric images of unit cubes. We also show that these new representation systems are stable in a certain range of Sobolev spaces. The construction is performed in such a way that, similar to the wavelet setting, the frame elements, the so-called quarklets, possess a certain number of vanishing moments. This enables us to generalize the basic building blocks of adaptive wavelet algorithms to the quarklet case. The applicability of the new approach is demonstrated by numerical experiments for the Poisson equation on L-shaped domains.
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