High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

Abstract

In the present work, we analyze a technique designed by Geraci et al. in [1,11] named the Truncate and Encode (TE) strategy. It was presented as a non-intrusive method for steady and non-steady Partial Differential Equations (PDEs) in Uncertainty Quantification (UQ), and as a weakly intrusive method in the unsteady case.We analyze the TE algorithm applied to the approximation of functions, and in particular its performance for piecewise smooth functions. We carry out some numerical experiments, comparing the performance of the algorithm when using different linear and non-linear interpolation techniques and provide some recommendations that we find useful in order to achieve a high performance of the algorithm.

Multiresolution analysis as a criterion for effective dynamic mesh adaptation – A case study for Euler equations in the SAMR framework AMROC

“…It is also possible to use a constant threshold value for all levels. For instance, in [31] it was found that for multiresolution techniques applied for UQ using a truncation and encode approach thresholding with constant values could be more efficient than the level dependent choice, proposed by Harten [32]. In the context of adaptive stochastic problems Abgrall et al [33] used however a level dependent threshold.…”

A Multiresolution approach to solve large-scale optimization problems

“…Harten's MRF has been successfully applied for different purposes and in several scenarios (see e.g. [18,1,5,14,13,7]) but, to the best of our knowledge, the applications carried out in [19,28] constitute the first attempt to use Harten's MRF in connection with constrained/unconstrained optimization. The aim of this paper is to provide a complete mathematical description of the technique used in [19,28] to solve large-scale optimization problems of the type eq.…”

A Harten's Multiresolution Framework for Subdivision Schemes

“…Despite that the subdivision was conceived with a geometrical purpose in CAGD [12], other applications have been adopted it because of its easy implementation and flexibility to reach special properties. We are interested in the presence of subdivision in multiresolution algorithms, with applications in image processing [2,6], optimization [10,14] and uncertainty quantification [8], among others.…”

“…Originally, Harten's Mulitiresolution Framework (HMR-F) [13] provided a set of tools that allows to define a consistent multi-scale structure for numerical methods for conservation laws. Nevertheless, this theory is prepared for very general multi-scale scenarios and over the years it were found applications in other mathematical fields, for instance in the above mentioned applications [2,6,10,8,14], where it was combined with subdivision schemes.…”

“…In other applications, however, the prediction operators are subdivision schemes and they seem to be more relevant than the decimation operators. In fact, in [8,10,14], the decimation does not appear in the implementation of the methods, despite being needed to describe the multiresolution structure. The aim of this paper is to give theoretical support to this procedure.…”