A nonlinear multiresolution scheme within Harten's framework [12], [13] is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.
This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200-212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2): 198-206, 2005; Serna and Marquina, J Comput Phys 194:632-658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165-187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).
This letter is devoted to the stability of the so-called piecewise polynomial harmonic (PPH) multiresolution transform that belongs to the class of data dependent nonlinear multiresolution algorithms. The presentation of the PPH multiresolution as some specific perturbation of a linear multiresolution allows to establish a two step contraction property that leads first to a convergence result and finally to the stability.
Laser doppler anemometry (LDA) measurements and numerical calculations have been made for a laminar boundary layer on triangular riblets. Calculated mean velocity distributions along the riblet contour are in good agreement with the measured ones. The results show that no transversal motion exists above and within the riblet valleys (e.g., no secondary motion). It is found that despite the large wetted area increase, the frictional drag is not increased on riblets relative to a smooth wall. This result suggests that the viscous effects are at play in the drag reduction for a turbulent boundary layer, in the sense that they compensate for the increase in wetted area.
Gaussian process (GP) models have become popular for approximating and exploring nonlinear systems using scarce input/output samples and prior hypotheses done through mean and covariance functions. While it is common to make stationarity assumptions and use variance-based criteria for exploration, in realistic cases it is not rare that systems under study exhibit a heterogeneous behavior depending on regions of the parameter space. We consider a class of problems where high variations occur along unknown noncanonical directions and we tackle the problem of accommodating nonstationarity from two angles. First we define a novel class of covariances (WaMI-GP) that simultaneously generalizes kernels of multiple index and of tensorized warped GPs, and second, we introduce derivative-based sampling criteria dedicated to the exploration of high-variation regions. The novel GP class is investigated through both mathematical analysis and numerical experiments, and it is shown that it allows encoding much expressiveness while keeping the number of parameters to be inferred moderate. Criteria and models are compared on a mechanical test case from safety studies conducted by IRSN. On this application some of the proposed criteria outperform usual variance-based criteria in the case of a stationary GP model; however, variance-based criteria with WaMI-GP perform even better. Our method is also compared with the treed Gaussian processes (TGP) on this application and on a NASA test case. In the IRSN application, WaMI-GP dominates TGP in static and sequential settings. In the NASA application, while TGP clearly dominates in the static case, for small designs it is outperformed by WaMI-GP in the sequential setup.
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