Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

A characterization of nonhomogeneous wavelet bi-frames for reducing subspaces of Sobolev spaces

For nonhomogeneous wavelet bi-frames in a pair of dual spaces $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$ ( H s ( R d ) , H − s ( R d ) ) with $s\neq 0$ s ≠ 0 , smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other one for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in $L^{2}(\mathbb{R}^{d})$ L 2 ( R d ) . In this paper, we introduce the reducing subspaces of Sobolev spaces, and characterize the nonhomogeneous wavelet bi-frames under the setting of a general pair of dual reducing subspaces of Sobolev spaces.

Design Space Exploration of Stagnation Temperature Probes via Dimension Reduction

The measurement of stagnation temperature is important for turbomachinery applications as it is used in the calculation of component efficiency and engine specific fuel consumption. This paper examines the use of polynomial variable projection to identify dimension reducing subspaces for stagnation temperature probes. As an example application we focus on a simplified Kiel probe geometry, but the proposed data-centric approach could be readily applied to new datasets with different geometries, boundary conditions and design objectives. The design of Kiel probes is non-trivial, with a large design space, complex flow physics, and competing design objectives. Two design objectives are considered: (1) the stagnation pressure loss, to reduce instrumentation losses; (2) the change in recovery ratio with respect to Mach number, to reduce temperature measurement uncertainty. Subspaces are obtained for the two design objectives, allowing the influence of seven design parameters to be understood. The entropy generation rate is used to provide physical insights into loss mechanisms. The recovery ratio subspace indicates that for the present probe there is an optimum vent-to-inlet area which minimises the change in recovery ratio with respect to Mach number, and design modifications that yield further small improvements are explored. Finally, the uncertainty in recovery ratio due to manufacturing variability is shown to be important. In comparison to global sensitivity measures, the use of an active subspace is shown to provide important information on what manufacturing tolerances are important for specific designs. New designs can also be selected that are insensitive to given manufacturing tolerances.

Supporting multi-point fan design with dimension reduction

Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations. First, we show that these subspaces exist across different blades—under the same parametrisation—largely independent of their Mach number or Reynolds number. This is demonstrated via a numerical study on three different blades. Then, in an attempt to reduce the computational cost of identifying a suitable dimension reducing subspace, we examine statistical sufficient dimension reduction methods, including sliced inverse regression, sliced average variance estimation, principal Hessian directions and contour regression. Unsatisfied by these results, we evaluate a new idea based on polynomial variable projection—a non-linear least-squares problem. Our results using polynomial variable projection clearly demonstrate that one can accurately identify dimension reducing subspaces for turbomachinery functionals at a fraction of the cost associated with prior methods. We apply these subspaces to the problem of comparing design configurations across different flight points on a working line of a fan blade. We demonstrate how designs that offer a healthy compromise between performance at cruise and sea-level conditions can be easily found by visually inspecting their subspaces.

Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

It can be seen from the literature that nonhomogeneous wavelet frames are much simpler to characterize and construct than homogeneous ones. In this work, we address such problems in reducing subspaces of L2ℝd. A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.