Quasi-Interpolation in a Space of C2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class C2 on the whole domain but fourth-order regularity is required at vertices and C3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

On the Raşa Inequality for Higher Order Convex Functions

We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for qth convolution power of the difference of probability distributions $$\mu $$ μ and $$\nu $$ ν $$\begin{aligned} (\nu -\mu )^{*q}\ge _{(q-1)cx} 0 , \quad q\ge 2, \end{aligned}$$ ( ν - μ ) ∗ q ≥ ( q - 1 ) c x 0 , q ≥ 2 , and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

$$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

A finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$ H 1 -inner product is introduced. It yields $$H^1$$ H 1 -conforming finite element spaces with exterior derivatives in $$H^1$$ H 1 . We use a tensor product construction to obtain $$L^2$$ L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.

Fast Linear Interpolation

We present fast implementations of linear interpolation operators for piecewise linear functions and multi-dimensional look-up tables. These operators are common for efficient transformations in image processing and are the core operations needed for lattice models like deep lattice networks, a popular machine learning function class for interpretable, shape-constrained machine learning. We present new strategies for an efficient compiler-based solution using MLIR to accelerate linear interpolation. For real-world machine-learned multi-layer lattice models that use multidimensional linear interpolation, we show these strategies run 5-10× faster on a standard CPU compared to an optimized C++ interpreter implementation.