Related papers: Hard thresholding hyperinterpolation over general …
We prove that the classic approximation guarantee for the higher-order singular value decomposition (HOSVD) is tight by constructing a tensor for which HOSVD achieves an approximation ratio of $N/(1+\varepsilon)$, for any $\varepsilon > 0$.…
This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz…
This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional…
We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial…
Iterative thresholding algorithms seek to optimize a differentiable objective function over a sparsity or rank constraint by alternating between gradient steps that reduce the objective, and thresholding steps that enforce the constraint.…
This paper focuses on developing a framework for constructing quasi-interpolation with the highest achievable approximation order from generalized Gaussian kernels with the help of kernel restriction trick and periodization technique. We…
A new generalization of shifted thin plate splines $$\varphi(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ is presented to increase the accuracy of quasi-interpolation further. With…
The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and…
In this paper, a new class of \emph{Taylor-accelerated neural network interpolation operators} is introduced on quasi-uniform irregular grids. These operators improve existing neural network interpolation operators by incorporating Taylor…
In this paper, we consider a non-convex problem which is the sum of $\ell_0$-norm and a convex smooth function under box constraint. We propose one proximal iterative hard thresholding type method with extrapolation step used for…
We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from…
We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every…
We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
In this paper, we consider the $\ell_0$ minimization problem whose objective function is the sum of $\ell_0$-norm and convex differentiable function. A variable metric type method which combines the PIHT method and the skill in quasi-newton…
Tensor Train (TT) decompositions provide a powerful framework to compress grid-structured data, such as sampled function values, on regular Cartesian grids. Such high compression, in turn, enables efficient high-dimensional computations.…
This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific…
We consider the well-known method of least squares on an equidistant grid with $N+1$ nodes on the interval $[-1,1]$ with the goal to approximate a function $f\in\mathcal{C}\left[-1,1\right]$ by a polynomial of degree $n$. We investigate the…