Published in Advances in Computational Mathematics
In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimate...
Published in Advances in Computational Mathematics
This paper considers multivariate deconvolution density estimations under the local Hölder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attai...
Published in Advances in Computational Mathematics
Polynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requir...
Published in Advances in Computational Mathematics
A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical te...
Published in Advances in Computational Mathematics
In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The beauty of the method lies in the fact that the same operator is used for the approximat...
Published in Advances in Computational Mathematics
A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions fro...
Published in Advances in Computational Mathematics
We consider a primal-dual algorithm for minimizing f(x)+h□l(Ax)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f({\mathbf {x}})+h\square l({\mathbf {A}}{\mathbf {x}})$\e...
Published in Advances in Computational Mathematics
In this work, we consider the tensor completion problem of an incomplete and noisy observation. We introduce a novel completion model using bilevel minimization. Therefore, bilevel model-based denoising for the tensor completion problem is proposed. The denoising and completion tasks are fully separated. The upper-level directly addresses the compl...
Published in Advances in Computational Mathematics
We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimens...
Published in Advances in Computational Mathematics
In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the l∞(0,T;H)\documentclass[12pt]{m...
