Density order of Parseval wavelet frames from extension principles

We characterize approximation order and density order of those Parseval wavelet frames obtained from Oblique Extension Principle. These notions are closely related to approximation order and density order by a quasi-projection operator. To give our characterizations, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of a refinable function. In particular, we invoke the classical notion of approximate continuity. We write our results in the multivariate context of Parseval wavelet frames associated to A, an expansive linear map preserving the integer lattice

Energy conservative isogeometric techniques for the wave equation

We analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is replaced by a strong form, written in terms of a projection operator. The system of equations is discretized with B-splines forming a De Rham complex along with suitable commutative projectors for the approximation of the second equation. The discrete scheme is energy conservative when discretized in time with a conservative method such as Crank-Nicolson. We propose a convergence analysis of the method to study the dependence with respect to the mesh size $h$, with focus on the consistency error. Numerical results show optimal convergence of the error in energy norm, and a relative error in energy conservation for long-time simulations of the order of machine precision

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors