Low-rank tensor representations can provide highly compressed approximations of
functions. These concepts, which essentially amount to generalizations of classical …

We present a survey of the fundamentals and the applications of sparse grids, with a focus
on the solution of partial differential equations (PDEs). The sparse grid approach, introduced …

Adaptive Finite Element Methods for numerically solving elliptic equations are used often in
practice. Only recently [12],[17] have these methods been shown to converge. However, this …

Data oscillation is intrinsic information missed by the averaging process associated with
finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data …

This paper reviews the state of the art and discusses recent developments in the field of
adaptive isogeometric analysis, with special focus on the mathematical theory. This includes …

Adaptive finite element methods (FEMs) have been widely used in applications for over 20
years now. In practice, they converge starting from coarse grids, although no mathematical …

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random
coefficients on a bounded domain D⊂ ℝ d are introduced and their convergence rates are …

Parametrized families of PDEs arise in various contexts such as inverse problems, control
and optimization, risk assessment, and uncertainty quantification. In most of these …

Wavelet analysis is a branch of applied mathematics that has produced a collection of tools
designed to process certain signals and images. This new book is devoted to describing …

Finite element solvers are a basic component of simulation applications; they are common in
computer graphics, engineering, and medical simulations. Although adaptive solvers can be …