This paper presents a review of the current state-of-the-art of numerical methods for
stochastic computations. The focus is on efficient high-order methods suitable for practical …

We formulate a general framework for hp-variational physics-informed neural networks (hp-
VPINNs) based on the nonlinear approximation of shallow and deep neural networks and …

Uncertainty quantification serves a central role for simulation-based analysis of physical,
engineering, and biological applications using mechanistic models. From a broad …

Uncertainty quantification in CFD computations is receiving increased interest, due in large
part to the increasing complexity of physical models, and the inherent introduction of random …

The@ first graduate-level textbook to focus on fundamental aspects of numerical methods
for stochastic computations, this book describes the class of numerical methods based on …

Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold
particular importance in recent advances in subjects such as orthogonal polynomials …

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 …

Recently there has been a growing interest in designing efficient methods for the solution of
ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin …

We present new and review existing algorithms for the numerical integration of multivariate
functions defined over d-dimensional cubes using several variants of the sparse grid method …

We study polynomial interpolation on ad-dimensional cube, where d is large. We suggest to
use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The …