One of the open problems in scientific computing is the long-time integration of nonlinear
stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We …

Over seventy years ago, Richard Bellman coined the term “the curse of dimensionality” to
describe phenomena and computational challenges that arise in high dimensions. These …

We give an overview of four different ensemble‐based techniques for uncertainty
quantification and illustrate their application in the context of oil plume simulations. These …

Polynomial approximations constructed using a least-squares approach form a ubiquitous
technique in numerical computation. One of the simplest ways to generate data for least …

This work proposes and analyzes a compressed sensing approach to polynomial
approximation of complex-valued functions in high dimensions. In this context, the target …

In this work we focus on the construction of numerical schemes for the approximation of
stochastic mean--field equations which preserve the nonnegativity of the solution. The …

We propose, theoretically investigate, and numerically validate an algorithm for the Monte
Carlo solution of least-squares polynomial approximation problems in a collocation …

We investigate a gradient enhanced ℓ 1-minimization for constructing sparse polynomial
chaos expansions. In addition to function evaluations, measurements of the function …

In recent years, the use of sparse recovery techniques in the approximation of high-
dimensional functions has garnered increasing interest. In this work we present a survey of …

We introduce and analyse a framework for function interpolation using compressed sensing.
This framework—which is based on weighted ℓ^ 1 ℓ 1 minimization—does not require a …