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 …

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

DeepONets have recently been proposed as a framework for learning nonlinear operators
mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove …

We develop a general framework for data-driven approximation of input-output maps
between infinitedimensional spaces. The proposed approach is motivated by the recent …

This text provides a framework in which the main objectives of the field of uncertainty
quantification (UQ) are defined and an overview of the range of mathematical methods by …

These lecture notes highlight the mathematical and computational structure relating to the
formulation of, and development of algorithms for, the Bayesian approach to inverse …

We estimate the expressive power of certain deep neural networks (DNNs for short) on a
class of countably-parametric, holomorphic maps u: U→ ℝ on the parameter domain U=[− 1 …

Parametric partial differential equations are commonly used to model physical systems.
They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo …

Abstract In Monte Carlo methods quadrupling the sample size halves the error. In
simulations of stochastic partial differential equations (SPDEs), the total work is the sample …

For a parameter dimension d∈ N, we consider the approximation of many-parametric maps
u:[-1, 1] d→ R by deep ReLU neural networks. The input dimension d may possibly be large …