Breaking the Curse of Dimensionality: Polynomial Chaos for Exploratory Modelling

In light of worsening climate change and an increased interest in adapting infrastructure to cope with its effects, model-based decision support has become an essential tool for policy makers. In conditions of deep uncertainty, models may be used to explore a large space of possible system behaviours and so encourage a wider consideration of the...

A Novel Method for Solving High-Dimensional Backward Stochastic Differential Equations Using Malliavin Calculus and Deep Learning

Backward stochastic differential equations (BSDE) are known to be a powerful tool in mathematical modeling due to their inherent connection with second-order parabolic partial differential equations (PDE) established by the non-linear Feynman-Kac relations. The fundamental power of BSDEs lies in the fact that with them one does not merely obtain...

Tensor Network B-splines for high-dimensional function approximation

B-splines are basis functions for the spline function space and are extensively used in applications requiring function approximation. The generalization of B-splines to multiple dimensions is done through tensor products of their univariate basis functions. The number of basis functions and weights that define a multivariate B-spline surface,...

Nonlinear system identification with regularized Tensor Network B-splines

This article introduces the Tensor Network B-spline (TNBS) model for the regularized identification of nonlinear systems using a nonlinear autoregressive exogenous (NARX) approach. Tensor network theory is used to alleviate the curse of dimensionality of multivariate B-splines by representing the high-dimensional weight tensor as a low-rank...

Tensor Network Kalman Filter for Large-Scale MIMO Systems: With Application to Adaptive Optics

For large-scale system with tens of thousands of states and outputs the computation in the conventional Kalman filter becomes time-consuming such that Kalman filtering in large-scale real-time application is practically infeasible. A possible mathematical framework to lift the curse of dimensionality is to lift the problem in higher dimensions...

Tensor network Kalman filter for LTI systems

An extension of the Tensor Network (TN) Kalman filter [2], [3] for large scale LTI systems is presented in this paper. The TN Kalman filter can handle exponentially large state vectors without constructing them explicitly. In order to have efficient algebraic operations, a low TN rank is required. We exploit the possibility to approximate the...