Low-rank tensor approaches for adaptive function approximation: algorithms and examples

In this talk, we present an adaptive method for approximating high-dimensional low-rank functions. Taking advantage of low-rank structure in approximation problems has been shown to prove advantageous for scaling numerical algorithms and computation to higher dimensions by mitigating the curse-of-dimensionality. The method we describe is an extension of the tensor-train cross approximation algorithm to the continuous case of multivariate functions that enables both global and local adaptivity. Our approach relies on a new adaptive algorithm for computing the CUR/skeleton decomposition of bivariate functions. We then extend this technique to the multidimensional case of the function-train decomposition. We demonstrate the benefits of our approach compared with the standard methodology that computes low-rank approximations by decomposing coefficients of tensor-product basis functions. We finish by demonstrating a wide range of applications that include machine learning, uncertainty quantification, stochastic optimal control, and Bayesian filtering.