Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Adaptive tensor product wavelet methods for solving PDEs

Tensor sparsity of solutions of high dimensional PDEs

We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators....

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a co...

Nonlinear Adaptive Algorithms on Rank-One Tensor Models

This work proposes a low complexity nonlinearity model and develops adaptive algorithms over it. The model is based on the decomposable—or rank-one, in tensor language— Volterra kernels. It may also be described as a product of FIR filters, which explains its low-complexity. The rank-one model is also interesting because it comes from a well-posed problem in approximation theory. The paper uses...

Infinite dimensional backstepping for nonlinear parabolic PDEs