Speaker
Description
Standard numerical methods for solving PDEs typically suffer from the curse of dimensionality: their computational cost scales exponentially with the dimension of the underlying domain, making them impractical even at low resolution. In many cases of interest, however, such limitations can be overcome by appropriately compressed representations of approximate solutions, for example, by low-rank tensor representations.
For time-dependent PDEs, several approaches to low-rank approximation exist that control ranks in different ways. These range from methods that keep ranks fixed, such as dynamical low-rank approximations, which may lead to uncontrolled errors, to methods that approximate standard time-stepping schemes to any desired accuracy but can produce unnecessarily large ranks.
We develop time integration methods in low‑rank tensor representations that adaptively adjust the approximation ranks to meet a prescribed accuracy while simultaneously maintaining control over the ranks of the computed approximations and all intermediate quantities. These ranks are a main determining factor in the computational costs of such methods. Our approach combines an iterative time-stepping scheme with soft thresholding of the iterates. In the matrix case, the proposed strategy yields iterates whose ranks remain comparable to natural benchmark quantities — namely, the best approximation ranks of the sought solutions at the achieved accuracy. In the higher-dimensional tensor case the algorithm can be adapted appropriately, leading to global error and rank bounds that depend only polynomially on the dimension. Numerical experiments illustrate the theory for linear time-dependent Schrödinger equations.