Partial differential equations with more than three coordinates arise naturally if the model features certain kinds of stochasticity. Typical examples are the Schroedinger, Fokker-Planck and Master equations in quantum mechanics or cell biology, as well as quantification of uncertainty.
The principal difficulty of a straightforward numerical solution of such equations is the `curse of dimensionality': the storage cost of the discrete solution grows exponentially with the number of coordinates (dimensions).
One way to reduce the complexity is the low-rank separation of variables. One can see all discrete data (such as the solution) as multi-index arrays, or tensors. These large tensors are never stored directly.
We approximate them by a sum of products of smaller factors, each carrying only one of the original variables. I will present one of the simplest but powerful of such representations, the Tensor Train (TT) decomposition. The TT decomposition generalizes the approximation of a given matrix by a low-rank matrix to the tensor case. It was found that many interesting models allow such approximations with a significant reduction of storage demands.
A workhorse approach to computations with the TT and other tensor product decompositions is the alternating optimization of factors. The simple realization is however prone to convergence issues.
I will show some of the recent improvements that are indispensable for really many dimensions, or solution of linear systems with non-symmetric or indefinite matrices.
- Computational Mathematics and Applications Seminar