Low-Rank Tucker Matrix Methods for Linear Systems and Eigenvalue Problems

In this article, we delve into the realm of high-order numerical methods and explore the concept of Truncated Polynomial Cubic Grids (TPCG). Developed by the author and their collaborators, TPCG represents a significant advancement in the field of tensor calculus. By leveraging the power of polynomial cubic grids, we can efficiently solve complex problems involving Tucker tensors, which are multilinear structures with multiple factors and ranks.
The core idea behind TPCG is to represent the solution tensor as an average of a coarse grid of points, evaluated at different levels of refinement. This allows us to truncate the tensor without significantly affecting its accuracy while significantly reducing computational cost. We employ block-wise relative truncation with tolerance ηk = βtol to maintain the multilinear rank of the iterates, ensuring that the solution remains accurate despite the truncation.
To illustrate the effectiveness of TPCG, we discuss several examples and applications, including the computation of optimal control systems, fluid dynamics, and image processing. We also highlight the advantages of TPCG over existing methods, such as its ability to handle complex geometry and non-Tucker tensors.
In conclusion, Truncated Polynomial Cubic Grids offer a powerful tool for solving tensor calculus problems with high accuracy and efficiency. By harnessing the power of polynomial cubic grids and block-wise relative truncation, we can tackle even the most challenging problems in numerical analysis and scientific computing.