Tensorized block rational Krylov methods for tensor Sylvester equations (2306.00705v1)

Abstract: We introduce the definition of tensorized block rational Krylov subspaces and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in [Kressner D., Tobler C., Krylov subspace methods for linear systems with tensor product structure, SIMAX, 2010]. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or Tensor Train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis, some strategies for pole selection, and techniques to efficiently compute the residual.

• The paper develops tensorized block rational Krylov methods specifically designed to efficiently solve tensor Sylvester equations where the right-hand side has a low multilinear or Tensor Train rank.
• Key methodological contributions include the introduction of adaptive pole selection strategies crucial for enhancing convergence rates and an efficient block rational Arnoldi process for residual computation.
• Numerical results demonstrate the method's effectiveness, showing that the required number of Arnoldi iterations remains nearly constant as problem dimensions increase, making it suitable for large-scale applications like discretized PDEs.

Overview of Tensorized Block Rational Krylov Methods for Tensor Sylvester Equations

The paper "Tensorized block rational Krylov methods for tensor Sylvester equations" by Angelo A. Casulli addresses the development and analysis of efficient numerical algorithms for solving tensor Sylvester equations, where the right-hand side of the equation is represented with low multilinear or Tensor Train (TT) rank. Building upon the foundational work by Kressner and Tobler, the paper extends the utility of Krylov subspace methods, adapting them through the incorporation of rational block Krylov techniques tailored to tensor structures.

Description of Tensor Sylvester Equations

A tensor Sylvester equation is a multidimensional extension of matrix Sylvester equations and can be succinctly expressed as:

$$\mathcal{X}\times_1 A_1 + \mathcal{X}\times_2 A_2 + \cdots + \mathcal{X}\times_d A_d = \mathcal{C},$$

where $\mathcal{X}$ denotes the unknown tensor, $A_i$ are known square matrices for each mode, and $\mathcal{C}$ is a tensor given with low rank structure either in Tucker or TT format. The focus of the paper is on efficiently solving such equations by leveraging the properties of tensorized block rational Krylov subspaces.

Methodological Advances

The key methodological contribution of this paper lies in the introduction of tensorized block rational Krylov subspaces that extend the standard Krylov subspace methods to handle the unique challenges posed by the tensor Sylvester equation. These subspaces allow for:

1. Block Rational Krylov Subspaces Definition: Extending rational Krylov subspaces to block versions where matrix polynomials are used for projecting the problem into a lower-dimensional subspace conducive for computation.
2. Adaptive Pole Selection: Implementing pole selection strategies that are crucial for enhancing convergence rates. By choosing poles effectively in the context of rational Krylov methods, the convergence can be significantly improved, as analyzed in the work through various adaptive strategies.
3. Efficient Residual Computation: Utilizing the block rational Arnoldi process, the method introduces a systematic approach for efficiently calculating residuals, allowing the monitoring of convergence without excessive computational overheads.

Numerical Results and Performance

The paper provides a detailed examination through numerical experiments comparing several methods of pole selection such as the use of adaptive strategies denoted as det and det2. The performance of these strategies was pivotal in demonstrating the efficacy of tensorized block rational Krylov methods, particularly in higher-dimensional cases. Numerical experiments indicated that the number of Arnoldi iterations required remained nearly constant as the problem dimension increased, although the overall runtime could grow due to operations on increasingly large matrices.

Implications and Future Directions

This work not only holds significant implications for the practical solution of large-scale multilinear algebra problems that arise in applications like discretized PDEs but also pushes the boundary of what Krylov-based methods can achieve in the context of tensors. The adaptability and efficiency of the proposed methods make them suitable candidates for broader applications in scientific computing, particularly in computational physics and engineering domains where tensor structures are prevalent.

A prospective avenue for future research would be the exploration of these methods for even more complex and computationally demanding problems, potentially involving tensors of even higher dimensions or incorporating stochastic elements. Moreover, the intersection with machine learning applications, where tensor structures are increasingly common, might provide fruitful, contemporary domains for these methodologies.

In conclusion, the paper delivers a robust framework for tackling tensor Sylvester equations via a novel adaptation of numerical linear algebra techniques. The emphasis on adaptive strategies for optimal pole selection and the innovative computation of the residuals mark significant contributions to the field, offering both theoretical and practical advancements in tensor computations.