A Lanczos-type procedure for tensors

Abstract

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

“…This may be exploited using projection methods with low-rank techniques (see, e.g., [11,12]). In [10], we also show that matrix A M in (8) can be compressed by the Tensor Train decomposition (note that [10] uses a different family of orthogonal functions instead of the Legendre polynomials). A Tensor Train approach may further reduce the memory and computational cost of the method.…”

“…with D, B sparse matrices described in [10]. In our experiments, we set T = 10 −3 , ν = 10 4 , k = 4, 7, 10, so obtaining three systems with exponentially increasing sizes.…”

“…Fig.3: Singular values of the matrix X, with x = vec(X) the solution of(10) for the examples in Section 2.1. From left to right, k = 4, 7, 10.…”

A ⋆‐product solver with spectral accuracy for non‐autonomous ordinary differential equations

“…The key to going from the ⋆-algebra to the matrix algebra is discretizing the ⋆-product. One way to discretize the ⋆-product is by the use of quadrature rules [1]. Another way is by expansion in a basis of orthonormal polynomials (ONPs), which is the topic of this paper.…”

“…Therefore, a numerical procedure is proposed that computes a discretization of this solution in the matrix algebra, equipped with the usual matrix-matrix product. The key to going from the ⋆-algebra to the matrix algebra is finding a suitable discretization of the ⋆-product, which can be based on a quadrature rule [1] or on the expansion of the bivariate distributions in a basis of orthonormal polynomials. The latter approach is followed in this paper, a basis of orthonormal Legendre polynomials is chosen and the resulting discretization is discussed in Section 3.…”