Numerical solution to a linear equation with tensor product structure

Abstract

We consider the numerical solution of a c-stable linear equation in the tensor product space R n 1 ×· · ·×n d , arising from a discretized elliptic partial differential equation in R d . Utilizing the stability, we produce an equivalent d-stable generalized Stein-like equation, which can be solved iteratively. For large-scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of O( ∑ i n i ) + O(n s ) computational complexity, under appropriate assumptions (with n s being the flop count for solving a linear system associated with A i − I n i ). Illustrative numerical examples will be presented. KEYWORDS Cayley transform, elliptic partial differential equation, Kronecker product, large-scale problem, linear equation, Stein equation, Sylvester equationfor a constant c ∈ R d . In many similar applications, A i can be formulated to be c-stable (with eigenvalues in the open left-half plane). We make use of the stability of A i and transform the corresponding LET_d (also described as c-stable) to an equivalent d-stable Stein-like equation with tensor product structure (SET), which are defined by matrices with spectra inside the unit Numer Linear Algebra Appl. 2017;24:e2106.wileyonlinelibrary.com/journal/nla

Inexact Newton Method for M-Tensor Equations

“…In this section, we list the detailed numerical results of the proposed methods compared with QCA method. The results are listed in Tables 3,4,5,6,7,8,9,10,11 and 12, where the columns 'Iter', 'Time', 'Res' and 'Ls-iter' stand for the total number of iterations, the computational time (in second) used for the method, the residual Âx From the data in the Tables 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, we can see that the proposed methods are effective for all test problems. In terms of the number of iterations and CPU time, Inexact Newton method and Regularized Newton method are better than QCA method, and the number of linear search of the Regularized Newton method are far less than that of the QCA method.…”

“…Tensor equation is also called multilinear equation. It appears in many practical fields including data mining and numerical partial differential equations [5,8,9,10,14,15,16,32]. The study in numerical methods for solving tensor equations has begun only a few years ago.…”

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“…Tensor equation is also called multi-linear equation. It appears in many practical fields including data mining and numerical partial equations [4, 7,8,9,13,15,16,27]. The study in the numerical methods for solving tensor equation has begun only a few years ago.…”

Finding a Nonnegative Solution to an M-Tensor Equation

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