Exploiting the Kronecker product structure of φ−functions in exponential integrators

Abstract

Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use 𝜑−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the 𝜑−function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum.For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of 𝜑−functions to compute this auxiliary function's action and then recover the original 𝜑−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen-Cahn equation.

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ-functions of matrices via quadrature

“…We now consider the 2D Eriksson-Johnson problem over Ω = (−1, 0) × (−0.5, 0.5) for 0 ≤ t ≤ T = 1 as presented in [23]. Here, the matrix A comes from the semidiscretization of the advection-diffusion operator phiquadmv gauss() phiquadmv cc() .…”

“…We now set the vector b with the nodal values of the initial condition u(x, y, 0) = 10x y 2 − 0.25 + e r 1 x −e r 2 x e −r 1 −e −r 2 cos(πy). As in [23], we select a Sishkin mesh (i.e. a graded, piecewise-uniform mesh in the x direction designed to capture the boundary layer, cf.…”

“…In fact, ϕ-functions of Kronecker sums do not simply separate into the Kronecker product of ϕ-functions of 1D matrices, making the tensor structure of the problem difficult to exploit in exponential integrators. Authors in [23] recently proposed an algorithm that circumvents this problem by building on recurrence relations between ϕ-functions to recast the evaluation problem in terms of the action of 1D ϕ-matrix-functions. However, this algorithm does not generalize easily to the 3D case and is numerically unstable for high-order exponential integrators.…”

A New Method of Kronecker Product Decomposition

“…For the sake of simplicity, we set n � 4, q � 2, and p � m, in which it is only to make the optimization method ( 6) easier that we take p � m. Also, each row of A m×4 is [1,2,3,4] and that of B m×2 is [1,1], where m � 1, 2, 3, • • •. For example,…”

“…Yang et al [3] researched the generalized Kronecker product linear system associated with a class of consecutiverank-descending matrices arising from bivariate interpolation problems. Muñoz-Matute et al [4] introduced an algorithm to speed up the computation of the φ-function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum using Kronecker products of one-dimensional matrices. More literature studies can refer to Rifa and Zinoviev [5], Enríquez and Rosas-Ortiz [6], Hao et al [7], Marco et al [8], Chen and Kressner [9], and the reference cited in.…”