**6.3. Discussion of efficiency**

in the fifth-order case, both of these methods are still significantly slower than KSS-EPI, and a

**Figure 10.** Relative error plotted against execution time for solving Burgers' equation (32) using the fifth-order EPI method (30). Matrix function-vector products are computed using KSS-EPI with denoising (solid curves), Krylov-EPI (dashed curves), AKP (dashed-dotted curves), and LEJA (dotted curves), on grids with *N* = 500 ('+' markers), 1500 ('x'

**Figure 11.** Average number of matrix-vector products, shown on a logarithmic scale, per matrix function-vector prod‐ uct evaluation for each method when solving Burgers' equation (32) using the fifth-order EPI method (30). For KSS and KSS denoised, FFTs are also included. For each method, bars correspond to grid sizes of *N* = 500, 1500, 3000 points,

The difference in scalability among the four approaches to computing matrix function-vector products is more apparent in **Figures 9** and **11**. Denoising applied to KSS-EPI is advantageous for this problem, unlike with the Allen-Cahn equation. As can be seen in these figures, for KSS-

, for *p* = 0, 1, 2, 3, 4.

markers), and 3000 ('o' markers) points. Time steps used are *Δt* = (0.01)2<sup>−</sup>*<sup>p</sup>*

from left to right. Left plot: *Δt* = 0.01. Right plot: *Δt* = 0.000625.

similar scalability gap can also be observed.

22 Applied Linear Algebra in Action

The major components of the computational cost of KSS-EPI stem from Krylov projection that is applied to low-frequency parts, and FFTs that are applied to both the low- and highfrequency parts. Specifically, suppose the EPI method is of order *p*, and *q* Krylov projection steps are needed for convergence of the low-frequency part. Then, the task of evaluating *φ*(*Aτ*)*b* requires *q* + *p* matrix-vector multiplications, (*p* + 3)/2 FFTs and 2 inverse FFTs if denoising is not used, and *q* + *p* matrix-vector multiplications, *q* + (*p* + 3)/2 FFTs and *q* + 2 inverse FFTs if denoising is used. Denoising, therefore, is only worthwhile if the value of *q* can be substantially reduced, as in the case of Burgers' equation.
