## 7 Parallel Scalability  
For large-scale problems, the efficient implementation of computing systems with parallel architecture becomes an increasingly important issue. Solving dense linear systems with block LU-based solvers is a commonly accepted standard for measuring the performance of supercomputers. This approach is used in the well-known LINPACK TOP 500 ranking [33]. And although numerical linear algebra is the backbone of computing in general, evaluating the overall performance of a real-life problem is always a more complex  
Figure 4: Sequential computational times of adaptive scheme with LU factorization and HSS compression.  
Figure 5: Adaptive time step size over the whole time interval for \(N=2131\) and \(\tau_{0}=1.5625\times 10^{-5}\) with LU solver.  
task than benchmarking a single routine. Ensuring a transparent comparison, we follow the steps from Section 6 here, now analyzing the parallel performance. Let us also recall that MKL and STRUMPACK were developed as parallel software packages.  
In Figure 6, we present the parallel times for the backward Euler method with a uniform time step and the adaptive forward-backward Euler time stepping scheme. The block LU-based solver from MKL is used. The tests were run on a single server with 16 threads. Similar to the sequential tests, the adaptive scheme has better times for all \(N\) values at smaller \(\tau_{0}\); see the graph on the right-hand side. However, we also observe a relative improvement in parallel times in the case of \(\tau_{0}=1.5625\times 10^{-5}\). This is because the solution with the already factorized system has a lower parallel speed-up, which can be seen in Figure 7. With 16 threads, the adaptive method achieves up to \(\sim\)14 times better computation time than the sequential one, while the speed-up for the uniform time step method is only \(\sim\)5. Further details on the parallel speed-up of the LU factorization step and then of the solution step with the factorized matrix system can be found in [22].  
Figure 6: Parallel computational times of the uniform and adaptive LU solvers from MKL.  
Figure 7: Parallel speed-up (parallel time divided by sequential time) of the uniform and adaptive methods with an LU factorization-based solver. The speed-up is calculated for \(\tau_{0}=4.8828125\times 10^{-7}\) experiments but is similar to the other value of \(\tau_{0}\).  
In Figure 8, we present the parallel times with LU factorization and HSS-compression-based solvers for the adaptive method. For all experiments, HSS compression outperforms LU factorization. This means that the stiffness matrix obtained from the FEM discretization of the fractional Laplacian is suitable for hierarchical semi-separable compression. Overall, the adaptive algorithm with the HSS compression solver has a lower parallel speed-up of \(\sim\)6 (see Figure 9) compared to the adaptive method with LU factorization (see Figure 7), but its better performance is due to the substantially lower computational complexity.  
Figure 8: Parallel times of the uniform and adaptive schemes with HSS compression and ULV-like factorization from the STRUMPACK package.  
Figure 9: Parallel speed-up of the HSS compression and ULV-like factorization-based solver for the adaptive algorithm.