### Speaker

### Description

Delay differential equations (DDE) appear in several branches of science and engineering. Possible applications include the modelling and forecasting of epidemics, the modelling of stability loss in control systems and many more. The delays in the differential equations can be caused by the incubation time of a virus in epidemic models or by the time which the computer needs to carry out the necessary calculation in case of computer control.

The numerical solution of problems described by DDEs is inevitable in several cases, and sometimes a large number of the same delay differential equations must be solved due to the many possible parameter combinations or initial conditions. **The serial solution of millions of equations is usually not viable**, thus some kind of parallelization is necessary; however, **general purpose DDE solvers for GPUs do not exist at present**. Compared to ordinary differential equation (ODE) solvers, DDE solvers use values from the past because of the delay in the equation; thus, every timestep must be saved. However, it cannot be guaranteed that the required previous time instance is available in the global memory. Therefore, interpolation between the past values is necessary to maintain the order of the used numerical method.

GPUs were found extremely efficient for the numerical solution of large number of ODEs. **The most efficient method for the parallelization of numerical solution is assigning each equation to one thread** (also called *per-thread* approach). In the present work, the same strategy is applied for the acceleration of DDE solvers. The traditional 4th order fixed-timestep Explicit—Runge—Kutta (ERK) method is implemented with the extension of 3rd order Hermite-interpolation. The code is written in CUDA C++ language. This solver can be applied for a wide range of possible problems on most GPUs. However, an efficient implementation is difficult since it requires the intensive use of global memory, while the goal is to reach the maximum possible FLOP efficiency. The bottleneck of such problems is the memory bandwidth; thus, finding an optimal memory structure is necessary. Furthermore, the global memory size can also be a limiting factor, as each thread saves thousands of states in the global memory. This can lead to several gigabytes of global memory usage limiting the total number of residing threads and throttling the performance.

In the study, the basic idea of the solver is shown. **An efficient memory structure is proposed and tested, with aligned and coalesced memory access pattern**. To minimise the global memory usage, each thread works on a fixed-size array, when this array fills up, the older saved timesteps will be overwritten in a circular fashion. Problem specific codes with the previously described structure are tested on several simple test cases, and the FLOP efficiency, memory load/store efficiency and further important metrics are measured. **It is found that 40% FLOP efficiency can be reached with 95% memory efficiency.** The FLOP efficiency of the problem specific codes can be regarded as the maximal possible efficiency of my proposed solver. The implementation of a general purpose DDE solver is presented in the already existing GPU ODE solver called MPGOS and the metrics of the implementation is measured. A general solver must work for every possible problem, which requires extra computations and memory operations; consequently, its efficiency in my case only reaches half of the problem specific solvers.

Finally, the limitations of a fixed-timestep ERK method is presented, and the possibilities of adaptive ERK solvers for GPUs is discussed. However, for adaptive ERK methods, aligned and coalesced memory access patterns are not possible with the *per-thread* approach, thus a heterogeneous CPU-GPU solver is proposed which may be a viable solution for later implementations.