Talks

Spectral deferred correction (SDC) is a time-stepping method where fully implicit Runge-Kutta methods (RKM) are solved iteratively. The method is only marginally more complicated to implement than the more ubiquitous diagonally implicit RKM, and it is often simpler for obtaining high-order solutions. We present numerical experiments that show SDC to be a modern and HPC capable method with various advantages over other RKM, including efficient time-parallelisation extensions. To this end, we present adaptive step size selection algorithms for SDC and demonstrate that they boost computational efficiency and resilience against soft faults at the same time. Then, we show that the parallel-in-time algorithm diagonal SDC can be used to extend strong-scaling capabilities beyond the saturation point of space-only scaling. This enables our space-time parallel Python code for the Gray-Scott equation to scale to the entirety of the JUWELS booster machine.

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Fluid flow problems can be modeled by the Navier-Stokes or Oseen equations. Their discretization results in saddle point problems. These systems of equations are typically of large scale and thus need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schurcomplement. Such an approximation can be obtained by a hierarchical matrix (H-Matrix) LU factorization for which the Schur complement is computed explicitly.

We present two strategies to improve the preconditioner set-up time. The first is a problem-dependent construction of the hierarchical block structure for the involved sparse matrices. These block structures are obtained from a partitioning of the velocity index set based on the connection with the pressure index set and results in a sparser block structure of the off-diagonal blocks of the saddle point system matrix.The second strategy are different approaches to the H-matrix multiplication which an important part of the H-LU factorzation and is used directly for the computation of the Schur complement. We briefly describe two variants introduced in [1] and [2] and examine their effectiveness for our problem with results from numerical experiments.

[1] S. Börm. “Hierarchical matrix arithmetic with accumulated updates”. In: Comput. Vis. Sci. 20.3-6 (2019), pp. 71–84. issn: 1432-9360. doi: 10 . 1007 /s00791 - 019 - 00311-3. url: https://doi.org/10.1007/s00791-019-00311-3.

[2] J. Dölz, H. Harbrecht, and M. D. Multerer. “On the best approximation of the hierarchical matrix product”. In: SIAM J. Matrix Anal. Appl. 40.1 (2019), pp. 147–174. issn: 0895-4798. doi: 10.1137/18M1189373. url: https://doi.org/10.1137/18M1189373.

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Spectral Deferred Correction (SDC) methods are an iterative technique for numerically solving initial value problems. SDC methods can be viewed as applying a suitable preconditioner to a Picard iteration, leading to faster and more reliable convergence to a collocation solution. Multi-level SDC (MLSDC) is an extension of SDC by computing the correction sweeps on a hierarchy of levels and the solutions are coupled through a Full Approximation Scheme (FAS) correction term inspired by nonlinear multigrid methods.

In this talk, we introduce Micro-Macro Multi-Level SDC (M3LSDC), a new extension of MLSDC for second-order ordinary differential equations (ODEs). Unlike MLSDC, which uses coarser discretizations on the coarse level, M3LSDC employs a reduced-order model on the coarse level. We will illustrate the benefits of this approach through numerical examples.

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The goal of parallel-in-time methods is to employ parallelism in the time direction in addition to the space direction, in the hope of obtaining further parallel speedups at the limits of what is possible due to spatial parallelism with domain decomposition alone. Recently diagonalisation techniques have emerged as a way of solving the coupled system for the solution of a differential equation at several timesteps simultaneously. One approach, sometimes referred to as “ParaDiag II” involves preconditioning this “all-at-once” system obtained from time discretisation of a linear constant coefficient ODE (perhaps obtained as the space discretisation of a time dependent PDE) with a nearby system that can be diagonalised in time, allowing the solution of independent blocks in parallel. For nonlinear PDEs this approach can form the basis of a preconditioner within a Newton-Krylov method for the all-at-once system after time averaging the (now generally time dependent) Jacobian system. After some preliminary description of the ParaDiag II approach, I will present results from our investigation of ParaDiag II applied to some testcases from the hierarchy of models used in the development of dry dynamical cores for atmosphere models, including performance benchmarks. Using these results I will identify the key challenges in obtaining further speedups and identify some directions to address these.

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