This paper investigates the use of overlapping Schwarz domain decomposition preconditioners to solve fully implicit three-dimensional ocean circulation models efficiently. These are essential for understanding climate-relevant ocean dynamics but pose formidable computational challenges due to their spatial and temporal dimensions combined with ill-conditioned systems typical of realistic ocean climate models.

The paper begins by describing an established ocean model which solves the primitive equations coupled with temperature and salinity convection, subject to wind stress, heat, and salt flux forcing. The model is discretized using a second-order control volume scheme on a hybrid B-C Lorenz grid. This results in a large non-linear system whose Jacobian features tightly coupled momentum, mass, and tracer blocks. Solving this system relies on Newton-Krylov iterations, whose convergence depends heavily on effective preconditioning.

The paper presents numerical experiments. It first examines a Laplace system to systematically study how the overlap size and number of subdomains affect solver convergence. Results confirm that increased overlap reduces iterations and that two-level Schwarz methods with coarse corrections outperform one-level approaches, especially as mesh refinement and subdomain counts increase.

Building on these insights, the study extends to a fully implicit ocean model that solves the primitive equations with coupled temperature and salinity dynamics under wind, heat, and salt flux forcing. Applying a one-level Schwarz preconditioner, it becomes clear that as temperature and salinity forcing intensify, the Jacobian’s conditioning deteriorates sharply which leads to a steep rise in GMRES iterations. Additional experiments isolating individual tracers indicate that the temperature exerts a particularly strong influence on convergence, though buoyancy interactions complicate this attribution.

The work concludes that while one-level Schwarz methods improve performance on simpler problems, fully implicit ocean models under strong tracer forcing require advanced two-level Schwarz preconditioners with robust coarse spaces to maintain efficiency and scalability. Future work is encouraged to incorporate coarse grid corrections explicitly to better handle these challenges and enable more detailed and computationally feasible ocean simulations.

Sparse triangular solves (SpTRSV) form the latency–critical inner loop of many direct and iterative solvers, but strong data dependencies limit thread–level parallelism and make the kernel dominated by memory–latency.
This thesis explores whether redundant computation can be exchanged for improved data locality to accelerate SpTRSV on cache-based multi-core CPUs.

The proposed two-phase algorithm first uses the Recursive Algebraic Colouring Engine (RACE) to permute an arbitrary sparse triangular factor into a block-lower–bidiagonal form whose diagonal blocks fit into a private L2 cache.
Each block is then solved twice: an independent provisional pass followed by a lightweight correction that re-uses the still-resident data.
The resulting task graph halves the critical path, exposes ample parallelism, and leaves total memory traffic unchanged.

An OpenMP5.0 implementation employs the new affinity clause so that producer–consumer task pairs are likely to run on the same core.
Performance is compared against IntelMKL’s sparse triangular routine and the Kokkos-Kernels sptrsv, while LIKWID counters validate cache behaviour.

On a 24-core Cascade Lake node of the DelftBlue supercomputer the task graph achieves a strong-scaling speed-up between 1.25 and 1.45 that saturates after roughly six threads; Kokkos attains similar absolute time only beyond sixteen threads.
With 24 threads the solver is up to an order of magnitude faster than single-threaded MKL, and the affinity hint alone accelerates execution by 1.1–2 times, confirming the importance of cache reuse.
MKL remains preferable for small or highly structured SPD matrices, but on irregular, memory-bound factors the new solver rivals state-of-the-art libraries despite its simple prototype status.

A fundamental difficulty of describing few-body systems is that the Schrödinger equation has no analytic solution for more than two interacting particles. Hence, we must resort to numerical methods to obtain a description of the system. The Schrödinger equation is discretised using the pseudo-spectral method. As a result, the Schrödinger equation is transformed into an eigenvalue problem. The resulting eigenvalue problem contains a large sparse linear system. For this reason, the Jacobi Davidson QR (JDQR) eigenvalue solver has been chosen. It excels at finding eigenpairs of large sparse matrices. Within JDQR a system of equations must be solved using an iterative solver such as Generalized minimal residual method (GMRES). Therefore, with increasing grid sizes solving the system becomes more troublesome for iterative methods. This problem is exacerbated in the 2D case, where the size of the matrices grows much faster than in the 1D case. In the 2D case the system matrix has a size of 𝑛^4 × 𝑛^4 instead of 𝑛^2 × 𝑛^2 in the one-dimensional case. As a result, a direct method for approximating this system is imperative, giving rise to the need for preconditioners. The purpose of the preconditioner is to approximate the inverse of the system matrix, thereby significantly reducing the computational time required to solve the system. In contrast to the 1D scenario[14], there is no direct method to precondition the system matrix in two dimensions. In this report we propose a preconditioner for the quantum-mechanical three-body problem in two spatial dimensions. The proposed preconditioner, called the ’2D inexact preconditioner’ reduces the amount of time needed for convergence and the number of iterations. The computation time was decreased by 86% on the largest grid size we have considered. Additionally, we attempt to expand upon the preconditioner proposed for the 1D case by [14]. Their approach neglects the potential in the eigenvalue problem. The idea is to incorporate the potential for an improved convergence. We derive two algorithms that attempt to achieve this. Lastly, the bosonic ground states of the three-body system are plotted and compared to those already present in literature