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Alternating direction multiplication is a powerful technique for solving convex optimisation problems. When challenging subproblems are encountered in the real world, it is useful to solve them by introducing neighbourhood terms. When the…
At the heart of the Met Office climate and weather forecasting capabilities lies a sophisticated numerical model which solves the equations of large-scale atmospheric flow. Since this model uses semi-implicit time-stepping, it requires the…
We present the derivation, implementation, and analysis of a multiresolution adaptive grid framework for numerical simulations on octree-based 3D block-structured collocated grids with distributed computational architectures. Our approach…
The alternating direction method of multipliers (ADMM) is an effective method for solving wide fields of convex problems. At each iteration, the classical ADMM solves two subproblems exactly. However, in many applications, it is expensive…
Methods for solving hyperbolic systems typically depend on unknown ordering (e.g., Gauss-Seidel, or sweep/wavefront/marching methods) to achieve good convergence. For many discretisations, mesh types or decompositions these methods do not…
In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother…
Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves,…
A number of different multiscale methods have been developed as a robust alternative to upscaling and as a means for accelerated reservoir simulation of high-resolution geomodels. In their basic setup, multiscale methods use a restriction…
Vertex-patch smoothers are essential for the robust convergence of geometric multigrid methods in high-order finite element applications, yet their adoption is traditionally hindered by the prohibitive cost of solving local patch problems.…
This paper develops a new algebraic multigrid (AMG) method for sparse least-squares systems of the form $A=G^TG$ motivated by challenging applications in scientific computing where classical AMG methods fail. First we review and relate the…
A fast multigrid solver is presented for high-order accurate Stokes problems discretised by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an element-wise block Gauss-Seidel smoother.…
Even nowadays, where Deep Learning (DL) has achieved state-of-the-art performance in a wide range of research domains, accelerating training and building robust DL models remains a challenging task. To this end, generations of researchers…
The problem of finding the solution of Partial Differential Equations (PDEs) plays a central role in modeling real world problems. Over the past years, Multigrid solvers have showed their robustness over other techniques, due to its high…
Alternating Direction Method of Multipliers (ADMM) is a popular method for solving large-scale Machine Learning problems. Stochastic ADMM was proposed to reduce the per iteration computational complexity, which is more suitable for big data…
Many problems in computational science and engineering involve partial differential equations and thus require the numerical solution of large, sparse (non)linear systems of equations. Multigrid is known to be one of the most efficient…
The Alternating Direction Method of Multipliers (ADMM) has now days gained tremendous attentions for solving large-scale machine learning and signal processing problems due to the relative simplicity. However, the two-block structure of the…
The implementation of efficient multigrid preconditioners for elliptic partial differential equations (PDEs) is a challenge due to the complexity of the resulting algorithms and corresponding computer code. For sophisticated finite element…
Many problems in machine learning and other fields can be (re)for-mulated as linearly constrained separable convex programs. In most of the cases, there are multiple blocks of variables. However, the traditional alternating direction method…
We study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the used computational units (cores). The overall procedure generates coarse…
Current Adaptive Mesh Refinement (AMR) simulations require algorithms that are highly parallelized and manage memory efficiently. As compute engines grow larger, AMR simulations will require algorithms that achieve new levels of efficient…