This is the second part of CoS-2 School on HPC architectures and numerical methods and will develop the ideas presented in the first part of CoS-2 in more detail. The focus will be on three “case studies” of large-scale problems and existing code bases that will be review in some detail. These code bases will be chosen from the research areas of the project. Successful completion of any one of the programmes will be worth 5 ECTS accreditation units. The School will offer 15 extra positions for Students and Researchers that are not HPC-LEAP fellows.

## Captured

## Parallel Algorithms

A tour of a few classic numerical algorithms will be given, with a review of how they can be implemented effectively on parallel architectures. To begin, solving sparse linear systems such as those that arise from a finite-difference approximation to the Laplace equation and numerical integration of statistical systems will be reviewed before more advanced topics, such as the fast Fourier transform are described.

Lectures by Prof. Michael Peardon (TCD).

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## Symplectic Integrators

Presents discussion of Hamiltonian dynamics, explaining the role of the Hamiltonian, the fundamental 2-form, phase space (the cotangent bundle) energy conservation, and Liouville’s theorem. The presentation introduces the leapfrog (Verlet) integrator and show that it is area-preserving, reversible, and good at conserving energy al- though it is not particularly good a following the true classical trajectory. The presentation also explains how higher-order symmetric symplectic integrators may be constructed following Campostrini et al. I will then introduce Hamiltonian vector fields, and establish that commutators of Hamiltonian vector fields are themselves Hamiltonian vector fields ex- pressible in terms of Poisson brackets. Introduces the Baker-Campbell-Hausdorff formula show how it leads to an asymptotic expansion for the Shadow Hamiltonian, a quantity that is exactly conserved by a symplectic integrator. Shows how examination of Poisson brackets may provide new symplectic integrator steps, such as the force-gradient. Explains how (for extensive systems) measurement of the appropriate Poisson brackets may be used to optimize symplectic integrators.

Lecture by Prof. Anthony Kennedy (University of Edinburgh).

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## Multigrid solvers

Multigrid solvers are used in situations where basic iterative solvers are plagued by critical slowing down, i.e., the required number of iterations is diverging in the region of interest. Understanding the origin of this problem com- mon to all basic iterative solvers directly leads to the development of the ingredients of multigrid methods. While the main concepts will be introduced using a simple model problem (Poisson equation), also the application to more complex problems (QCD) will be sketched.

Lecture by Dr. Björn Leder (Humboldt Universität Berlin).

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## Data Analysis

Data analysis is a central part of obtaining and interpreting the results of Monte Carlo simulations. A brief review of probability and statistics will be given, before focusing on applications. There will be a particular focus on Markov Chain Monte Carlo.

Lecture by Prof. John Bulava (TCD).

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## MPI

This course will cover some advanced topics in MPI programming, beginning with topics in parallel IO using both native MPI calls and HDFS. Then, PMPI, the profiling interface to MPI will be discussed, with software examples presented.

Lectures by Dermot Frost (TCHPC) and Louise Spellacy.

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## Optimisation

The course will describe the architectural features of the Intel KNC/KNL architectures and describe how to use the AVX512 instruction set for software optimisation and the best strategies for memory management. The day-long course will conclude with some case studies.

Lecture by Dr. Michael Lysaght (ICHEC).

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## Visualisation

This course will describe basic scientific visualisation techniques. A general introduction will be given, focusing on different types of datasets and different techniques that can be used to visualize them.

Lecture by Jose Refojo (TCHPC).

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