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We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi-Newton methods regardless of the scale of the calculation or the machine.

Arm usage has substantially grown in the High-Performance Computing (HPC) community. Japanese supercomputer Fugaku, powered by Arm-based A64FX processors, held the top position on the Top500 list between June 2020 and June 2022, currently sitting in the second position. The recently released 7th generation of Amazon EC2 instances for compute-intensive workloads (C7g) is also powered by Arm Graviton3 processors. Projects like European Mont-Blanc and U.S. DOE/NNSA Astra are further examples of Arm irruption in HPC. In parallel, over the last decade, the rapid improvement of genomic sequencing technologies and the exponential growth of sequencing data has placed a significant bottleneck on the computational side. While the majority of genomics applications have been thoroughly tested and optimized for x86 systems, just a few are prepared to perform efficiently on Arm machines, let alone exploit the advantages of the newly introduced Scalable Vector Extensions (SVE). This thesis presents GenArchBench, the first genome analysis benchmark suite targeting Arm architectures. We have selected a set of computationally demanding kernels from the most widely used tools in genome data analysis and ported them to Arm-based A64FX and Graviton3 processors. The porting features the usage of the novel Arm SVE instructions, algorithmic and code optimizations, and the exploitation of Arm-optimized libraries. All in all, the GenArch benchmark suite comprises 13 multi-core kernels from critical stages of widely-used genome analysis pipelines, including base-calling, read mapping, variant calling, and genome assembly. Moreover, our benchmark suite includes different input data sets per kernel (small and large), each with a corresponding regression test to verify the correctness of each execution automatically. In this work, we present the optimizations implemented in each kernel and a detailed performance evaluation and comparison of their performance on four different architectures (i.e.

In molecular dynamics simulations we can often increase the time step by imposing constraints on bond lengths and bond angles. This allows us to extend the length of the time interval and therefore the range of physical phenomena that we can afford to simulate. We examine the existing algorithms and software for solving nonlinear constraint equations in parallel and we explain why it is necessary to advance the state-of-the-art. We present ILVES-PC, a new algorithm for imposing bond constraints on proteins accurately and efficiently. It solves the same system of differential algebraic equations as the celebrated SHAKE algorithm, but ILVES-PC solves the nonlinear constraint equations using Newton's method rather than the nonlinear Gauss-Seidel method. Moreover, ILVES-PC solves the necessary linear systems using a specialized linear solver that exploits the structure of the protein. ILVES-PC can rapidly solve constraint equations as accurately as the hardware will allow. The run-time of ILVES-PC is proportional to the number of constraints. We have integrated ILVES-PC into GROMACS and simulated proteins of different sizes. Compared with SHAKE, we have achieved speedups of up to 4.9× in single-threaded executions and up to 76× in shared-memory multi-threaded executions. Moreover, ILVES-PC is more accurate than P-LINCS algorithm. Our work is a proof-of-concept of the utility of software designed specifically for the simulation of polymers., This work has been partially supported by the Spanish Ministry of Science and Innovation MCIN/AEI/10.13039/501100011033 (contracts PID2019-107255GB-C21, PID2019-105660RB-C21, and PID2019-107293GB-I00), by the Generalitat de Catalunya (contract 2017-SGR-1328), by the Gobierno de Aragón (E45_20R T58_20R research groups), and by Lenovo-BSC Contract-Framework Contract (2020). Santiago Marco was supported by the Agencia Estatal de Investigación (Spain) under Juan de la Cierva fellowship grant IJC2020-045916-I. Miquel Moretó was partially supported by the Agencia Estatal de Investigación (Spain) under Ramón y Cajal fellowship number RYC-2016-21104. Carl Christian Kjelgaard Mikkelsen is supported by eSSENCE, a collaborative e-Science programme funded by the Swedish Research Council within the framework of the strategic research areas designated by the Swedish Government., Peer Reviewed, Postprint (published version)

We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi-Newton methods regardless of the scale of the calculation or the machine.