Fast 3D Block Parallelisation for the Matrix Multiplication Prefix Problem

Waldherr, Konrad; Huckle, Thomas; Auckenthaler, T.; Sander, Uwe; Schulte‐Herbrüggen, Thomas

doi:10.1007/978-3-642-13872-0_4

“…Gradl et al [24] presented a parallel prefix algorithm for accumulation of matrix multiplications in quantum control. Waldherr et al [25] and Auckenthaler [26] showed later that prefix scan parallelization of this operation is outperformed by a sequential prefix scan with parallel matrix multiplication operator. These applications of prefix scan resulted in neither tuning nor designing a scan algorithm for operators where computation time is significantly larger than communication.…”

Section: Specific Prefix Scan Operatorsmentioning

confidence: 99%

Work-Stealing Prefix Scan: Addressing Load Imbalance in Large-Scale Image Registration

Copik

¹

,

Grosser

²

,

Hoefler

³

et al. 2022

IEEE Trans. Parallel Distrib. Syst.

6

0

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Parallelism patterns (e.g., map or reduce) have proven to be effective tools for parallelizing high-performance applications.In this paper, we study the recursive registration of a series of electron microscopy images -a time consuming and imbalanced computation necessary for nano-scale microscopy analysis. We show that by translating the image registration into a specific instance of the prefix scan, we can convert this seemingly sequential problem into a parallel computation that scales to over thousand of cores. We analyze a variety of scan algorithms that behave similarly for common low-compute operators and propose a novel work-stealing procedure for a hierarchical prefix scan. Our evaluation shows that by identifying a suitable and well-optimized prefix scan algorithm, we reduce time-to-solution on a series of 4,096 images spanning ten seconds of microscopy acquisition from over 10 hours to less than 3 minutes (using 1024 Intel Haswell cores), enabling derivation of material properties at nanoscale for long microscopy image series.

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“…Gradl et al [23] presented a parallel prefix algorithm for accumulation of matrix multiplications in quantum control. Waldherr et al [24] and Auckenthaler [25] showed later that prefix scan parallelization of this operation is outperformed by a sequential prefix scan with parallel matrix multiplication operator. These applications of prefix scan resulted in neither tuning nor designing a scan algorithm for operators where computation time is significantly larger than communication.…”

Section: Specific Prefix Scan Operatorsmentioning

confidence: 99%

Work-stealing prefix scan: Addressing load imbalance in large-scale image registration

Copik¹,

Grosser²,

Hoefler³

et al. 2020

Preprint

0

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Parallelism patterns (e.g., map or reduce) have proven to be effective tools for parallelizing high-performance applications.In this paper, we study the recursive registration of a series of electron microscopy images -a time consuming and imbalanced computation necessary for nano-scale microscopy analysis. We show that by translating the image registration into a specific instance of the prefix scan, we can convert this seemingly sequential problem into a parallel computation that scales to over thousand of cores. We analyze a variety of scan algorithms that behave similarly for common low-compute operators and propose a novel work-stealing procedure for a hierarchical prefix scan. Our evaluation shows that by identifying a suitable and well-optimized prefix scan algorithm, we reduce time-to-solution on a series of 4,096 images spanning ten seconds of microscopy acquisition from over 10 hours to less than 3 minutes (using 1024 Intel Haswell cores), enabling derivation of material properties at nanoscale for long microscopy image series.

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“…While this reduces the wall clock time, the CPU time will be increased to handle the overheads of parallelisation. This will affect the observed speed-up in a way that depends upon matrix size [19] as matrix exponentiation is more easily parallelised than our efficient algorithm. The speed gain achievable will depend on both the spin system chosen and the precise code used, but our MATLAB simulations for the 3-spin homonuclear system corresponding to the three 13 C nuclei in alanine [20] indicate that the calculation of subpropagators can be sped up by a factor of around 18, while full propagators (which require an additional full matrix multiplication for each sub-propagator) are sped up by a factor of around 13.…”

Section: Errors and Speed Gainsmentioning

confidence: 99%

Practical pulse engineering: Gradient ascent without matrix exponentiation

Bhole

¹

,

Jones

²

2018

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Since 2005 there has been a huge growth in the use of engineered control pulses to perform desired quantum operations in systems such as NMR quantum information processors. These approaches, which build on the original gradient ascent pulse engineering (GRAPE) algorithm, remain computationally intensive because of the need to calculate matrix exponentials for each time step in the control pulse. Here we discuss how the propagators for each time step can be approximated using the Trotter-Suzuki formula, and a further speed up achieved by avoiding unnecessary operations. The resulting procedure can give a substantial speed gain with negligible cost in propagator error, providing a more practical approach to pulse engineering. PACS numbers:Quantum information processors encode information in two-level quantum systems (qubits) and manipulate this through a series of elementary unitary transformations (quantum logic gates) [1,2]. Quantum control seeks to implement some target unitary propagator U in a quantum system with background Hamiltonian H 0 by applying some time-dependent Hamiltonian H 1 (t). The resulting operator can be written aswhere T is the Dyson time-ordering operator. To make progress beyond this formal solution it is usually necessary to replace this continuously varying Hamiltonian by a piecewise constant form, so thatand to write the time-varying portion of the Hamiltonian as a weighted sum of a set of p distinct control fieldsAny particular control pulse can then be described by the corresponding set of amplitudes, a k j , and the time step δt, here taken as fixed. The quality of a control pulse can be measured by its fidelity with the desired operation Uwhere the Hilbert-Schmidt inner product is defined by U |V = tr(U † V ), possibly normalised by the dimension of the operators [2]. The optimal control problem is then to find the set of amplitudes which maximises this fidelity, usually in the presence of practical constraints on the magnitudes of the amplitudes and the total length * Electronic address: jonathan.jones@physics.ox.ac.uk of the sequence. This is computationally challenging as the dimension of the underlying Hilbert space rises exponentially with the number of qubits to be controlled, although this difficulty can in some cases be reduced by using subsystems to simplify the calculations [3]. One recent approach [4] is to use subsystem methods to find approximate control pulses and then optimise these directly using the quantum system itself. Whatever approach is adopted, it is important to perform any computations as efficiently as possible.

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Fast 3D Block Parallelisation for the Matrix Multiplication Prefix Problem

Cited by 4 publications

References 8 publications

Work-Stealing Prefix Scan: Addressing Load Imbalance in Large-Scale Image Registration

Work-Stealing Prefix Scan: Addressing Load Imbalance in Large-Scale Image Registration

Work-stealing prefix scan: Addressing load imbalance in large-scale image registration

Practical pulse engineering: Gradient ascent without matrix exponentiation

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