Meeting the Real-Time Challenges of Ground-Based Telescopes Using Low-Rank Matrix Computations

Adaptive Optics (AO) is a technology that permits to measure and mitigate the distortion effects of atmospheric turbulence on optical beams. AO must operate in real-time by controlling thousands of actuators to shape the surface of deformable mirrors deployed on ground-based telescopes to compensate for these distortions. The command vectors that trigger how each individual actuator should act to bend a portion of the mirror are obtained from Matrix-Vector Multiplications (MVM). We identify and leverage the data sparsity structure of these control matrices coming from the MAVIS instruments for the European Southern Observatory's Very Large Telescope. We provide performance evaluation on x86 and accelerator-based systems. We present the impact of tile low-rank (TLR) matrix approximations on time-to-solution for the MVM and assess the produced image quality. We achieve performance improvement up to two orders of magnitude for TLR-MVM compared to regular dense MVM, while maintaining the image quality.

[1]  W. Hackbusch,et al.  Introduction to Hierarchical Matrices with Applications , 2003 .

[2]  洋一 中西,et al.  2012: , 2012, Disasters and Social Reproduction.

[3]  J. Hardy,et al.  Real-time atmospheric compensation , 1977 .

[4]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[5]  A. Sevin,et al.  GPUs for adaptive optics: simulations and real-time control , 2012, Other Conferences.

[6]  Leslie Greengard,et al.  Fast Direct Methods for Gaussian Processes , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  Jacques M. Beckers,et al.  Detailed Compensation Of Atmospheric Seeing Using Multiconjugate Adaptive Optics , 1989, Defense, Security, and Sensing.

[8]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[9]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[10]  Roberto Ragazzoni,et al.  MAORY RTC, a status update , 2020, Astronomical Telescopes + Instrumentation.

[11]  Gianpietro Marchiori,et al.  ELT design status: the most powerful ground telescope , 2018, Astronomical Telescopes + Instrumentation.

[12]  David E. Keyes,et al.  Tile Low-Rank GEMM Using Batched Operations on GPUs , 2018, Euro-Par.

[13]  Eric Darve,et al.  A fast block low-rank dense solver with applications to finite-element matrices , 2014, J. Comput. Phys..

[14]  David E. Keyes,et al.  Tile Low Rank Cholesky Factorization for Climate/Weather Modeling Applications on Manycore Architectures , 2017, ISC.

[15]  Eugene E. Tyrtyshnikov,et al.  Matrix‐free iterative solution strategies for large dense linear systems , 1997 .

[16]  Ronald Kriemann,et al.  H-LU Factorization on Many-Core Systems , 2014 .

[17]  Jean-Yves L'Excellent,et al.  Improving Multifrontal Methods by Means of Block Low-Rank Representations , 2015, SIAM J. Sci. Comput..

[18]  Claude-Pierre Jeannerod,et al.  Improving the Complexity of Block Low-Rank Factorizations with Fast Matrix Arithmetic , 2019, SIAM J. Matrix Anal. Appl..

[19]  Nicolas Doucet,et al.  Predictive learn and apply: MAVIS application - learn , 2020, Astronomical Telescopes + Instrumentation.

[20]  Alexander Kmentt.  2017 , 2018, The Treaty Prohibiting Nuclear Weapons.

[21]  Malcolm Smith,et al.  NFIRAOS adaptive optics for the Thirty Meter Telescope , 2018, Astronomical Telescopes + Instrumentation.

[22]  Pieter Ghysels,et al.  A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization , 2015, ACM Trans. Math. Softw..

[23]  David E. Keyes,et al.  Exploiting Data Sparsity for Large-Scale Matrix Computations , 2018, Euro-Par.

[24]  David E. Keyes,et al.  Pipelining Computational Stages of the Tomographic Reconstructor for Multi-Object Adaptive Optics on a Multi-GPU System , 2014, SC14: International Conference for High Performance Computing, Networking, Storage and Analysis.

[25]  John Shalf,et al.  The International Exascale Software Project roadmap , 2011, Int. J. High Perform. Comput. Appl..

[26]  S. Börm.  Efficient Numerical Methods for Non-local Operators , 2010 .

[27]  Hatem Ltaief,et al.  Solving Acoustic Boundary Integral Equations Using High Performance Tile Low-Rank LU Factorization , 2020, ISC.

[28]  George Z. Angeli,et al.  An overview and status of GMT active and adaptive optics , 2018, Astronomical Telescopes + Instrumentation.

[29]  David E. Keyes,et al.  Extreme Computing for Extreme Adaptive Optics: The Key to Finding Life Outside our Solar System , 2018, PASC.

[30]  Nicolas Doucet,et al.  Mixed-Precision Tomographic Reconstructor Computations on Hardware Accelerators , 2019, 2019 IEEE/ACM 9th Workshop on Irregular Applications: Architectures and Algorithms (IA3).

[31]  Théo Mary,et al.  Block Low-Rank multifrontal solvers: complexity, performance, and scalability. (Solveurs multifrontaux exploitant des blocs de rang faible: complexité, performance et parallélisme) , 2017 .

[32]  A. Sevin,et al.  A novel fast and accurate pseudo-analytical simulation approach for MOAO , 2014, Astronomical Telescopes and Instrumentation.

[33]  Thomas Hérault,et al.  Flexible Development of Dense Linear Algebra Algorithms on Massively Parallel Architectures with DPLASMA , 2011, 2011 IEEE International Symposium on Parallel and Distributed Processing Workshops and Phd Forum.

[34]  G. Turkiyyah,et al.  Hierarchical algorithms on hierarchical architectures , 2020, Philosophical Transactions of the Royal Society A.

[35]  Jack Dongarra,et al.  Numerical linear algebra on emerging architectures: The PLASMA and MAGMA projects , 2009 .

[36]  H. Ltaief,et al.  Scalable soft real-time supervisor for tomographic AO , 2018, Astronomical Telescopes + Instrumentation.

[37]  G. Rousset,et al.  Tomography approach for multi-object adaptive optics. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[38]  Ronald Kriemann,et al.  $${{\fancyscript{H}}} $$H-LU factorization on many-core systems , 2013, Comput. Vis. Sci..

[39]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[40]  Julien Bernard,et al.  Hard real-time core software of the AO RTC COSMIC platform: architecture and performance , 2020, Astronomical Telescopes + Instrumentation.

[41]  M. Puech,et al.  MOSAIC at the E-ELT: A multi-object spectrograph for astrophysics, IGM and cosmology , 2014, Astronomical Telescopes and Instrumentation.

[42]  Per-Gunnar Martinsson,et al.  An O(N) Direct Solver for Integral Equations on the Plane , 2013, 1303.5466.

[43]  David E. Keyes,et al.  Batched QR and SVD Algorithms on GPUs with Applications in Hierarchical Matrix Compression , 2017, Parallel Comput..

[44]  Daniel Kressner,et al.  A literature survey of low‐rank tensor approximation techniques , 2013, 1302.7121.

[45]  David E. Keyes,et al.  Real-Time Massively Distributed Multi-object Adaptive Optics Simulations for the European Extremely Large Telescope , 2018, 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS).

[46]  Alex Yu. Yeremin,et al.  Matrix-free iterative solution strategies for large dense linear systems , 1997, Numer. Linear Algebra Appl..

[47]  Damien Gratadour,et al.  COMPASS: An Efficient GPU-based Simulation Software for Adaptive Optics Systems , 2018, 2018 International Conference on High Performance Computing & Simulation (HPCS).