The project inEXASCALE aims to change the way people think about designing and analyzing algorithms in the exascale era. On supercomputers that exist today, achieving even close to the peak performance is incredibly difficult if not impossible for many applications.

Techniques designed to improve performance - making computations less expensive by reorganizing an algorithm, making intentional approximations, and using lower precision - all introduce what we can generally call "inexactness". The question is, with all this inexactness involved, does the algorithm still get close enough to the right answer?

The effects of these sources of inexactness have been studied separately, but never together in a holistic way. By studying the combination of different sources of inexactness, we will reveal not only the limitations of these techniques, but also reveal new opportunities for developing algorithms that are both fast and provably accurate.

inEXASCALE

Analyzing and Exploiting Inexactness in Exascale Matrix Computations

PI: Erin Carson
Charles University
Prague, Czech Republic

Publications

Journal Papers
  • E. Carson, X. Chen, and X. Liu, Mixed Precision HODLR Matrices, SIAM Journal on Scientific Computing (accepted; in press), 2025. [preprint]
  • E. Carson, K. Lund, Y. Ma, and E. Oktay, Reorthogonalized Pythagorean Variants of Block Classical Gram-Schmidt, SIAM Journal on Matrix Analysis and Applications (accepted; in press), 2025. [preprint]
  • P. Vacek, E. Carson, and K. M. Soodhalter, The Effect of Approximate Coarsest-Level Solves on the Convergence of Multigrid V-Cycle Methods, SIAM Journal on Scientific Computing 46(4), pp. A2634-A2659, 2024. [link][preprint]
  • E. Carson, J. Liesen, and Z. Strakoš, Towards Understanding CG and GMRES Through Examples, Linear Algebra and its Applications 692, pp. 241-291, 2024. [link]
  • E. Carson and I. Daužickaitė, Single-pass Nyström Approximation in Mixed Precision, SIAM Journal on Matrix Analysis and Applications 45(3), pp. 1361-1391, 2024. [link][preprint]
  • E. Carson and I. Daužickaitė, The Stability of Split-Preconditioned FGMRES in Four Precisions, Electronic Transactions on Numerical Analysis 60, pp. 40-48, 2024. [link]
  • S. Thomas, E. Carson, M. Rozložník, A. Carr, and K. Świrydowicz, Iterated Gauss-Seidel GMRES, SIAM Journal on Scientific Computing 46(2), pp. S254-S279, 2024. [link][preprint]
  • E. Oktay and E. Carson, Mixed Precision Rayleigh Quotient Iteration for Total Least Squares Problems, Numerical Algorithms, 2023. [link][preprint]
  • E. Carson and N. Khan, Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning, SIAM Journal on Scientific Computing 45(3), pp. C131-C153, 2023. [link][preprint]
Preprints and Technical Reports
  • E. Carson and Y. Ma, A Stable One-Synchronization Variant of Reorthogonalized Block Classical Gram-Schmidt, 2024. [preprint]
  • E. Carson and Y. Ma, On the Backward Stability of s-step GMRES, 2024. [preprint]
  • E. Carson and I. Daužickaitė, Mixed Precision Sketching for Least-Squares Problems and its Application in GMRES-based Iterative Refinement, 2024. [preprint]
  • E. Carson and J. Hercik, The Detection and Correction of Silent Errors in Pipelined Krylov Subspace Methods, 2024. [preprint]
  • E. Carson, X. Chen, and X. Liu, Computing k-means in Mixed Precision, 2024 [preprint]
  • E. Carson and I. Daužickaitė, A Comparison of Mixed Precision Iterative Refinement Approaches for Least-Squares Problems, 2024. [preprint]

People

PI: Erin C. Carson

Postdocs:

PhD Students:

  • Ioannis Thanasis
  • Thomas Bake Arenas

Graduated PhD Students:

  • Eda Oktay (currently at MPI Magdeburg)
  • Petr Vacek (currently at IFPEN)

Former Postdocs:

Open Positions



Postdoctoral Researcher

The Faculty of Mathematics and Physics at Charles University invites applicants for a postdoctoral researcher position in numerical linear algebra and high-performance computing. The positions are funded by the ERC in the framework of the ERC STG project inEXASCALE.

The project will error analysis, algorithm development, and implementation. The candidate should ideally have a PhD in mathematics, computer science, or a related field, with a strong background in one or more of the following:

  • numerical linear algebra
  • Krylov subspace methods
  • finite precision analysis
  • randomized numerical linear algebra
  • data science and machine learning
  • parallel algorithms and high-performance computing

The start date is negotiable. By the start date the applicants must hold a PhD degree.

The application will remain open until the submission is filled, but for full consideration, applications should be received by March 1, 2025.

To apply, candidates should submit the following documents to carson@karlin.mff.cuni.cz:

  • Curriculum Vitae
  • Cover Letter explaining motivation and interest
  • List of publications
  • Brief summary of PhD thesis (including pdf file of PhD thesis if available)

In addition, the candidate should arrange for two letters of recommendation to be sent directly to the same e-mail address before March 1, 2025.

Questions regarding the application can be directed to carson@karlin.mff.cuni.cz




PhD Positions

Funded PhD positions are available within the framework of the ERC project "Analyzing and Exploiting Inexactness in Exascale Matrix Computations", led by Dr. Erin Carson at the Faculty of Mathematics and Physics at Charles University. Applications are invited from candidates who have strong background in numerical linear algebra, numerical analysis, parallel computing, or computational/data science application domains. The start date is October 2025, by which time the applicants must hold a Master's degree. Successful candidates must formally enroll in the PhD program at Charles University.

Potential PhD topics include, but are not limited to:

  • Development and analysis of mixed precision numerical linear algebra algorithms
  • Parallel algorithms and implementations for new mixed precision algorithms
  • Numerical linear algebra applied to data analysis and machine learning applications
  • Stability and convergence analysis of asynchronous methods
  • Application of new algorithms within a particular computational or data science discipline

Further details about the project can be found at the project website: https://www.karlin.mff.cuni.cz/~carson/inexascale

Application deadline: March 1, 2025

Interested candidates should submit the following documents to carson@karlin.mff.cuni.cz:

  • Curriculum Vitae
  • Cover letter explaining motivation and interest to obtain the position of PhD student
  • Brief summary of Master's thesis (including pdf file of Master's thesis if available)
In addition, the candidate should arrange for two letters of recommendation to be sent directly (i.e., not sent by the applicant) to the same e-mail address before the application deadline.

Selected finalists will be invited to officially apply through the university system; official acceptance in the program will be pending an entrance examination (which can be done virtually) and an English examination (which can be waived in many circumstances). For complete details on the admissions procedure, see https://www.mff.cuni.cz/en/admissions/admission-procedure-in-phd-programmes/2024-2025.

Questions regarding the application can be directed to:
Dr. Erin C. Carson
Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University
Sokolovska 83, 18675 Prague
Email: carson@karlin.mff.cuni.cz
Homepage: https://www.karlin.mff.cuni.cz/~carson





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