Parallel Algorithms for High Performance Computing
Domain decompostion, parallel-in-time, waveform relaxation

Parallel in space and time

Partial or differential-algebraic equations are commonly solved in time domain by integration methods like the backward Euler scheme. Those methods subdivide the time axis into small time steps, replace for each step sequentially the derivatives by difference quotients and solve in each time step for all unknowns a nonlinear equation system. This may become very time-consuming or impossible for long time intervals and large systems of equations, in particular if the system stems from a coupled problem that consists of subproblems with different properties. Often each subproblem describes a different physical effect (multiphysics), for example electromagnetic fields and heat distribution. The subproblems are mutually connected by coupling conditions (connecting ‘inputs’ and ‘outputs’). Often, the various phenomena evolve on different time and spatial scales (multiscale). This given decomposition in subproblems can be exploited to parallelize the simulation (in space and time) and thus reduce the time to solution considerably.

We are working on parallel-in-time methods, e.g. co-simulation, waveform-relaxation (opens in new tab) or Parareal methods. They aim at solving problems more efficiently. The ideas are similar to iterative methods for linear equation systems.

Pels, Andreas ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2021):
Parallel-in-Time Simulation of Power Converters Using Multirate PDEs.
In: Scientific Computing in Electrical Engineering SCEE 2020, Mathematics in Industry. Springer. ARXIV: 2006.06544, forthcoming. [In Proceedings]

Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2021):
Towards a Parallel-in-Time Calculation of Time-Periodic Solutions with Unknown Period.
In: Scientific Computing in Electrical Engineering SCEE 2020, Mathematics in Industry. Springer. ARXIV: 2004.01612, forthcoming. [In Proceedings]

Kulchytska-Ruchka, Iryna ; Schöps, Sebastian ; Hinze, Michael ; Friedhoff, Stephanie ; Ulbrich, Stefan ; Rain, Oliver. PASIROM: Parallel Simulation and Robust Optimization of Electro-Mechanical Energy Converters. In German Success Stories on Mathematics in Industry, volume 35 of Mathematics in Industry. Springer, 2021, forthcoming. [In Collection]

Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2021):
Efficient Parallel-in-Time Solution of Time-Periodic Problems Using a Multi-Harmonic Coarse Grid Correction.
In: SIAM Journal on Scientific Computing, 43, (1), pp. C61–C88, ISSN: 1064-8275, DOI: 10.1137/20M1314756, ARXIV: 1908.05245. [Article]

Cortes Garcia, Idoia ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2021):
Parareal for Higher Index Differential Algebraic Equations.
Cornell University, ARXIV: 2107.07755. [Preprint].

Cortes Garcia, Idoia ; Pade, Jonas ; Schöps, Sebastian ; Tischendorf, Caren (2021):
Waveform relaxation for low frequency coupled field/circuit differential-algebraic models of index 2.
In: Scientific Computing in Electrical Engineering SCEE 2020, Mathematics in Industry. Springer. ARXIV: 2005.13272, forthcoming. [In Proceedings]

Gander, Martin J. ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2020):
A New Parareal Algorithm for Time-Periodic Problems with Discontinuous Inputs.
In: Domain Decomposition Methods in Science and Engineering XXV, volume 138 of Lecture Notes in Computational Science and Engineering. Springer. ISBN: 978-3-030-56749-1, DOI: 10.1007/978-3-030-56750-7_27, ARXIV: 1810.12372. [In Proceedings]

Cortes Garcia, Idoia ; Kulchytska-Ruchka, Iryna ; Clemens, Markus ; Schöps, Sebastian (2020):
Parallel-in-Time Solution of Eddy Current Problems Using Implicit and Explicit Time-stepping Methods.
In: IEEE 19th Biennial Conference on Electromagnetic Field Computation (CEFC 2020). DOI: 10.1109/CEFC46938.2020.9451465, ARXIV: 2012.07603. [In Proceedings]

Cortes Garcia, Idoia ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2020):
Efficient Simulation of Field/Circuit Coupled Systems with Parallelised Waveform Relaxation.
In: IEEE Transactions on Magnetics, 56, (2), pp. 1–4, ISSN: 0018-9464, DOI: 10.1109/TMAG.2019.2952695, ARXIV: 1909.08895. [Article]

Bolten, Matthias ; Friedhoff, Stephanie ; Hahne, Jens ; Schöps, Sebastian (2020):
Parallel-in-Time Simulation of an Electrical Machine using MGRIT.
In: Computing and Visualization in Science, ISSN: 1432-9360, DOI: 10.1007/s00791-020-00333-2, ARXIV: 1912.03106. [Article]

Bast, Denys ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian ; Rain, Oliver (2020):
Accelerated Steady-State Torque Computation for Induction Machines using Parallel-In-Time Algorithms.
In: IEEE Transactions on Magnetics, 56, (2), pp. 1–9, ISSN: 0018-9464, DOI: 10.1109/TMAG.2019.2945510, ARXIV: 1902.08277. [Article]

Kulchytska-Ruchka, Iryna ; De Gersem, Herbert ; Schöps, Sebastian (2019):
An efficient steady-state analysis of the eddy current problem using a parallel-in-time algorithm.
In: The Tenth International Conference on Computational Electromagnetics (CEM 2019). ISBN: 978-1-83953-066-1, DOI: 10.1049/cp.2019.0113, ARXIV: 1905.13076. [In Proceedings]

Gander, Martin J. ; Kulchytska-Ruchka, Iryna ; Niyonzima, Innocent ; Schöps, Sebastian (2019):
A New Parareal Algorithm for Problems with Discontinuous Sources.
In: SIAM Journal on Scientific Computing, 41, (2), pp. B375–B395, ISSN: 1064-8275, DOI: 10.1137/18M1175653, ARXIV: 1803.05503. [Article]

Friedhoff, Stephanie ; Hahne, Jens ; Kulchytska-Ruchka, Iryna ; Schöps, Sebastian (2019):
Exploring Parallel-in-Time Approaches for Eddy Current Problems.
In: Progress in Industrial Mathematics at ECMI 2018, volume 30 of The European Consortium for Mathematics in Industry, 373–379. Springer. ISBN: 9783030275495, DOI: 10.1007/978-3-030-27550-1_47, ARXIV: 1810.13263. [In Proceedings]

Schöps, Sebastian ; Niyonzima, Innocent ; Clemens, Markus (2018):
Parallel-in-time Simulation of Eddy Current Problems using Parareal.
In: IEEE Transactions on Magnetics, 54, (3), pp. 1–4, ISSN: 0018-9464, DOI: 10.1109/TMAG.2017.2763090, ARXIV: 1706.05750. [Article]

Dutiné, Jennifer Susanne ; Clemens, Markus ; Schöps, Sebastian (2018):
Survey on semi-explicit time integration of eddy current problems.
In: Scientific Computing in Electrical Engineering SCEE 2016, volume 28 of Mathematics in Industry. Springer. ISBN: 978-3-319-75537-3, DOI: 10.1007/978-3-319-75538-0_13. [In Proceedings]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2017):
GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields.
In: IEEE Transactions on Magnetics, 53, (6), pp. 1–4, ISSN: 0018-9464, DOI: 10.1109/TMAG.2017.2662234, ARXIV: 1612.09447. [Article]

Merkel, Melina ; Niyonzima, Innocent ; Schöps, Sebastian (2017):
ParaExp using Leapfrog as Integrator for High-Frequency Electromagnetic Simulations.
In: Radio Science, 52, (12), pp. 1558–1569, ISSN: 0048-6604, DOI: 10.1002/2017RS006357, ARXIV: 1705.08019. [Article]

Dutiné, Jennifer Susanne ; Clemens, Markus ; Schöps, Sebastian (2017):
Explicit time integration of eddy current problems using a selective matrix update strategy.
In: COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 36, (5), pp. 1364–1371, ISSN: 0332-1649, DOI: 10.1108/COMPEL-02-2017-0100. [Article]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2016):
GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields.
In: Proceedings of 17th Biennial IEEE Conference on Electromagnetic Field Computation (CEFC 2016). ISBN: 978-1-5090-1032-5, DOI: 10.1109/CEFC.2016.7816199, URL: http://cefc2016.org, Digest. [In Proceedings]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2016):
Multi-GPU Acceleration of Algebraic Multigrid Preconditioners.
In: Scientific Computing in Electrical Engineering SCEE 2014, number 23 in Mathematics in Industry, 83–90. Springer. ISBN: 978-3-319-30399-4, DOI: 10.1007/978-3-319-30399-4_9. [In Proceedings]

Richter, Christian ; Schöps, Sebastian ; Dutiné, Jennifer Susanne ; Schreiber, Robert ; Clemens, Markus (2016):
Transient Simulation of Nonlinear Electro-Quasistatic Field Problems Accelerated by Multiple GPUs.
In: IEEE Transactions on Magnetics, ISSN: 0018-9464, DOI: 10.1109/TMAG.2015.2466602, Article #9401204. [Article]

Niyonzima, Innocent ; Clemens, Markus ; Schöps, Sebastian (2016):
Investigation of the Time Integration Methods on the Parareal Method for Field Computation of Eddy Currents Problems.
In: Proceedings of 17th Biennial IEEE Conference on Electromagnetic Field Computation (CEFC 2016). ISBN: 978-1-5090-1032-5, DOI: 10.1109/CEFC.2016.7816372, URL: http://cefc2016.org, Digest. [In Proceedings]

Merkel, Melina ; Niyonzima, Innocent ; Schöps, Sebastian (2016):
An Application of ParaExp to Electromagnetic Waves.
In: 2016 URSI International Symposium on Electromagnetic Theory (EMTS), 121–124. IEEE. ISBN: 978-1-5090-2503-9, DOI: 10.1109/URSI-EMTS.2016.7571330, ARXIV: 1607.00368. [In Proceedings]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2015):
Multi-GPU Acceleration of Algebraic Multigrid Preconditioners for Elliptic Field Problems.
In: IEEE Transactions on Magnetics, 51, (3), pp. 1–4, ISSN: 0018-9464, DOI: 10.1109/TMAG.2014.2357332. [Article]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus. Sykulski, Jan K. (editor) (2014):
GPU-Accelerated Mixed Precision Algebraic Multigrid Preconditioners for Discrete Elliptic Field Problems.
In: 9th IET International Conference on Computation in Electromagnetics (CEM 2014). IET. DOI: 10.1049/cp.2014.0185. [In Proceedings]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2014):
GPU-Acceleration of Algebraic Multigrid Preconditioners for Discrete Elliptic Field Problems.
In: IEEE Transactions on Magnetics, 50, (2), pp. 461–464, ISSN: 0018-9464, DOI: 10.1109/TMAG.2013.2283099. [Article]

Bartel, Andreas ; Hülsmann, Timo ; Kühn, Jan ; Pulch, Roland ; Schöps, Sebastian (2014):
Influence of Measurement Errors on Transformer Inrush Currents Using Different Material Models.
In: IEEE Transactions on Magnetics, 50, (2), pp. 485–488, ISSN: 0018-9464, DOI: 10.1109/TMAG.2013.2284072. [Article]

Scholz, Eike ; Ye, Hanyu ; Schöps, Sebastian ; Clemens, Markus (2013):
A Parallel FEM Matrix Assembly for Electro-Quasistatic Problems on GPGPU Systems.
In: IEEE Transactions on Magnetics, 49, (5), pp. 1801–1804, ISSN: 0018-9464, DOI: 10.1109/TMAG.2013.2239624. [Article]

Richter, Christian ; Schöps, Sebastian ; Clemens, Markus (2013):
GPU Acceleration of Finite Difference Schemes Used in Coupled Electromagnetic/Thermal Field Simulations.
In: IEEE Transactions on Magnetics, 49, (5), pp. 1649–1652, ISSN: 0018-9464, DOI: 10.1109/TMAG.2013.2238662. [Article]