10 research outputs found
Multilevel convergence analysis of multigrid-reduction-in-time
This paper presents a multilevel convergence framework for
multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid
estimates. The framework provides a priori upper bounds on the convergence of
MGRIT V- and F-cycles, with different relaxation schemes, by deriving the
respective residual and error propagation operators. The residual and error
operators are functions of the time stepping operator, analyzed directly and
bounded in norm, both numerically and analytically. We present various upper
bounds of different computational cost and varying sharpness. These upper
bounds are complemented by proposing analytic formulae for the approximate
convergence factor of V-cycle algorithms that take the number of fine grid time
points, the temporal coarsening factors, and the eigenvalues of the time
stepping operator as parameters.
The paper concludes with supporting numerical investigations of parabolic
(anisotropic diffusion) and hyperbolic (wave equation) model problems. We
assess the sharpness of the bounds and the quality of the approximate
convergence factors. Observations from these numerical investigations
demonstrate the value of the proposed multilevel convergence framework for
estimating MGRIT convergence a priori and for the design of a convergent
algorithm. We further highlight that observations in the literature are
captured by the theory, including that two-level Parareal and multilevel MGRIT
with F-relaxation do not yield scalable algorithms and the benefit of a
stronger relaxation scheme. An important observation is that with increasing
numbers of levels MGRIT convergence deteriorates for the hyperbolic model
problem, while constant convergence factors can be achieved for the diffusion
equation. The theory also indicates that L-stable Runge-Kutta schemes are more
amendable to multilevel parallel-in-time integration with MGRIT than A-stable
Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material
Enabling Detailed, Biophysics-Based Skeletal Muscle Models on HPC Systems
Realistic simulations of detailed, biophysics-based, multi-scale models often require very high resolution and, thus, large-scale compute facilities. Existing simulation environments, especially for biomedical applications, are typically designed to allow for high flexibility and generality in model development. Flexibility and model development, however, are often a limiting factor for large-scale simulations. Therefore, new models are typically tested and run on small-scale compute facilities. By using a detailed biophysics-based, chemo-electromechanical skeletal muscle model and the international open-source software library OpenCMISS as an example, we present an approach to upgrade an existing muscle simulation framework from a moderately parallel version toward a massively parallel one that scales both in terms of problem size and in terms of the number of parallel processes. For this purpose, we investigate different modeling, algorithmic and implementational aspects. We present improvements addressing both numerical and parallel scalability. In addition, our approach includes a novel visualization environment which is based on the MegaMol framework and is capable of handling large amounts of simulated data. We present the results of a number of scaling studies at the Tier-1 supercomputer HazelHen at the High Performance Computing Center Stuttgart (HLRS). We improve the overall runtime by a factor of up to 2.6 and achieve good scalability on up to 768 cores
Time-periodic steady-state solution of fluid-structure interaction and cardiac flow problems through multigrid-reduction-in-time
In this paper, a time-periodic MGRIT algorithm is proposed as a means to
reduce the time-to-solution of numerical algorithms by exploiting the time
periodicity inherent to many applications in science and engineering. The
time-periodic MGRIT algorithm is applied to a variety of linear and nonlinear
single- and multiphysics problems that are periodic-in-time. It is demonstrated
that the proposed parallel-in-time algorithm can obtain the same time-periodic
steady-state solution as sequential time-stepping. It is shown that the
required number of MGRIT iterations can be estimated a priori and that the new
MGRIT variant can significantly and consistently reduce the time-to-solution
compared to sequential time-stepping, irrespective of the number of dimensions,
linear or nonlinear PDE models, single-physics or coupled problems and the
employed computing resources. The numerical experiments demonstrate that the
time-periodic MGRIT algorithm enables a greater level of parallelism yielding
faster turnaround, and thus, facilitating more complex and more realistic
problems to be solved.Comment: 29 pages; 7 pages Supplementary Material
Non-invasive estimation of relative pressure for intracardiac flows using virtual work-energy
Intracardiac blood flow is driven by differences in relative pressure, and assessing these is critical in understanding cardiac disease. Non-invasive image-based methods exist to assess relative pressure, however, the complex flow and dynamically moving fluid domain of the intracardiac space limits assessment. Recently, we proposed a method, ?WERP, utilizing an auxiliary virtual field to probe relative pressure through complex, and previously inaccessible flow domains. Here we present an extension of ?WERP for intracardiac flow assessments, solving the virtual field over sub-domains to effectively handle the dynamically shifting flow domain. The extended ?WERP is validated in an in-silico benchmark problem, as well as in a patient-specific simulation model of the left heart, proving accurate over ranges of realistic image resolutions and noise levels, as well as superior to alternative approaches. Lastly, the extended ?WERP is applied on clinically acquired 4D Flow MRI data, exhibiting realistic ventricular relative pressure patterns, as well as indicating signs of diastolic dysfunction in an exemplifying patient case. Summarized, the extended ?WERP approach represents a directly applicable implementation for intracardiac flow assessments.Funding agencies: D.M. holds a Knut and Alice Wallenberg Foundation scholar-ship for postdoctoral studies at Massachusetts Institute of Technology. M.B. acknowledges funding from King’s College London and Imperical College London ESPRC Centre for Doctoral Training in Medical Imaging (EP/L015226/1). D.N. would like to acknowledge funding from Engineering and Physical Sciences Research Council (EP/N011554/1 and EP/R003866/1). P.L. holds a Wellcome Trust Senior Research Fellowship (g.a. 209450/Z/17/Z). T.E. would like to acknowledge funding from the Swedish Research Council (2018–04454) and the Swedish Heart-Lung Foundation (2018-0657). This work was also supported by the Wellcome ESPRC Centre for Medical Engineering at King’s College London (WT 203148/Z/16/Z) and the British Heart Foundation (TG/17/3/33406). E.R.E. was funded in part by NIH R01 49039.</p