11 research outputs found
An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs
To solve optimization problems with parabolic PDE constraints, often methods
working on the reduced objective functional are used. They are computationally
expensive due to the necessity of solving both the state equation and a
backward-in-time adjoint equation to evaluate the reduced gradient in each
iteration of the optimization method. In this study, we investigate the use of
the parallel-in-time method PFASST in the setting of PDE constrained
optimization. In order to develop an efficient fully time-parallel algorithm we
discuss different options for applying PFASST to adjoint gradient computation,
including the possibility of doing PFASST iterations on both the state and
adjoint equations simultaneously. We also explore the additional gains in
efficiency from reusing information from previous optimization iterations when
solving each equation. Numerical results for both a linear and a non-linear
reaction-diffusion optimal control problem demonstrate the parallel speedup and
efficiency of different approaches
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
Time integration for complex fluid dynamics
2021 Fall.Includes bibliographical references.Efficient and accurate simulation of turbulent combusting flows in complex geometry remains a challenging and computationally expensive proposition. A significant source of computational expense is in the integration of the temporal domain, where small time steps are required for the accurate resolution of chemical reactions and long solution times are needed for many practical applications. To address the small step sizes, a fourth-order implicit-explicit additive Runge-Kutta (ARK4) method is developed to integrate the stiff chemical reactions implicitly while advancing the convective and diffusive physics explicitly in time. Applications involving complex geometry, stiff reaction mechanisms, and high-order spatial discretizations are challenged by stability issues in the numerical solution of the nonlinear problem that arises from the implicit treatment of the stiff term. Techniques for maintaining a physical thermodynamic state during the numerical solution of the nonlinear problem, such as placing constraints on the nonlinear solver and the use of a nonlinear optimizer to find valid thermodynamic states, are proposed and tested. Verification and validation are performed for the new adaptive ARK4 method using lean premixed flames burning hydrogen, showing preservation of 4th-order error convergence and recovery of literature results. ARK4 is then applied to solve lean, premixed C3H8-air combustion in a bluff-body combustor geometry. In the two-dimensional case, ARK4 provides a 70× speedup over the standard explicit four-stage Runge-Kutta method and, for the three-dimensional case, three-orders-of-magnitude-larger time step sizes are achieved. To further increase the computational scaling of the algorithms, parallel-in-time (PinT) techniques are explored. PinT has the dual benefit of providing parallelization to long temporal domains as well as taking advantage of hardware trends towards more concurrency in modern high-performance computing platforms. Specifically, the multigrid reduction-in-time (MGRIT) method is adapted and enhanced by adding adaptive mesh refinement (AMR) in time. This creates a space-time algorithm with efficient solution-adaptive grids. The new MGRIT+AMR algorithm is first verified and validated using problems dominated by diffusion or characterized by time periodicity, such as Couette flow and Stokes second problem. The adaptive space-time parallel algorithm demonstrates up to a 13.7× speedup over a time-sequential algorithm for the same solution accuracy. However, MGRIT has difficulties when applied to solve practical fluid flows, such as turbulence, governed by strong hyperbolic partial differential equations. To overcome this challenge, the multigrid operations are modified and applied in a novel way by exploiting the space-time localization of fine turbulence scales. With these new operators, the coarse-scale errors are advected out of the temporal domain while the fine-scale dynamics iterate to equilibrium. This leads to rapid convergence of the bulk flow, which is important for computing macroscopic properties useful for engineering purposes. The novel multigrid operations are applied to the compressible inviscid Taylor-Green vortex flow and the convergence of the low-frequency modes is achieved within a few iterations. Future work will be focused on a performance study for practical highly turbulent flows
Schwarz waveform relaxation with adaptive pipelining
Schwarz waveform relaxation (SWR) methods have been developed to solve a wide range of diffusion-dominated and reaction-dominated equations. The appeal of these methods stems primarily from their ability to use nonconforming space-time discretizations; SWR methods are consequently well-adapted for coupling models with highly varying spatial and time scales. The efficacy of SWR methods is questionable, however, since in each iteration, one propagates an error across the entire time interval. In this manuscript, we introduce an adaptive pipeline approach wherein one subdivides the computational domain into space-time blocks, and adaptively selects the waveform iterates which should be updated given a fixed number of computational workers. Our method is complementary to existing space and time parallel methods, and can be used to obtain additional speedup when the saturation point is reached for other types of parallelism. We analyze these waveform relaxation with adaptive pipelining (WRAP) methods to show convergence and the theoretical speedup that can be expected. Numerical experiments on solutions to the linear heat equation, the advection-diffusion equation, and a reaction-diffusion equation illustrate features and efficacy of WRAP methods for various transmission conditions
Parallel-in-time integration of astro- and geo- physical flows; application of Parareal to kinematic dynamos and Rayleigh-Bénard convection
The precise mechanisms responsible for the natural dynamos in the Earth and Sun
are still not fully understood. Numerical simulations of natural dynamos are extremely
computationally intensive, and are carried out in parameter regimes many
orders of magnitude away from real conditions. Parallelization in space is a common
strategy to speed up simulations on high performance computers, but eventually hits
a scaling limit. Additional directions of parallelization are desirable to utilise the
high number of processor cores now available. Parallel-in-time methods can deliver
speed up in addition to that offered by spatial partitioning but have not yet
been applied to dynamo simulations. This thesis investigates the feasibility of using
Parallel-in-time integration to speed up numerical simulations of dynamos.
We concentrate on applying the non-intrusive Parareal algorithm to two sub-problems
of natural dynamos: kinematic dynamos and Rayleigh-Bénard convection (RBC).
We perform real-world scaling tests on high performance computing (HPC) facilities
using the open source Dedalus spectral solver.
The kinematic dynamo problem prescribes a fluid flow and observes how the magnetic field changes over time. We investigate the time independent Roberts and
time dependent Galloway-Proctor 2.5D dynamos over a range of magnetic Reynolds
numbers. Speed ups beyond those possible from spatial parallelisation are found
in both cases. Results for the Galloway-Proctor flow are promising, with Parareal
efficiency found to be close to 0.3, while Roberts flow results are less efficient, with
efficiencies < 0.1. Parallel in space and time speed ups of 300 were found for 1600
cores for the Galloway-Proctor flow, with total parallel efficiency of 0.16.
Convective motions are thought to be the source of dynamo action in the Earth
and Sun. RBC is the archetypal problem for convection studies, and is also a
fundamental problem of fluid dynamics, with many applications to geophysical,
astrophysical, and industrial flows. We investigate Parareal for Rayleigh numbers
Ra = 10⁵, 10⁶ and 10⁷, finding limited speed up in all cases for up to ~20 processors,
whilst performance and convergence of Parareal degrades as Ra increases.
We summarise our results for the kinematic dynamos + RBC, and discuss their relevance
and implications on Parallel-in-time simulations for the full dynamo problem
Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference
The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA