3,568 research outputs found
Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems
We discuss how âparallel-in-space & simultaneous-in-timeâ Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypicalapplication, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finitedifference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearizedproblems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case
The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
Space-Time Isogeometric Analysis of Parabolic Evolution Equations
We present and analyze a new stable space-time Isogeometric Analysis (IgA)
method for the numerical solution of parabolic evolution equations in fixed and
moving spatial computational domains. The discrete bilinear form is elliptic on
the IgA space with respect to a discrete energy norm. This property together
with a corresponding boundedness property, consistency and approximation
results for the IgA spaces yields an a priori discretization error estimate
with respect to the discrete norm. The theoretical results are confirmed by
several numerical experiments with low- and high-order IgA spaces
A parallel nearly implicit time-stepping scheme
Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem. These schemes have attractive stability properties and they are also very well parallelizable.\ud
The purpose of this article is to give an overall assessment of the parallelism of the method
A multigrid perspective on the parallel full approximation scheme in space and time
For the numerical solution of time-dependent partial differential equations,
time-parallel methods have recently shown to provide a promising way to extend
prevailing strong-scaling limits of numerical codes. One of the most complex
methods in this field is the "Parallel Full Approximation Scheme in Space and
Time" (PFASST). PFASST already shows promising results for many use cases and
many more is work in progress. However, a solid and reliable mathematical
foundation is still missing. We show that under certain assumptions the PFASST
algorithm can be conveniently and rigorously described as a multigrid-in-time
method. Following this equivalence, first steps towards a comprehensive
analysis of PFASST using block-wise local Fourier analysis are taken. The
theoretical results are applied to examples of diffusive and advective type
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