31 research outputs found
Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain
We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm,
recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently
solving subdiffusion equations with heterogeneous coefficients in long time.
This algorithm combines the benefits of multiscale methods, which can handle
heterogeneity in the spatial domain, and the strength of parareal algorithms
for speeding up time evolution problems when sufficient processors are
available. Our algorithm overcomes the challenge posed by the nonlocality of
the fractional derivative in previous parabolic problem work by constructing an
auxiliary problem on each coarse temporal subdomain to completely uncouple the
temporal variable. We prove the approximation properties of the correction
operator and derive a new summation of exponential to generate a single-step
time stepping scheme, with the number of terms of
independent of the final time, where
is the fine-scale time step size. We establish the convergence rate of our
algorithm in terms of the mesh size in the spatial domain, the level parameter
used in the multiscale method, the coarse-scale time step size, and the
fine-scale time step size. Finally, we present several numerical tests that
demonstrate the effectiveness of our algorithm and validate our theoretical
results.Comment: arXiv admin note: text overlap with arXiv:2003.1044
Learning Coarse Propagators in Parareal Algorithm
The parareal algorithm represents an important class of parallel-in-time
algorithms for solving evolution equations and has been widely applied in
practice. To achieve effective speedup, the choice of the coarse propagator in
the algorithm is vital. In this work, we investigate the use of learned coarse
propagators. Building upon the error estimation framework, we present a
systematic procedure for constructing coarse propagators that enjoy desirable
stability and consistent order. Additionally, we provide preliminary
mathematical guarantees for the resulting parareal algorithm. Numerical
experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn
model, and viscous Burgers model, show that learning can significantly improve
parallel efficiency when compared with the more ad hoc choice of some
conventional and widely used coarse propagators.Comment: 24 page
Towards Faster-than-real-time Power System Simulation Using a Semi-analytical Approach and High-performance Computing
This dissertation investigates two possible directions of achieving faster-than-real-time simulation of power systems. The first direction is to develop a semi-analytical solution which represents the nonlinear dynamic characteristics of power systems in a limited time period. The second direction is to develop a parallel simulation scheme which allows the local numerical solutions of power systems to be developed independently in consecutive time intervals and then iteratively corrected toward the accurate global solution through the entire simulation time period.
For the first direction, the semi-analytical solution is acquired using Adomian decomposition method (ADM). The ADM assumes the analytical solution of any nonlinear system can be decomposed into the summation of infinite analytical expressions. Those expressions are derived recursively using the system differential equations. By only keeping a finite number of those analytical expressions, an approximation of the analytical solution is yielded, which is defined as a semi-analytical solution. The semi-analytical solutions can be developed offline and evaluated online to facilitate the speedup of simulations. A parallel implementation and variable time window approach for the online evaluation stage are proposed in addition to the time performance analysis.
For the second direction, the Parareal-in-time algorithm is tested for power system simulation. Parareal is essentially a multiple shooting method. It decomposes the simulation time into coarse time intervals and then fine time intervals within each coarse interval. The numerical integration uses a computational cheap solver on the coarse time grid and an expensive solver on the fine time grids. The solution within each coarse interval is propagated independently using the fine solver. The mismatch of the solution between the coarse solution and fine solution is corrected iteratively. The theoretical speedup can be achieved is the ratio of the coarse interval number and iteration number. In this dissertation, the Parareal algorithm is tested on the North American eastern interconnection system with around 70,000 buses and 5,000 generators
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Recommended from our members
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations