Related papers: Parareal Algorithm Implementation and Simulation i…
The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We…
This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation…
Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time'…
Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial…
In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have…
This paper proposes a parallel in time (called also time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the same spirit as the domain decomposition that consists of breaking…
In view of the existing limitations of sequential computing, parallelization has emerged as an alternative in order to improve the speedup of numerical simulations. In the framework of evolutionary problems, space-time parallel methods…
The polylogarithm function is one of the constellation of important mathematical functions. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics.…
The parareal in time algorithm allows to efficiently use parallel computing for the simulation of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where…
Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a…
In problems of mathematical physics, to study the structures of spaces using the Cayley-Klein models in theoretical calculations, the use of generalized complex numbers is required. In the case of computational experiments, such tasks…
We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from spectral approximations of the transfer operators mapping initial values for a given time interval to…
We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the…
In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation…
The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal…
We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow…
The time parallel solution of optimality systems arising in PDE constraint optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the…
Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion,…
We present new algorithms for the parallelization of Eulerian-Lagrangian interaction operations in the immersed boundary method. Our algorithms rely on two well-studied parallel primitives: key-value sort and segmented reduce. The use of…
The subject of the paper is to verify the convergence conditions for the parareal algorithm using Gander and Hairer's theorem . The analysis is conducted in the case where the coarse integrator is the Euler method and the high-accuracy…