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Nonlinear Functional Analysis And Its Applications

The heat equation also shows what a PDE is in general: We are given some kind of relationship between a function u(x,t) and/or its partial derivatives. In the one dimensional case this can be partial derivatives in time t or in space x. The problem is to find functions u(x,t) which satisfy the given relationship and sometimes satisfy additional conditions (initital value conditions or boundary conditions).

The One-Dimensional Heat Equation

In this paper, we present an idea toward parallel coarse grid correction (CGC) for the parareal algorithm. It is well known that such a CGC procedure is often the bottleneck of speedup of the parareal algorithm. For an ODE system with initial-value condition $u(0)=u_0$ the idea can be explained as follows. First, we apply the $\mathcal{G}$-propagator to the same ODE system but with a special condition $u(0)=\alpha u(T)$, where $\alpha\in\mathbb{R}$ is a crux parameter. Second, in each iteration of the parareal algorithm the CGC procedure will be carried out by the so-called diagonalization technique established recently. The parameter $\alpha$ controls both the roundoff error arising from such a diagonalization technique and the convergence rate of the resulting parareal algorithm. We show that there exists some threshold $\alpha^*$ such that the parareal algorithm with diagonalization-based CGC possesses the same convergence rate as that of the parareal algorithm with classical CGC if $|\alpha|\leq \alph...

Toward Parallel Coarse Grid Correction for the Parareal Algorithm

One-level domain-decomposition methods are in general not scalable, and coarse corrections are needed to obtain scalability. It has however recently been observed in applications in computational chemistry that the classical one-level parallel Schwarz method is surprizingly scalable for the solution of one- and two-dimensional chains of fixed-sized subdomains. We first review some of these recent scalability results of the classical one-level parallel Schwarz method, and then prove similar results for other classical one-level domain-decomposition methods, namely the optimized Schwarz method, the Dirichlet–Neumann method, and the Neumann–Neumann method. We show that the scalability of one-level domain decomposition methods depends critically on the geometry of the domain-decomposition and the boundary conditions imposed on the original problem. We illustrate all our results also with numerical experiments.

On the Scalability of Classical One-Level Domain-Decomposition Methods

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem $\mathbf{u}'+A\mathbf{u}=g$. The coarse propagator $\mathcal{G}$ is fixed to the backward-Euler method and three time integrators are chosen for the $\mathcal{F}$-propagator: the trapezoidal rule, the third-order diagonal implicit Runge--Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for $\mathcal{F}$ and $\mathcal{G}$ converges rapidly, but less is known when one uses for $\mathcal{F}$ the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio $J(=\Delta T/\Delta t)$ is small. We show that for a specified $\lambda_{\max}$(the maximal eigenvalue of $A$ or its upper bound), there exists some critical $J_{\rm cri}$ such that the parareal solvers derived from these three choices of $\mathcal{F}$ converge as fast as Parareal-Euler, provided $J\geq J_{\rm cri}$....

Convergence analysis for three parareal solvers

The parareal Schwarz waveform relaxation algorithm is a new space-time parallel algorithm for the solution of evolution partial differential equations. It is based on a decomposition of the entire ...

A Superlinear Convergence Estimate for the Parareal Schwarz Waveform Relaxation Algorithm

The additive Schwarz method does not converge in general when used as a stationary iterative method, it must be used as a preconditioner for a Krylov method. In the two level variant of the additive Schwarz method, a coarse grid correction is added to make the method scalable. We introduce a new coarse space for the additive Schwarz method which makes it convergent when used as a stationary iterative method, and show that an optimal choice even makes the method nilpotent, i.e. it converges in one iteration, independently of the overlap and the number of subdomains. We then show how this optimal choice can be approximated leading to a spectral harmonically enriched coarse space, based on interface eigenvalue problems. We present a convergence analysis of our new coarse corrections in the two subdomain case in two spatial dimensions, and also compare them with GenEO recently proposed by Spillane, Dolean, Hauret, Nataf, Pechstein, and Scheichl, and the local spectral multiscale coarse space proposed by Galvis and Efendiev.

Complete, Optimal and Optimized Coarse Spaces for Additive Schwarz

The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are non-overlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of space and time parallel strategies, we present and analyze parareal Dirichlet-Neumann and parareal Neumann-Neumann waveform relaxation for parabolic problems. Between these two algorithms, parareal Neumann-Neumann waveform relaxation is a space-time parallel algorithm, which increases the parallelism both in space and time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments.

Coupling Parareal and Dirichlet-Neumann/Neumann-Neumann Waveform Relaxation Methods for the Heat Equation

We propose a new domain decomposition waveform relaxation algorithm, which enables using different time steps across subdomains for parallel solving initial-boundary-value problems of linear parabolic equations. Distinct with the classical Schwarz waveform relaxation, the algorithm includes preconditioners to accelerate convergence and has greatly reduced memory requirements. A specific implementation of the local time stepping is also presented and its well-posedness is proved. The preconditioned system is analyzed at the continuous level. Finally, the efficiency of the algorithm is shown through numerical experiments.

A new domain decomposition waveform relaxation algorithm with local time-stepping

Dirichlet-Neumann and Neumann-Neumann methods are not only the parallel strategies in the spatial domain, but also can be used as a class of parallel methods in time. In this paper, we propose the Dirichlet-Neumann and Dirichlet-Neumann algorithms for time-periodic parabolic optimal control problems. By the Lagrange multiplier approach, a coupled system is obtained with a special time-periodic condition. For this coupled system, the Dirichlet-Neumann and Neumann-Dirichlet algorithms and their three variants are derived. We present the convergence analysis for all proposed algorithms. The numerical performance of the convergence factors is shown to illustrate our theoretical analysis. Based on our analysis, there is a class of algorithms with the better convergence compared with the 
natural Dirichlet-Neumann algorithm. Finally, numerical experiments are provided to illustrate the theoretical results.

Analysis of Dirichlet-Neumann and Neumann-Dirichlet Methods for Time-Periodic Parabolic Optimal Control Problems

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