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<span id="AppelEtAl2024">M. Appel and J. Alexandersen, “One-shot Parareal Approach for Topology Optimisation of Transient Heat Flow,” arXiv:2411.19030v1 [cs.CE], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2411.19030v1" target="_blank">http://arxiv.org/abs/2411.19030v1</a></span>
This paper presents a method of performing topology optimisation of transient heat conduction problems using the parallel-in-time method Parareal. To accommodate the adjoint analysis, the Parareal method was modified to store intermediate time steps. Preliminary tests revealed that Parareal requires many iterations to achieve accurate results and, thus, achieves no appreciable speedup. To mitigate this, a one-shot approach was used, where the time history is iteratively refined over the optimisation process. The method estimates objectives and sensitivities by introducing cumulative objectives and sensitivities and solving for these using a single iteration of Parareal, after which it updates the design using the Method of Moving Asymptotes. The resulting method was applied to a test problem where a power mean of the temperature was minimised. It achieved a peak speedup relative to a sequential reference method of 5\times using 16 threads. The resulting designs were similar to the one found by the reference method, both in terms of objective values and qualitative appearance. The one-shot Parareal method was compared to the Parallel Local-in-Time method of topology optimisation. This revealed that the Parallel Local-in-Time method was unstable for the considered test problem, but it achieved a peak speedup of 12\times using 32 threads. It was determined that the dominant bottleneck in the one-shot Parareal method was the time spent on computing coarse propagators.
Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block α-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the convergence of the preconditioned iteration
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by studying the spectral radius of the iteration matrix, and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified convergence analysis for the algorithm applied to u’+Au=f, where σ(A)⊂\mathbbC^+ with σ(A) being the spectrum of A∈\mathbbC^m\times m. For any one-step method (such as the Runge-Kutta methods) with stability function \mathcalR(z), we prove that the decay rate of the global error is bounded by α/(1-α), provided the method is stable, i.e., \max_λ∈σ(A)|\mathcalR(∆tλ)|\leq1. For any linear multistep method, such a bound becomes cα/(1-cα), where c\geq1 is a constant specified by the multistep method itself.
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Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block α-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the convergence of the preconditioned iteration
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+
by studying the spectral radius of the iteration matrix, and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified convergence analysis for the algorithm applied to u’+Au=f, where σ(A)⊂\mathbbC^+ with σ(A) being the spectrum of A∈\mathbbC^m\times m. For any one-step method (such as the Runge-Kutta methods) with stability function \mathcalR(z), we prove that the decay rate of the global error is bounded by α/(1-α), provided the method is stable, i.e., \max_λ∈σ(A)|\mathcalR(∆tλ)|\leq1. For any linear multistep method, such a bound becomes cα/(1-cα), where c\geq1 is a constant specified by the multistep method itself.
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Our proof only relies on the stability of the time-integrator and the estimate is independent of the step size ∆t and the spectrum σ(A).
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