Abstract: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 recently been proposed, based on splitting the projected vector field or by applying projected Runge–Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Pa… Show more
“…Proof. The proof is in the spirit of existing results from the literature [7,21,28,29] with similar techniques. If 𝑘 = 0, using definition (14) for 𝑢 𝑛 0 , we have for 0 ≤ 𝑛 ≤ 𝑁 − 1,…”
Section: Preliminary Resultsmentioning
confidence: 71%
“…Proof. The proof is in the spirit of existing results from the literature [7,21,28,29] with similar techniques based on the use of generating functions (A.4). We also refer to [23] for the convergence study of several parallel in time algorithms with generating functions.…”
Section: The Main Convergence Theoremmentioning
confidence: 76%
“…where we have used (7) and (8) to derive the second to last inequality. For 𝑘 ≥ 1, starting from ( 14), we have…”
In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections
computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver’s accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.
“…Proof. The proof is in the spirit of existing results from the literature [7,21,28,29] with similar techniques. If 𝑘 = 0, using definition (14) for 𝑢 𝑛 0 , we have for 0 ≤ 𝑛 ≤ 𝑁 − 1,…”
Section: Preliminary Resultsmentioning
confidence: 71%
“…Proof. The proof is in the spirit of existing results from the literature [7,21,28,29] with similar techniques based on the use of generating functions (A.4). We also refer to [23] for the convergence study of several parallel in time algorithms with generating functions.…”
Section: The Main Convergence Theoremmentioning
confidence: 76%
“…where we have used (7) and (8) to derive the second to last inequality. For 𝑘 ≥ 1, starting from ( 14), we have…”
In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections
computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver’s accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.
“…However, TDB-CUR ultimately achieves lower error than PRK4 as the rank is increased. Recently, a parallel in-time integrator is introduced that can accelerate the computation of DLRA [55]. Figure 6Stochastic Burgers equation with nonlinear diffusion: ( a ) Error versus time.…”
Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.
“…Low-rank integrators that do not depend on the manifold's curvature have been designed to move only along flat subspaces on the manifold, thereby inheriting the stability region of the original, full-rank problem. Existing robust integrators are the projector-splitting (PS) integrator [40,31], basis-update & Galerkin (BUG) integrators [9,7,8,6], and projection methods [32,11,53,4]. While the projector-splitting integrator has proven to yield accurate solutions for various problems, the fact that it includes a substep that evolves the solution backward in time can lead to instabilities not only for parabolic problems but also for numerical stabilization terms in hyperbolic problems [37] and problems exhibiting strong scattering [17].…”
The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L 2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.
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