Stability of the Parareal Time Discretization for Parabolic Inverse Problems

Daoud, Daoud S.

doi:10.1007/978-3-540-34469-8_32

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…In several cases, parareal in time algorithm gives an impressive rate of convergence for the linear diffusion equations or with the same efficiency with non-linearity [7]. This algorithm is studied and shows stability and convergence results [8,9,10,11,12] particularly for diffusion system and others. Also it presents efficiency in parallel computer simulation.…”

Section: Introductionmentioning

confidence: 92%

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

Baudron

Lautard

Maday

et al. 2014

Journal of Computational Physics

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We present a parareal in time algorithm for the simulation of neutron diffusion transient model. The method is made efficient by means of a coarse solver defined with large time steps and steady control rods model. Using finite element for the space discretization, our implementation provides a good scalability of the algorithm. Numerical results show the efficiency of the parareal method on large light water reactor transient model corresponding to the Langenbuch-Maurer-Werner (LMW) benchmark [1].

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Section: Introductionmentioning

confidence: 92%

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

Baudron

Lautard

Maday

et al. 2014

Journal of Computational Physics

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“…Theorem 4.1. Let , ( ) be the regularization solution of the problem (5) with the measured data satisfying (27). Let ( ) be the exact solution of the problem (1) and satisfy a priori condition (28) for any ≥ 0.…”

Section: Convergence Analysismentioning

confidence: 99%

“…In addition to the achieved high parallelism in space, a lot of recent advances in various parallel-in-time (PinT) algorithms for solving forward time-dependent PDE problems were reviewed in [12]. However, the application of such PinT algorithms to ill-posed inverse PDE problems were rarely investigated in the literature, except in a short paper [5] about the parareal algorithm for a different parabolic inverse problem and another earlier paper [22] based on numerical (inverse) Laplace transform techniques in time. One obvious difficulty is how to address the underlying regularization treatment in the framework of PinT algorithms, which seems to be highly dependent on the regularized problem structure and discretization schemes.…”

Section: Introductionmentioning

confidence: 99%

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

Jiang¹,

Li²,

Wang³

2022

Preprint

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Inverse source problems arise often in real-world applications, such as localizing unknown groundwater contaminant sources. Being different from Tikhonov regularization, the quasi-boundary value method has been proposed and analyzed as an effective way for regularizing such inverse source problems, which was shown to achieve an optimal order convergence rate under suitable assumptions. However, fast direct or iterative solvers for the resulting all-at-once large-scale linear systems have been rarely studied in the literature. In this work, we first proposed and analyzed a modified quasi-boundary value method, and then developed a diagonalization-based parallel-in-time (PinT) direct solver, which can achieve a dramatic speedup in CPU times when compared with MATLAB's sparse direct solver. In particular, the time-discretization matrix is shown to be diagonalizable, and the condition number of its eigenvector matrix is proven to exhibit quadratic growth, which guarantees the roundoff errors due to diagonalization is well controlled.Several 1D and 2D examples are presented to demonstrate the very promising computational efficiency of our proposed method, where the CPU times in 2D cases can be speedup by three orders of magnitude.

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“…Besides the achieved high parallelism in space, we have seen a lot of recent advances in various parallel-in-time (PinT) algorithms 1 for solving forward time-dependent PDE problems [14]. However, the application of such PinT algorithms to ill-posed backward heat conduction problems were rarely investigated in the literature, except in one short paper [10] about the parareal algorithm for a different parabolic inverse problem and another earlier paper [26] based on numerical (inverse) Laplace transform techniques in time. One obvious difficulty is how to address the underlying regularization treatment in the framework of PinT algorithms, which seems to be highly dependent on the problem structure.…”

Section: Introductionmentioning

confidence: 99%

Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat Conduction Problems

Li¹

2021

Preprint

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In this paper we proposed two new quasi-boundary value methods for regularizing the ill-posed backward heat conduction problems. With a standard finite difference discretization in space and time, the obtained all-at-once nonsymmetric sparse linear systems have the desired block ω-circulant structure, which can be utilized to design an efficient parallel-in-time (PinT) direct solver that built upon an explicit FFT-based diagonalization of the time discretization matrix. Convergence analysis is presented to justify the optimal choice of the regularization parameter. Numerical examples are reported to validate our analysis and illustrate the superior computational efficiency of our proposed PinT methods.

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Stability of the Parareal Time Discretization for Parabolic Inverse Problems

Cited by 4 publications

References 12 publications

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat Conduction Problems

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