On Transformation Methods and the Induced Parallel Properties for the Temporal Domain

Lai, Choi-Hong

doi:10.4203/csets.24.3

Cited by 7 publications

(5 citation statements)

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“…linear problems with constant coefficients in the time direction, and one needs a solver that works with complex numbers for (45). It is however a very successful and efficient time parallel solver when it can be used, see [74,78,52,17].…”

Section: Sheen Sloan and Thomée 1999mentioning

confidence: 99%

50 Years of Time Parallel Time Integration

Gander

2015

Contributions in Mathematical and Computational Sciences

281

226

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Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today's processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is affected (it is even determined) by the solution earlier in time, but not the other way round. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account. We show in this chapter how time domain decomposition methods were invented, and give an overview of the existing techniques. Time parallel methods can be classified into four different groups: methods based on multiple shooting, methods based on domain decomposition and waveform relaxation, space-time multigrid methods and direct time parallel methods. We show for each of these techniques the main inventions over time by choosing specific publications and explaining the core ideas of the authors. This chapter is for people who want to quickly gain an overview of the exciting and rapidly developing area of research of time parallel methods.

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Section: Sheen Sloan and Thomée 1999mentioning

confidence: 99%

50 Years of Time Parallel Time Integration

Gander

2015

Contributions in Mathematical and Computational Sciences

281

226

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“…Since the solutions at intermediate steps of Eq (2) are usually not of interest for in-painting problems, it is possible to apply the Laplace transform [2]. To evolve the solution to Eq (2) from u(x, y, T j ) to u(x, y, T j+1 ) linearisation technique is needed due to the non-linearity.…”

Section: Using Laplace Transform For 2d In-painting Modelmentioning

confidence: 99%

“…The most recent approach is to use the concept of the transformation methods theory to recast time domain problems into a transformed space that does not involve the time. There are several transform methods that have been investigated such as the Laplace transform, similarity transform, Henkel transform and the Boltzmann transform [1][2][3][4][5]. The Laplace transform solution methods for time dependent problems, which transform parabolic problems into elliptic problems in the Laplace transformed space, have been considered by many authors [7][8].…”

Section: Introductionmentioning

confidence: 99%

A Laplace Transform Method for the Image In-Painting

Kokulan

Chen

2015

Journal of Algorithms & Computational Technology

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There are many image processing techniques based on partial differential equations that perform well, but they consume much computational time. It's vital that rapid and efficient ways of solving these equations are developed. Use of the Laplace transforms permits solution to the time dependent problems in a parallel environment. The solution procedure requires numerical computation of an inverse Laplace transform of which the Stehfest method was examined in the tests. We investigated the performance and efficiency of using the Laplace transform technique for the solution of a mathematical model related to image in-painting and compared the results with temporal integration.

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“…Obviously, Laplace transform belongs to the direct methods where [12] and [36] are cited as pioneering works in this category. Our paper is based on the algorithm described in [12] that is called Gaver-Stehfest algorithm, which has a good accuracy on a fairly wide range of functions as illustrated in [13], thereafter they developed an iterative scheme in [26,25] through linearization of implied volatility, leading to a series of iterations throughout the time-stepping process.…”

Section: Introductionmentioning

confidence: 99%

Asynchronous time-parallel method based on Laplace transform

Magoulès

Zou

2020

International Journal of Computer Mathematics

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Laplace transform method has proved to be very efficient and easy to parallelize for the solution of time-dependent problems. However, the synchronization delay among processors implies an upper bound on the expectable acceleration factor, which leads to a lot of wasted time. In this paper, we propose an original asynchronous Laplace transform method formalized for quasilinear problems based on the well-known Gaver-Stehfest algorithm. Parallel experiments show the convergence of our new method, as well as several interesting properties compared with the classical algorithms.

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On Transformation Methods and the Induced Parallel Properties for the Temporal Domain

Cited by 7 publications

References 0 publications

50 Years of Time Parallel Time Integration

50 Years of Time Parallel Time Integration

A Laplace Transform Method for the Image In-Painting

Asynchronous time-parallel method based on Laplace transform

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