英文标题：
《Hybrid PDE solver for data-driven problems and modern branching》
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作者：
Francisco Bernal and Gon\\c{c}alo dos Reis and Greig Smith
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最新提交年份：
2017
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英文摘要：
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.
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中文摘要：
大规模偏微分方程的数值解，例如发生在数据驱动应用中的偏微分方程，不可避免地需要强大的并行计算机和定制的并行算法来尽可能充分地利用它们。事实上，对现实问题的并行化和可伸缩性的考虑通常非常关键，足以保证在建模阶段得到承认。本文的目的是推广概率区域分解（PDD）方法，这是一种新的PDE并行化方法，具有良好的可扩展性。该思想利用了PDE的随机表示及其通过蒙特卡罗逼近，并结合确定性高性能PDE解算器。我们描述了PDD的组成及其在数据科学领域的适用性。特别是，我们强调了使用分支扩散的非线性偏微分方程随机表示的最新进展，这大大拓宽了偏微分方程的范围。我们设想这项工作是一本词典，为大规模PDE从业者提供关于确定性和概率数值方法界面上非标准但高度并行的方法学的最新算法和技术的参考。在结束这项工作时，我们邀请大家讨论完全非线性的案例和开放的研究问题。
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分类信息：

一级分类：Mathematics 数学
二级分类：Numerical Analysis 数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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