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Rigorous Development of an Efficient Reduced Collocation Approach for High-Dimensional Parametric Partial Differential Equations

严格开发高维参数偏微分方程的高效简化配置方法

基本信息

批准号：
1719698
负责人：
Yanlai Chen
金额：
$ 15.85万
依托单位：
University of Massachusetts, Dartmouth
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719698&HistoricalAwards=false
关键词：
Rigorous Development Efficient Reduced Collocation

项目摘要

Many physical phenomena depend on a range of parameters, and to understand the phenomena many repeated simulations for different parameter values are required. Such parameters may describe properties of a material, wave frequencies, uncertainties in measured data, physical states at the boundaries, or domain geometry, among others. These massively repeated simulations require significant computer time, and are frequently computationally prohibitive. Reduced basis methods were developed to resolve this issue by providing efficient and accurate surrogate solutions for the large space of parameter values, based on a relatively small number of carefully selected parameters and their related pre-computed highly accurate "snapshot" solutions. Once these "snapshot" solutions are pre-computed, computing the surrogate solutions for any parameter values is quick and efficient. Moreover, their accuracy is certified by a mathematically rigorous error bound. A variant of the reduced basis method, called the reduced collocation method, was recently introduced by the PIs that is more efficient for nonlinear problems. The goals of this project are to develop this new method to be more efficient, both in precise identification of the "snapshot" solutions and in computing the surrogate solutions once the "snapshot" solutions are found, to integrate reduced basis and reduced collocation methods with uncertainly quantification techniques to efficiently handle complex problems, and to ensure that certain strong stability properties satisfied by the underlying snapshot solutions are preserved by the reduced collocation methods. These goals will enable an efficient and powerful reduced collocation method that can be applied to a wide range of parameter-dependent phenomena. Reduced basis methods were originally developed for use with a Galerkin formulation of a partial differential equations, and recently extended by the PIs for collocation formulations, which are frequently preferred for nonlinear problems. In fact, this new reduced collocation method (RCM) is more efficient than the typical reduced basis method (RBM) for a large class of partial differential equations (PDEs). The goal of this project is to rigorously develop the new RCM in order to: (1) improve the offline stage by introducing novel approaches to building a more optimal reduced space faster; (2) make the online stage of the RCM more robust and efficient for nonlinear problems through a variety of mathematical approaches to selecting the collocation points, developing pre-conditioners, and creating a co-prime multi-grid approach; (3) integrate the RBM/RCM with uncertainty quantification approaches to efficiently handle problems with high dimensional random spaces; and (4) enhance the RBM/RCM approaches to guarantee that the surrogate solutions preserve the strong stability properties satisfied by the underlying snapshot solutions. Rigorous mathematical analysis and innovative and efficient algorithm design will be combined to improve the reduced basis approaches thereby making them more efficient and robust for large classes of problems. This project will transform the RCM to make it efficient and robust, for linear as well as nonlinear problems. Furthermore, this project will combine knowledge from the field of reduced order modeling for linear and nonlinear PDEs and the field of uncertainty quantification to create powerful new methods, hybrids of the generalized polynomial chaos method and reduced basis methods. The resulting algorithms will impact the field of uncertainty quantification by improving the efficiency and robust handling of high parameter dimensions. Finally, the RBM/RCM will be improved to guarantee the preservation of nonlinear properties such as positivity. This novel development will resolve a major concern about the quality of the surrogate solution.

许多物理现象依赖于一系列参数，为了理解这些现象，需要对不同参数值进行多次重复模拟。这些参数可以描述材料的特性、波频率、测量数据的不确定性、边界处的物理状态或域几何等。这些大规模重复的模拟需要大量的计算机时间，并且经常在计算上令人望而却步。为了解决这一问题，开发了约基方法，通过相对少量的精心选择的参数及其相关的预先计算的高精度“快照”解，为大空间的参数值提供高效、准确的替代解。一旦预先计算了这些“快照”解决方案，计算任何参数值的代理解决方案都是快速而有效的。此外，它们的准确性由数学上严格的误差界来证明。pi最近引入了一种简化基法的变体，称为简化配置法，它对非线性问题更有效。该项目的目标是发展这种新方法，使其在精确识别“快照”解和在找到“快照”解后计算替代解方面更有效，将减少基和减少搭配方法与不确定性量化技术相结合，以有效地处理复杂问题。并通过简化的配置方法保证了底层快照解所满足的强稳定性。这些目标将使一个有效和强大的约简搭配方法，可以应用于广泛的参数依赖现象。约基方法最初是为偏微分方程的伽辽金公式而开发的，最近由pi扩展到配置公式，这通常是非线性问题的首选。事实上，对于一类较大的偏微分方程，这种新的约简配置方法比典型的约简基方法更有效。该项目的目标是严格开发新的RCM，以便：(1)通过引入新的方法来更快地构建更优化的缩减空间，从而改善离线阶段；(2)通过多种数学方法选择配点、开发预调节器和创建共素数多网格方法，使RCM在线阶段对非线性问题更加鲁棒和高效；(3)将RBM/RCM与不确定性量化方法相结合，有效处理高维随机空间问题；(4)增强RBM/RCM方法，以保证替代解保持底层快照解所满足的强稳定性。将严谨的数学分析与创新高效的算法设计相结合，改进简化基方法，从而使其在处理大类别问题时更加高效和鲁棒。该项目将改变RCM，使其对线性和非线性问题都有效和健壮。此外，本项目将结合线性和非线性偏微分方程的降阶建模领域和不确定性量化领域的知识，创建强大的新方法，即广义多项式混沌方法和降基方法的混合方法。由此产生的算法将通过提高效率和鲁棒性来处理高参数维度，从而影响不确定度量化领域。最后，对RBM/RCM进行改进，以保证正性等非线性性质的保留。这一新颖的发展将解决关于替代溶液质量的主要问题。