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

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属性、波频、测量数据中的不确定性、边界处的物理状态或域几何等。这些大量重复的模拟需要大量的计算机时间，并且通常在计算上是困难的。为解决这一问题，根据相对较少的精心选择的参数及其相关的预先计算的高精度“快照”解决方案，开发了缩小基数方法，为大量参数值提供高效和准确的替代解决方案。一旦预计算出这些“快照”解决方案，就可以快速高效地计算任何参数值的代理解决方案。此外，它们的精确度由数学上严格的误差界来证明。PIS最近引入了一种简化基法，称为简化配置法，该方法对非线性问题更有效。该项目的目标是开发这种新的方法，使其在精确识别“快照”解和找到“快照”解后的计算代理解方面都更加高效，将降基和降维配置方法与不确定量化技术相结合，以有效地处理复杂问题，并确保简化配置方法保持了底层快照解所满足的某些强稳定性。这些目标将使一种有效和强大的简化配置法能够应用于广泛的参数依赖现象。减基方法最初是为偏微分方程组的Galerkin公式而发展的，最近被PI扩展用于配置公式，这通常是非线性问题的首选。事实上，对于一大类偏微分方程(PDE)，这种新的约化配置法(RCM)比典型的约化基法(RBM)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约化空间来改进离线阶段；(2)通过各种数学方法来选择配置点、开发预条件和创建共素多重网格方法，使RCM的在线阶段更加健壮和高效；(3)将RBM/RCM与不确定性量化方法相结合，以有效地处理高维随机空间中的问题；(4)改进RBM/RCM方法，保证代理解保持底层快照解所满足的强稳定性。严格的数学分析和创新高效的算法设计将结合起来，以改进缩减基数的方法，从而使它们对大类问题更加有效和稳健。这个项目将改变RCM，使其对于线性和非线性问题都是有效和健壮的。此外，该项目将结合线性和非线性偏微分方程降阶建模领域以及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缩减基方法的混合方法。由此产生的算法将通过改进高参数维度的效率和健壮性处理来影响不确定性量化领域。最后，对RBM/RCM进行改进，以保证正性等非线性性质的保持。这一新颖的发展将解决人们对替代解决方案质量的主要担忧。

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.