Numerical Methods for Parametric Partial Differential Equations

本世纪最重大的科学挑战之一是准确描述和计算复杂的过程，如气候变化、污染物流动、基因组学，甚至社交媒体和金融。虽然可以为这些过程创建数学模型，但模型中的大量参数抑制了使用传统计算工具进行快速可靠的预测。此外，还有数学模型的有效性问题。拟议的研究提出了新的数学思想，主要是基于模型简化，以确定各种参数的重要性，并推导出更简单的模型，仍然忠实地描述了基本过程。这反过来又导致更准确和成本更低的计算模型，可以在当今现有的计算资源中实现。 该项目还研究了如何量化模型和过程数据观测参数的不确定性。该项目研究了参数偏微分方程（PDE）中三个要求苛刻的计算任务。其中第一个，所谓的前向问题，寻求创建快速和准确的在线求解器的偏微分方程时，给定的参数查询。这样的在线求解器被用在无数的应用程序中，这些应用程序试图通过参数选择来优化性能。第二个寻求最佳的方法来计算状态的偏微分方程的观测数据。与此相关的第三个问题是从观测数据估计偏微分方程的参数。由于大量的参数，用于这种高维问题的传统数值方法面临所谓的“维数灾难”，即，它们不能在合理的计算时间内获得期望的计算精度。该研究通过开发基于稀疏性和高度非线性近似（如n项字典近似）的模型降阶新方法来规避这一困难。还将建立逆参数估计的基础结果，证明Lipschitz光滑的前向和逆映射的最小光滑条件下的参数。这些基础性的结果再加上简化的建模，以创建参数估计和模型验证的数值方法。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

One of the most significant scientific challenges of this century is the accurate description and computation of complex processes such as climate change, contaminant flow, genomics, and even social media and finance. While one can create a mathematical model for these processes, the large number of parameters in the model inhibits the use of traditional computational tools for fast and reliable predictions. In addition, there is the question of the efficacy of the mathematical model. The proposed research puts forward new mathematical ideas, based primarily on model reduction, to determine the importance of the various parameters and derive simpler models that still faithfully describe the underlying process. This, in turn, leads to more accurate and less costly computational models that can be implemented within today's existing computing resources. The project also investigates how to quantify uncertainty in both the model and the parameters from data observations of the process.This project investigates three demanding computational tasks in parametric partial differential equations (PDEs). The first of these, called the forward problem, seeks the creation of fast and accurate online solvers for the PDE when given a parameter query. Such online solvers are used in a myriad of applications that seek to optimize performance through parameter selection. The second seeks optimal methods to compute the state of the PDE from observational data. Related to this is the third problem of estimating the parameters of the PDE from observational data. Because of the large number of parameters, traditional numerical methods for such high dimensional problems face the so-called "curse of dimensionality", i.e., they cannot obtain the desired accuracy of computation in a reasonable computational time. The proposed research circumvents this difficulty by developing novel methods of model reduction based on sparsity ad highly nonlinear approximation such as n-term dictionary approximation. Foundational results will also be established for inverse parameter estimation that prove Lipschitz smoothness for the forward and inverse maps under minimal smoothness conditions on the parameters. These foundational results are then coupled with reduced modeling to create numerical methods for parameter estimation and model verification.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.