Numerical Methods for Parametric Partial Differential Equations

参数偏微分方程的数值方法

基本信息

批准号：
1817603
负责人：
Ronald DeVore
金额：
$ 36.92万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1817603&HistoricalAwards=false
关键词：
Numerical Methods Parametric Partial Differential

项目摘要

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.

本世纪最重要的科学挑战之一是对复杂过程的准确描述和计算，例如气候变化，污染物流，基因组学甚至社交媒体和金融。尽管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传统的计算工具来快速可靠的预测。另外，还有一个数学模型功效的问题。拟议的研究主要基于模型降低提出了新的数学思想，以确定各种参数的重要性，并得出更简单的模型，这些模型仍然忠实地描述了基本过程。反过来，这将导致更准确，更昂贵的计算模型，这些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该过程数据观察的模型和参数中的不确定性。本项目研究了参数偏微分方程（PDES）中的三项要求的三个要求的计算任务。其中的第一个（称为远期问题）在给出参数查询时寻求为PDE快速准确的在线求解器创建。此类在线求解器用于多种应用程序，这些应用程序试图通过选择参数来优化性能。第二个寻求从观察数据中计算PDE状态的最佳方法。与此相关的是从观察数据估算PDE参数的第三个问题。由于参数大量，这种高维问题的传统数值方法面临着所谓的“维度的诅咒”，即，它们无法在合理的计算时间内获得所需的计算准确性。拟议的研究通过基于稀疏性AD的高度非线性近似（例如n期 - 词典近似值）开发新的模型还原方法来避免这种困难。还将为反参数估计而建立基础结果，该估计在最小的平滑度条件下，在参数上证明了Lipschitz的平滑度。然后，这些基本结果与减少的建模相结合，以创建用于参数估计和模型验证的数值方法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。