Approximation of Functions with Parameter-Dependent or Stochastic Shock Locations Arising from Hyperbolic Partial Differential Equations

由双曲偏微分方程产生的参数相关或随机冲击位置的函数逼近

基本信息

批准号：
1912703
负责人：
Gerrit Welper
金额：
$ 6万
依托单位：
The University of Central Florida Board of Trustees
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：
2018-08-29 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1912703&HistoricalAwards=false
关键词：
Approximation Functions Parameter Dependent Stochastic

项目摘要

Many mathematical models of scientific or engineering problems are influenced by design parameters and random parameters. An example is the modeling of flow around an airfoil. A design parameter is its shape, and random parameters include, for example, perturbations in the air before a plane travels through it. These parameters pose some natural questions for engineers and the mathematical models they use: What is the best possible shape? Can we compute safety tolerances of the wing and limit the probability of failure? With increasing computational power, such questions attracted considerable attention in recent years. However, for physical phenomena with sharp transitions, only brute-force approaches are available, which quickly become challenging even with advanced computer hardware. In the wing example, one such transition is the sonic boom, a rapid change in the pressure around the wing at high speed. This project aims to develop new algorithms that can handle such rapid transitions in parametric models, far more efficiently than currently. The method under development will be applicable to a range of other engineering problems as well, such as environmental questions in groundwater flows or the simulation of bio-molecules.The goal of the project is the development of new approximation schemes for functions with parameter-dependent jumps or kinks, motivated by solutions of parametric and stochastic hyperbolic partial differential equations. These singularities pose serious challenges by deteriorating the convergence rates for established methods such as reduced basis methods, proper orthogonal decomposition, or polynomial chaos expansions. Recent work offers a new method to approximate functions with jump discontinuities and achieves super-polynomial convergence rates for many problems. It serves as a proof of principle that high-order methods are advantageous, but it needs to be developed further to make it practical: In most realistic problems, jumps not only move but they also interact, and parameter spaces are typically high-dimensional. Addressing these issues is the goal of this project. The new algorithms will be tested numerically, in particular regarding the convergence rates that can be achieved. In addition, the investigator plans to prove convergence rates for model classes of parametric functions.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.

许多科学或工程问题的数学模型都受设计参数和随机参数的影响。一个例子是围绕机翼的流量进行建模。设计参数是其形状，随机参数包括，例如，在平面穿过它之前，空气中的扰动。这些参数为工程师和他们使用的数学模型提出了一些自然问题：最佳形状是什么？我们可以计算机翼的安全公差并限制故障的可能性吗？随着计算能力的增加，这些问题近年来引起了人们的关注。但是，对于具有急剧过渡的物理现象，只有蛮力的方法可用，即使使用先进的计算机硬件，也很快就变得具有挑战性。在机翼示例中，这样的过渡是声音繁荣，这是在高速下方的机翼周围压力的迅速变化。该项目旨在开发可以在参数模型中处理如此快速过渡的新算法，比目前要高得多。正在开发的方法还适用于其他一系列工程问题，例如地下水流中的环境问题或生物分子的模拟。该项目的目的是开发新的近似方案，用于具有参数依赖性跳跃或扭结的函数的新近似方案，该功能是由参数和定位型超元素差异的解决方案动机。这些奇异性通过恶化诸如较低的基础方法，适当的正交分解或多项式混乱膨胀等既定方法的收敛速度来提出严重的挑战。最近的工作提供了一种新的方法，可以通过跳跃不连续性近似功能，并在许多问题上达到超级顺序收敛率。它可以证明高阶方法是有利的，但是需要进一步开发它才能使其实用：在最现实的问题中，跳跃不仅可以移动，而且它们也相互作用，而且参数空间通常是高维的。解决这些问题是该项目的目标。新算法将进行数值测试，特别是关于可以实现的收敛速率。此外，调查人员计划证明参数功能模型类别的融合率。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。

项目成果

期刊论文数量（2）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Transformed Snapshot Interpolation with High Resolution Transforms

DOI：
10.1137/19m126356x
发表时间：
2019-01
期刊：
SIAM J. Sci. Comput.
影响因子：
0
作者：
G. Welper
通讯作者：
G. Welper

Universality of Gradient Descent Neural Network Training

DOI：
10.1016/j.neunet.2022.02.016
发表时间：
2020-07
期刊：
Neural networks : the official journal of the International Neural Network Society
影响因子：
0
作者：
G. Welper
通讯作者：
G. Welper