Inverse Problems for Nonlinear Partial Differential Equations

非线性偏微分方程的反问题

基本信息

批准号：
2111020
负责人：
William Rundell
金额：
$ 21.03万
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：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111020&HistoricalAwards=false
关键词：
Inverse Problems Nonlinear Partial Differential

项目摘要

This project will address three topics in the general area of inverse problems and those are the amount of data needed to recover the desired unknown, the stability of the result in terms of the data measurements, and the existence or not of an algorithm to go from the measured data to the desired unknowns. Each of these will be an essential component of the work for this proposal. The first of these is critical and from a mathematical viewpoint typically the most challenging. Without such a result we have no guarantee that even finding a solution to the mathematical problem allows us to correlate this with the actual physical solution. In each of the works we anticipate this will require the greatest effort and challenges. Answering the stability question will be essential for us to determine: "given a tolerance level between our constructed solution and the actual one, what is the allowed maximal error in the data measurements that will allow us to achieve this." Of course, to carry this out one needs a reconstruction method and each algorithm, even if it provides a solution, may require a different error bound on the data. Thus in some sense the question we have to answer is not only if a computational algorithm can be found, but in what sense is it near to being optimal? This latter question is one where the work of a mathematically strong undergraduate student can be engaged. Mentoring of such students will be an aspect of this work. This project will support 3 undergraduate students each year of the 3 year grant. Specifically, the recovery of the nonlinear terms in nonlinear reaction-diffusion equations and systems of parabolic type is sought; that is, coefficients such as the conductivity or the reaction or interaction terms that depends on the solution itself. An example here is a (spatially or environment variable) rate coefficient in a complex inter-species interaction term that itself has to be determined as is typical in sophisticated epidemic models. Also considered are nonlinear hyperbolic equations occurring in, for example, medical imaging. Nonlinear acoustics has a term that essentially represents the object to be reconstructed and this is coupled to a second term that arises from the nonlinear model and appears as a coefficient in the leading term of the partial differential operator. The simplest model that retains the nonlinear effects is to take this to be the identity operator but a more realistic case is to assume this is more complex and additionally seek its recovery. The damped or attenuated wave equation occurs in many areas of physics and engineering. The usual assumption is the damping mechanism is proportional to velocity so that a time-derivative term is incorporated into the basic equation. It is often been observed in applications such as acoustics, viscoelasticity, structural vibration and seismic wave propagation, that the magnitude of the damping is frequency dependent and obeys a power law behavior. A typical formulation involves operators of nonlocal type and these are usually based on fractional derivatives or fractional powers of differential operators. The aim is to explore these effects with particular emphasis on asking whether the inverse problems are more tractable (that is, in terms of ill-conditioning and convergence of numerical methods) for both types of damping. In all cases, the analysis of iterative schemes to recover the unknown terms is an essential feature of the work.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.

该项目将解决逆问题一般领域的三个主题，即恢复所需未知数所需的数据量、数据测量结果的稳定性以及是否存在算法将测量数据转换为所需的未知数。其中每一项都将是该提案工作的重要组成部分。第一个是至关重要的，从数学角度来看通常是最具挑战性的。如果没有这样的结果，我们就无法保证即使找到数学问题的解决方案也能让我们将其与实际的物理解决方案联系起来。我们预计每件作品都需要最大的努力和挑战。回答稳定性问题对于我们确定：“给定我们构建的解决方案与实际解决方案之间的容差水平，数据测量中允许的最大误差是多少，这将使我们能够实现这一目标。”当然，要执行此操作需要一种重建方法，并且每种算法即使提供了解决方案，也可能需要对数据进行不同的误差限制。因此，从某种意义上说，我们必须回答的问题不仅是是否可以找到一种计算算法，而且是在什么意义上它接近最优？后一个问题是数学能力很强的本科生可以参与的工作。对此类学生的指导将是这项工作的一个方面。该项目将每年资助 3 名本科生，为期 3 年。具体来说，寻求非线性反应扩散方程和抛物型系统中非线性项的恢复；即取决于溶液本身的电导率或反应或相互作用项等系数。这里的一个例子是复杂的物种间相互作用项中的（空间或环境变量）速率系数，其本身必须像复杂的流行病模型中的典型情况一样被确定。还考虑了例如医学成像中出现的非线性双曲方程。非线性声学有一项本质上代表要重建的对象，并且该项与非线性模型产生的第二项耦合，并作为偏微分算子首项中的系数出现。保留非线性效应的最简单模型是将其作为恒等算子，但更现实的情况是假设它更复杂并另外寻求其恢复。阻尼或衰减波动方程出现在物理和工程的许多领域。通常的假设是阻尼机制与速度成正比，因此时间导数项被纳入基本方程中。在声学、粘弹性、结构振动和地震波传播等应用中经常观察到阻尼的大小与频率相关并且服从幂律行为。典型的公式涉及非局部类型的算子，这些算子通常基于微分算子的分数导数或分数幂。目的是探索这些影响，特别强调询问两种类型阻尼的反演问题是否更容易处理（即，就病态和数值方法的收敛而言）。在所有情况下，对迭代方案进行分析以恢复未知项是这项工作的一个基本特征。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。