Inverse Problems for Nonlinear Partial Differential Equations

该项目将解决反问题一般领域的三个主题，即恢复所需未知量所需的数据量，数据测量结果的稳定性，以及从测量数据到所需未知量的算法的存在与否。其中每一项都将是本提案工作的重要组成部分。其中第一个是关键的，从数学的角度来看，通常是最具挑战性的。如果没有这样的结果，我们就不能保证，即使找到了数学问题的解，也能使我们把它与实际的物理解联系起来。 在每一项工作中，我们预计这将需要最大的努力和挑战。 解决稳定性问题对我们来说至关重要：“给定我们构建的解决方案和实际解决方案之间的公差水平，允许我们实现这一目标的数据测量的最大误差是多少。“当然，要实现这一点，需要一种重建方法，每种算法，即使它提供了一个解决方案，也可能需要不同的数据误差范围。 因此，在某种意义上，我们必须回答的问题不仅是是否可以找到计算算法，而且在什么意义上它接近最优？后一个问题是一个数学强的本科生可以从事的工作。 指导这些学生将是这项工作的一个方面。该项目将支持3个本科生每年的3年补助金。具体来说，恢复的非线性项在非线性反应扩散方程和系统的抛物型寻求;也就是说，系数，如电导率或反应或相互作用的条款，取决于解决方案本身。 这里的一个例子是一个复杂的物种间相互作用项中的（空间或环境变量）速率系数，它本身必须被确定，这在复杂的流行病模型中是典型的。还考虑了发生在例如医学成像中的非线性双曲方程。 非线性声学具有基本上表示要重建的对象的项，并且该项耦合到第二项，该第二项产生于非线性模型并且表现为偏微分算子的首项中的系数。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这更复杂，并额外寻求其恢复。阻尼或衰减波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比例，因此在基本方程中加入了时间导数项。 在诸如声学、粘弹性、结构振动和地震波传播的应用中经常观察到，阻尼的大小是频率相关的并且服从幂律行为。 一个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阶导数或分数阶幂。 我们的目的是探讨这些影响，特别强调问是否反问题更容易处理（即，在病态和收敛的数值方法）的两种类型的阻尼。 在所有情况下，迭代方案的分析，以恢复未知的条款是工作的一个基本特征。这个奖项反映了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。

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.