Inverse Problems for Nonlinear Partial Differential Equations

这个项目将解决反问题一般领域中的三个主题，即恢复期望未知所需的数据量，根据数据测量的结果的稳定性，以及是否存在从测量数据到期望未知的算法。其中每一项都将是本提案工作的重要组成部分。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，即使找到数学问题的解决方案，也能使我们将其与实际的物理解决方案相关联。在我们预计的每一部作品中，这将需要最大的努力和挑战。回答稳定性问题将是我们确定以下问题的关键：“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，使我们能够实现这一点的数据测量中允许的最大误差是多少。”当然，要实现这一点，需要一种重建方法，而每种算法，即使它提供了解决方案，也可能需要数据上的不同误差界。因此，在某种意义上，我们必须回答的问题不仅是能否找到一种计算算法，而且在什么意义上它接近于最优？后一个问题是一个数学能力很强的本科生可以参与的问题。对这类学生的辅导将是这项工作的一个方面。该项目每年将资助3名本科生获得3年助学金。具体地说，寻求恢复非线性反应扩散方程和抛物型系统中的非线性项，即依赖于解本身的系数，如电导率、反应或相互作用项。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，它本身必须像复杂的流行病模型中的典型那样被确定。还考虑了出现在例如医学成像中的非线性双曲型方程。非线性声学具有本质上表示要重建的对象的项，并且该项耦合到由非线性模型产生的第二项，并且作为偏微分算子的前导项中的系数出现。保留非线性效应的最简单模型是将其作为恒等式运算符，但更现实的情况是假设这更复杂，并另外寻求其恢复。衰减波动方程广泛存在于物理和工程的许多领域。通常的假设是，阻尼机构与速度成正比，因此在基本方程中加入了时间导数项。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，阻尼值的大小与频率有关，并服从幂函数行为。一个典型的公式涉及非局部类型的算子，这些算子通常基于分数导数或微分算子的分数次方。其目的是探索这些影响，特别强调询问对于两种类型的阻尼，反问题是否更容易处理(即，在病态和数值方法的收敛方面)。在所有情况下，分析迭代方案以恢复未知项是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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.