Inverse Problems Light: Numerical Differentiation
Inverse Problems Light: Numerical Differentiation
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DOI：
10.1080/00029890.2001.11919778
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发表时间：
2001-06
期刊：
影响因子：
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通讯作者：
M. Hanke;O. Scherzer
中科院分区：
文献类型：
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作者：
M. Hanke;O. Scherzer
涉及非线性扩散系数a: IR IR +。式（1.1）也可以作为液体流动使多孔介质饱和的模型，其中a (u)与孔隙的毛细压力有关。在某些工业应用中，数值模拟可能需要为u求解（1.1）。我们称之为直接问题。在这些模拟中，使用系数a(u)是至关重要的，它不仅在质量上是正确的，而且是合理准确的。不幸的是，文献中列出的a (u)值通常只能提供对真实系数的粗略猜测；在这种情况下，模拟不可能是可靠的。因此，从实验数据中识别扩散系数a （u)（对于某些横坐标x E（0,1）和0 < t < t）通常是u(x, t)）通常是要清除的第一个障碍。这是相关的逆问题。解决反问题的标准方法是输出最小二乘法，它试图使用梯度或牛顿型方法来更新扩散系数，使给定数据与模拟量相匹配。或者，可以将（1.1）视为a (u)的线性方程。要建立这个方程，需要对数据[6]进行数值微分。这种方法称为方程误差法。必须强调的是，逆问题通常是非常病态的：例如，a（.）的微小变化对（1.1）中的解u几乎没有影响，因此，在u存在测量误差的情况下，人们不能期望a的高分辨率重建。事实上，如果不适当地考虑u的小误差，u的小误差可能会导致计算a的大误差。数据的数值微分包含了复杂（线性）逆问题可能表现出的许多微妙之处和陷阱；然而，它很容易理解和分析。因此，我们可以说，数值微分本身就是基础数值分析课程中求解反问题的理想模型。为了支持这一说法，我们重新审视了一种众所周知的噪声数据数值微分算法，并提出了一种新的误差界。该方法和误差界可以解释为不适定问题正则化理论中最重要的结果之一的实例。尽管如此，我们的演示是在一个非常基础的水平上，除了标准的n维微积分和三次样条的概念之外，不需要任何先验知识。Groetsch的书[4]在基本技术层面上提出了其他现实的逆问题。在[1]中可以找到进一步的例子和对这些例子的稳定解计算的正则化理论的严格介绍。
involving a nonlinear diffusion coefficient a: IR iR+. Problem (1.1) also serves as a model for the saturation of porous media by liquid flow, in which case a (u) is related to the capillary pressure of the pores. In certain industrial applications a numerical simulation may require solving (1.1) for u. We call this the direct problem. In these simulations it is crucial that a coefficient a(u) be used that is not only qualitatively correct but also reasonably accurate. Unfortunately, tabulated values for a (u) from the literature often provide only a rough guess of the true coefficient; in this case simulations are not likely to be reliable. Consequently, identification of the diffusion coefficient a (u) from experimental data (typically, u(x, t) for some abscissa x E (0, 1) and 0 < t < T) is often the first hurdle to clear. This is the associated inverse problem. A standard method to solve the inverse problem is the output least squares method, which tries to match the given data with simulated quantities using a gradient or Newton type method for updating the diffusion coefficient. Alternatively, one can consider (1.1) as a linear equation for a (u). To set up this equation requires numerical differentiation of the data [6]. This approach is called the equation error method. It must be emphasized that inverse problems are often very ill-conditioned: for example, small changes in a (.) have little effect on the solution u in (1.1), and consequently one cannot expect high resolution reconstructions of a in the presence of measurement errors in u. Indeed, small errors in u may cause large errors in the computed a if they are not taken into account appropriately. Numerical differentiation of the data encompasses many subtleties and pitfalls that a complex (linear) inverse problem can exhibit; yet it is very easy to understand and analyze. For this reason one could say that numerical differentiation itself is an ideal model for inverse problems in a basic numerical analysis course. To support this statement we revisit a well-known algorithm for numerical differentiation of noisy data and present a new error bound for it. The method and the error bound can be interpreted as an instance of one of the most important results in regularization theory for ill-posed problems. Still, our presentation is on a very basic level and requires no prior knowledge besides standard n-dimensional calculus and the notion of cubic splines. Groetsch's book [4] presents other realistic inverse problems on an elementary technical level. Further examples and a rigorous introduction to regularization theory for the computation of stable solutions to these examples can be found in [1].