Nonlinear Least Squares for Inverse Problems
Nonlinear Least Squares for Inverse Problems
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DOI：
10.1007/978-90-481-2785-6
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
G. Chavent
中科院分区：
文献类型：
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作者：
G. Chavent
在计算能力的提高和数值建模的进步的推动下，反问题的领域经历了快速扩展。几年前，当我开始研究这个领域时，我不知何故开始担心我的朋友们正在进行建模，其中为他们的方程的解提供了存在性、唯一性和稳定性结果，但由于问题的非线性，我大部分时间都受到限制，无法证明我的最小二乘目标函数是di？但随着我经验的增长，我开始相信，在逆问题得到适当修正后，？最后的最小二乘问题，即在计算机上解决的问题，应该是二次（Q）适定的，即我们提出的和可优化的：可优化性确保最小二乘函数的全局最小化实际上可以使用 e?古老的局部优化算法，以及该最小化器对于数据扰动而言是稳定的适定性。但绝大多数反问题都是非线性的，可用于分析的经典数学工具无法为这些关键问题提供答案：例如，紧性将确保存在性，但不能提供唯一性结果，并且不能提供有关寄生局部最小或平稳点存在或不存在的信息……
The domain of inverse problems has experienced a rapid expansion, driven by the increase in computing power and the progress in numerical modeling. When I started working on this domain years ago, I became somehow fr-tratedtoseethatmyfriendsworkingonmodelingwhereproducingexistence, uniqueness, and stability results for the solution of their equations, but that I was most of the time limited, because of the nonlinearity of the problem, to provethatmyleastsquaresobjectivefunctionwasdi? erentiable.... Butwith my experience growing, I became convinced that, after the inverse problem has been properly trimmed, the? nal least squares problem, the one solved on the computer, should be Quadratically (Q)-wellposed, thatis, both we-posed and optimizable: optimizability ensures that a global minimizer of the least squares function can actually be found using e? cient local optimization algorithms, and wellposedness that this minimizer is stable with respect to perturbation of the data. But the vast majority of inverse problems are nonlinear, and the clas-cal mathematical tools available for their analysis fail to bring answers to these crucial questions: for example, compactness will ensure existence, but provides no uniqueness results, and brings no information on the presence or absenceofparasiticlocalminimaorstationarypoints....