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Introduction
Published in Mario Bertero, Patrizia Boccacci, Christine De Mol, Introduction to Inverse Problems in Imaging, 2021
Mario Bertero, Patrizia Boccacci, Christine De Mol
Other examples, however, are more appropriate for the modern applications of inverse methods. In scattering and diffraction theory, the direct problem is the computation of the scattered (or diffracted) waves from the knowledge of the sources and obstacles, while the inverse problem consists in the determination of the obstacles from the knowledge of the sources and of the scattered (or diffracted) waves. Inverse problems of this kind are fundamental for various methods of non-destructive evaluation (including medical imaging), which consist in sounding an object by means of a suitable radiation. Another example of direct problem in wave-propagation theory is the computation of the field radiated by a given source, for instance, the radiation pattern of a given antenna; then the inverse problem is the determination of the source from the knowledge of the radiated field (in this case, the determination of the current distribution in the antenna from the knowledge of the radiation pattern). Analogously in potential theory, which is basic in geodesy, the direct problem is the determination of the potential generated by a known mass distribution, while the inverse problem is the determination of the mass distribution from the values of the potential, and so on.
Basic Concepts
Published in M. Necati Özisik, Helcio R. B. Orlande, Inverse Heat Transfer, 2021
M. Necati Özisik, Helcio R. B. Orlande
Although the solution of inverse problems does not necessarily make use of optimization techniques, many popular methods are nowadays based on the minimization of an objective function. Despite their similarities, inverse and optimization problems are conceptually different. Inverse problems are concerned with the identification of unknown quantities appearing in the mathematical formulation of problems by using measurements of the system response. On the other hand, optimization problems generally deal with the minimization or maximization of a certain objective or cost function in order to find design variables that will result in desired state variables. Engineering applications of optimization techniques are very often concerned with the minimization or maximization of different quantities such as minimum weight (e.g., lighter airplanes), minimum fuel consumption (e.g., more economic cars) and maximum autonomy (e.g., longer range airplanes). In contrast to inverse problems, solution uniqueness may not be an important issue for optimization problems, as long as the solution obtained is physically feasible and can be practically implemented [58].
Tackling Heterogeneity in Groundwater Numerical Modeling: A Comparison of Linear and Inverse Geostatistical Approaches— Example of a Volcanic Aquifer in the East African Rift
Published in M. Thangarajan, Vijay P. Singh, Groundwater Assessment, Modeling, and Management, 2016
The use of strategies meant to transform an ill-posed problem into a well-posed problem is generally known as regularization. Regularization is a way of alleviating the ill-posedness of the inverse problem through incorporation of prior information into the objective function (Christensen and Doherty, 2008; Doherty, 2003). The mathematical background of this procedure is detailed in Doherty (2003), Doherty and Hunt (2010), Doherty et al (2010), and Tonkin and Doherty (2005). With regularization, the number of parameters can exceed the number of observations. The pilot point approach and regularization can be used in conjunction with the well known parameter estimation software PEST (Doherty, 2004). As a result, complex hydraulic conductivity distributions can be defined, resulting in extremely low residual error.
Alpert multi-wavelets for functional inverse problems: direct optimization and deep learning
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2023
A key input to inverse problems in general is the data used to inform the model about the unknown field. In such a heat transfer problem, we consider measurements of the temperature as a function of time at several points in the 1 D bar. We study the effect of the temperature data on the results by performing the optimization using data collected from different numbers of equidistant thermocouples mounted on the 1 D bar. We plot the heat source field estimated from several numbers of thermocouples at a given time in order to have a 1 D plot as shown in Figure 7. The plots show that a good agreement is obtained between the true and optimal heat source when data from more than 2 thermocouples are used for both copper and stainless steel materials. The optimization algorithm is able to accurately capture the location of the front (maximum value of f).
Least squares formulation for ill-posed inverse problems and applications
Published in Applicable Analysis, 2022
Eric Chung, Kazufumi Ito, Masahiro Yamamoto
Recall that (4) is a regularization problem and attempts to transform the original ill-posed problem into a well-posed one, so that it is stable and can be numerically solved. As we have analysed for the backward heat equation example (2) and for the Cauchy problem in Section 3.1 below, we choose the regularization term appropriately and verify that the condition (17) is satisfied for each inverse problem. The regularization term is chosen for approximate solutions to the ill-posed inverse problems (not necessary for the true solution ) and it aims to prevent instability. Thus, the solution to (4) depends continuously on and in the graph norm W in the sense of Theorem 2.1, i.e. (7) holds.
A reduced-order modelling for real-time identification of damages in multi-layered composite materials
Published in Inverse Problems in Science and Engineering, 2021
Yu Liang, Xiao-Wei Gao, Bing-Bing Xu, Miao Cui, Bao-Jing Zheng
As a type of inverse problems, inverse heat conduction problems (IHCPs), exist in aerospace field such as the measurement of boundary conditions for reentry flight vehicle [6] or a scramjet combustor [7], designing of thermal protection systems[8] and other engineering fields [9,10]. However, inverse problems are generally ill-posed, and thus, each new problem can include challenges relating to existence, uniqueness, and stability. The traditional way for solving inverse problems is to cast the inverse problem as an optimization problem. The results are obtained by repeating the global search and iteration, thus, it requires a lot of computing resources and very time-consuming. Several different techniques in references [11,12] could solve optimization problems and have been used in multilayered materials [13]. The techniques could be roughly divided into two categories, stochastic methods and gradient-based methods [14]. Compared with the gradient methods, the stochastic algorithm avoids local optimum to some extent. Genetic Algorithm (GA), one of the stochastic global optimal algorithms, was developed by reference to the natural selection and genetic evolution mechanism of living beings [15]. Individuals are created by three stochastic operations: selection, crossover and mutation. Thus, in order to improve computational efficiency while maintaining the excellent characteristics of random algorithms, the model reduction strategy is introduced into the calculation of inverse problem [16,17].