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Bridge damage detection utilizing dynamic force obtained from moving vehicle acceleration
Published in Joan-Ramon Casas, Dan M. Frangopol, Jose Turmo, Bridge Safety, Maintenance, Management, Life-Cycle, Resilience and Sustainability, 2022
S. Hasegawa, C.W. Kim, K.C. Chang, N. Toshi
The unknown dynamic vehicle force vector can be estimated with utilizing Equation 8 and Equation 12. A regularized least square minimization is used to estimate the dynamic vehicle force, as it is an ill-posed inverse problem in which the slight change in the measured data causes a big difference in the estimated results. In order to deal with the ill-posed problem and obtain the robust solution, Tikhonov regularization is used. The cost function (CF) to be minimized is defined as Equation 13. CF=∑j=1N[(dj−QZj,dj−QZj)+(△fv,j,λ△fv,j)]
Image Super-Resolution: Historical Overview and Future Challenges
Published in Peyman Milanfar, Super-Resolution Imaging, 2017
which is well-known as the Tikhonov regularization [95, 26, 63], the most commonly used method for regularization of ill-posed problems. Γ is usually referred as Tikhonov matrix. Hardie et al. [35] proposed a joint MAP framework for simultaneous estimation of the high-resolution image and motion parameters with Gaussian MRF prior for the HR image. Bishop and Tipping [96] proposed a simple Gaussian process prior where the covariance matrix Q is constructed by spatial correlations of the image pixels. The nice analytical property of Gaussian process prior allows a Bayesian treatment of the SR reconstruction problem, where the unknown high-resolution image is integrated out for robust estimation of the observation model parameters (unknown PSFs and registration parameters). Although the GMRF prior has many analytical advantages, a common criticism for it associated with super-resolution reconstruction is that the results tend to be overly smooth, penalizing sharp edges that we desire to recover.
Application of 3D Level Set based Optimization in Microwave Breast Imaging for Cancer Detection
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
Hardik N. Patel, Deepak K. Ghodgaonkar
Tikhonov regularization performs well in a situation where the solution has smooth changes. It will not perform well in a case where the solution has abrupt changes. Total variation regularization term is defined as shown in equation (2.39).
An Online Database of Benchmark Problems for Verification of Inverse Problems Computer Codes
Published in Heat Transfer Engineering, 2023
One feature of this database is that solutions for inverse test problems through different techniques will be available. Furthermore, the necessary intermediate steps in such techniques will also be disclosed. For example, when using Tikhonov regularization in linear function estimation problems, different methods are available for regularization parameter selection, such as Morozov criterion [13], L-curve and generalized cross-validation [14]. Parameter selection will be made with at least one of such methods, but preferably with many. On the other hand, the same problem can be solved e.g. via the truncated SVD method [4], on which one must select the number of singular values to be considered. In similar fashion, nonlinear problems can be solved using classical techniques such as the Levenberg-Marquardt method [15–17], the Box-Kanemasu method [18] or many other techniques available in the literature.
Least squares formulation for ill-posed inverse problems and applications
Published in Applicable Analysis, 2022
Eric Chung, Kazufumi Ito, Masahiro Yamamoto
The Tikhonov regularization techniques are widely used in many applications, e.g. [1, 3, 8–16] the references therein and they result in gaining the stability and enhancing solutions, e.g. sparsity [3, 7]. Least squares methods are very popular and commonly used. But, often the solution u is eliminated as a function of unknown x and then minimize the least squares fit criterion, i.e. function minimization of the form for certain observations y of solution x [3, 11]. For the case of the backward heat equation we have , which is a linear smoothing operator. As in the field of penalized regression and lasso [17, 18] one uses various penalizations for unknown x.
Sparsifying spherical radial basis functions based regional gravity models
Published in Journal of Spatial Science, 2022
Haipeng Yu, Guobin Chang, Shubi Zhang, Nijia Qian
Two important issues are in place concerning using the SRBF method. The first is determining the distribution or location of the kernel functions, including the number and depths of the layers, the number of the nodes on each layer. This issue is often tackled empirically in general, though in some works adaptivity is introduced to make the method data-driven to certain extent (Barthelmes 1986, Klees et al. 2008, Lin et al. 2019). On one side, too many kernel functions would make the parameter estimation ill-posed, namely the so-called over-fitting problem. On the other side, too few would make the model insufficient to represent the spatial variation of the real gravity signal. Careful tuning is needed in general which makes the problem rather complex and hard. The second issue is the parameter estimation. Based on the above analysis, sufficient kernel functions are generally included which often inevitably makes the problem badly conditioned. As a result, regularization is often needed in the parameter estimation (Tikhonov and Arsenin 1977). In geodetic community, L2-norm regularization, also called the Tikhonov regularization is often employed, see e.g. (Koch and Kusche 2002, Kusche and Klees 2002, Xu et al. 2006a, 2006b, Chang et al. 2018). Note that the Tikhonov regularization is also called ridge estimation in the literature (Hoerl and Kennard 1970).