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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.
Computational Methods and Software for Bioelectric Field Problems
Published in Joseph D. Bronzino, Donald R. Peterson, Biomedical Engineering Fundamentals, 2019
A number of techniques have arisen to deal with ill-posed inverse problems. ese techniques include truncated singular value decomposition (TSVD), generalized singular value decomposition (GSVD), maximum entropy, and a number of generalized least squares schemes, including Twomey and Tikhonov regularization methods. Since this section is concerned more with the numerical techniques for approximating bioelectric eld problems, the reader is referred to References 26, 32, 81, and 82 to further investigate the regularization of ill-posed problems. A particularly useful reference for discrete ill-posed problems is the MATLAB• package developed by Per Christian Hansen, which is available via his website [31].
The Electrocardiographic Inverse Problem
Published in Theo C. Pilkington, Bruce Loftis, Joe F. Thompson, Savio L-Y. Woo, Thomas C. Palmer, Thomas F. Budinger, High-Performance Computing in Biomedical Research, 2020
Several classes of techniques are available to regularize ill-posed problems and were reviewed previously.13 These include rank explicit methods using singular value decomposition or Gram−Schmidt orthogonalization, iterative techniques, constraints on the solution derivatives without the use of a penalty term, statistical constraints, and Tikhonov regularization. Although different in approach, many of these methods produce similar results and even reduce to a similar form for certain parameter values.
Least squares formulation for ill-posed inverse problems and applications
Published in Applicable Analysis, 2022
Eric Chung, Kazufumi Ito, Masahiro Yamamoto
Inverse problems are some of the most important mathematical problems in science and mathematics because they determine unknowns and parameters in models that we cannot directly observe. They have wide applications in many fields of science and engineering, see for example [1–3] and the references therein. Ill-posed inverse problems are those such that an inverse solution exists for a regular data but does not continuously depend on data, or there is no inverse solution for noisy data. In this paper we discuss the least squares formulation for ill-posed inverse problems. Our objective is to develop algorithms that accurately restore solutions and assure continuous dependence of solution on data by treating equations and constraints in the framework of the least squares method, i.e. we address the Hadamard's question on the well-posedness.
Unknown source identification problem for space-time fractional diffusion equation: optimal error bound analysis and regularization method
Published in Inverse Problems in Science and Engineering, 2021
Fan Yang, Qian-Chao Wang, Xiao-Xiao Li
Nowadays, it is a hot topic in the field of inverse problems to deal with ill-posed problems by regularization method. There are many regularization methods to deal with ill-posed problem. In the early days, most scholars used Tikhonov regularization method [11,12] to deal with inverse problems. It is one of the oldest regularization methods. Later, on the basis of Tikhonov regularization method, some scholars studied the modified Tikhonov regularization method [13], the fractional Tikhonov regularization method [14,15] and the simplified Tikhonov regularization method [16]. In dealing with the inverse problem of bounded domain, the regularization methods are as follows: quasi-boundary regularization method [17–19], the truncation regularization method [20,21], a modified quasi-boundary regularization method [22], a mollification regularization method [23], quasi-reversibility regularization method [24,25], Landweber iterative regularization method [26–28], etc. In solving the inverse problem in unbounded domain, the common regularization methods are Fourier truncation regularization method [29–32]. In [33], Liu and Feng considered a backward problem of spatial fractional diffusion equation, and constructed a regularization method based on the improved ‘kernel’ idea, namely the improved kernel method. The modified kernel method can also deal with more ill-posed problem [34–37].
Regularized ab initio molecular force fields for key biological molecules: melatonin and pyridoxal-5′-phosphate methylamine Shiff base (Vitamin B6)
Published in Inverse Problems in Science and Engineering, 2021
Gulnara M. Kuramshina, Igor V. Kochikov, Svetlana A. Sharapova
The system (9) can be extended to include additional experimental evidence when available (for example, data for isotopic species of a molecule sharing the same force field and equilibrium geometry). Due to the experimental errors, the lack of experimental data and the model limitations, this system of equations (that can be also treated as a finite-dimensional non-linear operator equation) usually fails to define the unique solution, often proves to be incompatible and does not provide stability with respect to the errors of input data. To avoid these unfavourable features characteristic of the ill-posed problems, it is necessary to implement a regularizing algorithm for its solution. We have suggested [20,21] to use a regularizing algorithm based on optimization of the next Tikhonov’s functional where in the last (‘stabilizer’) term F0 and R0 represent parameters of ab initio force field and equilibrium geometry, respectively. With the appropriate choice of regularization parameter α (that depends on the experimental errors characterized by some numerical parameter δ), it proves possible to obtain approximations converging to a normal pseudosolution of the system (10) when experimental errors tend to zero. These approximations are obtained as the extremals {Fα(δ), Rα(δ)}of Tikhonov’s functional.