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# Inverse Problems and Regularization

DOI link for Inverse Problems and Regularization

Inverse Problems and Regularization book

# Inverse Problems and Regularization

DOI link for Inverse Problems and Regularization

Inverse Problems and Regularization book

## ABSTRACT

Nonextensive Regularization .......................................................................................... 296 8.7 Intrinsic Regularization.................................................................................................... 300 8.8 Statistical Methods............................................................................................................ 301 8.9 Regularized Neural Networks ........................................................................................ 306 8.10 Conclusion.......................................................................................................................... 309 Appendix 8.A: Some Properties for Nonextensive Thermostatics...................................... 310 Nomenclature ............................................................................................................................. 311 References.................................................................................................................................... 312

Looking at light propagation, we ask ourselves: What is light? How is light propagated? On observing the sea movements, we again wonder: How does the sea wave travel? And how about the dynamics of sky bodies? The aim of science is to answer the above questions. More generally, the understanding

of the natural and cultural phenomena is the challenge in science. One historical point in the scientiﬁc development was in Isaac Newton’s book, Math-

ematical Principles of Natural Philosophy (written in Latin: Philosophiæ Naturalis Principia Mathematica), usually called the Principia, the ﬁrst edition of which was published in 1687. Actually, the Principia is made up of a set of three books: Book 1: De motu corporum (On the Motion of Bodies), Book 2: (On the Motion of Bodies, but Motion through Resisting Mediums), Book 3: De systemate mundi (On the System of the World). There are many contributions related to Newton’s approach. The Principia has a lot of solved problems, and it establishes many aspects of modern scientiﬁc thought. For example, the physics applied for the planet Earth is the same as for the other celestial bodies. Moreover, natural science should be a quantitative issue, that is, the quantities involved are connected

and calculus, the quantitative relations are expressed as differential and=or integral equations. Therefore, a long period was needed until we were able to describe the equations for

ﬂuid dynamics, for example, for understanding qualitatively and quantitatively the movement of a sea wave. The description of the mathematical equations, the material properties (constitutive equations), and the initial and boundary conditions constitute the direct problem (also called the forward problem). However, estimating a property from a natural phenomenon, taking into account the

quantity measured or desired, characterizes an inverse problem. The expression ‘‘inverse problem’’ is attributed to the Georgian astrophysicist Viktor Amazaspovich Ambartsumian (for more information about him, consult the Internet: http:==www.phys-astro. sonoma.edu=brucemedalists=ambartsumian=). One deﬁnition for inverse problems is attributed to the eminent Russian scientist Oleg Mikailivitch Alifanov: ‘‘Solution of an inverse problem entails determining unknown causes based on observation of their effects’’ (see in the Internet: http:==www.me.ua.edu=inverse=whatis.html). In the ﬁrst years of the twentieth century, a French mathematician, Jacques Hadamard, in

his studies on differential equations, established the concept of a well-posed problem, which comprises the following: (1) the solution exists; (2) the solution is unique; (3) there is a continuous dependence on the input data. Hadamard has derived some examples that fail to follow one or more conditions cited. Such problems are called ill-posed problems (also known as incorrectly posed or improperly posed problems). However, Hadamard believed that ill-posed problems are curiosities, and problems from the physical reality should be wellposed problems, because nature works in a stable way. Unfortunately, inverse problems belong to the class of ill-posed problems. This was a

great motivation to change the conception that improperly posed problems are only a pathological mathematical curiosity. This is the motivation to study mathematical methods to deal with this kind of problems. Researchers like David L. Phillips (1962) and Sean A. Twomey (1963a, 1963b) deserve special mention, but it was the scientiﬁc work from Andrei Nikolaevich Tikhonov that was the starting point of the general formulation to the illposed problems-the regularization method. Professor Tikhonov was a prominent Russian mathematician from the famous Steklov Mathematical Institute. He worked mainly on topology, functional analysis, mathematical physics, and computational mathematics. The regularization method is based on computing the smoothest approximated solution

consistent with available data. The search for the smoothest (or regular) solution is the additional information added to the problem, moving the original problem from the illposed problem to the well-posed one. Figure 8.1 gives an outline of the idea behind the scheme. This is one of the most powerful techniques for computing inverse solutions. One competitive strategy is to consider the function to be estimated as a realization of a

stochastic process. The methods that employ only the statistical properties of the noise (a permanent feature in the measured quantities) are the maximum likelihood (ML) estimators. Another statistical approach is the Bayesian method, where an a priori

argue a Bayesian scheme is a statistical justiﬁcation to the regularization methods. A standard reference on the regularization method is Tikhonov and Arsenin’s book

(1977). Another reference with some mathematical details is Engl et al. (1996). However, I would also like to suggest the books by Bertero and Boccacci (1998) and Aster et al. (2005). For statistical methods of inverse problems (IP), Tarantola (1987) and Kaipio and Somersalo (2005) are good references.