A general way to solve the nonuniqueness of geophysical inverse problems is first described in the regularization theory by Tikhonov and Arsenin (1977), which uses both data misfit and model stabilizing functionals to construct the parametric objective functional.
Secondly, from the regularization theory, the gradient fidelity term works as Tikhonov regularization in Sobolev functional space The problem of admits a unique solution characterized by the Euler-Lagrange equation Moreover, the function is called harmonic (subharmonic, superharmonic) if it satisfies.
The regularization theory has been developed since the pioneer work of Tikhonov [1] and Tikhonov and Arsénine [2] who had introduced a quadratic regularization terms to account for some prior properties of the solution (smoothness).
end{cases} By the standard regularization theory we may assume that (bar{v}_{i} stackrel{C^{1}}{rightarrow}bar{v}), and thus the limit equation of this equation is textstylebegin{cases} dDeltabar{v} +bbar{v}g z,0)=0,& xinOmega, frac{partialbar {v}}{partial n}+rbar{v}=0,& xinpartialOmega.
A new inverse method was presented according to the Tikhonov regularization theory.
end{cases} By using the standard elliptic regularization theory we have that (bar{w}_{i}stackrel{C^{1}}{rightarrow}bar{w}geq0,notequiv 0), and w̄ satisfies textstylebegin{cases} Deltabar{w}+a 1-q bar{w}h_{1}+a 1-q bar{w}h frac{partialbar{w}}{partial n}+rbar{w}=0,& xinpartialOmega.
Over the past decades, regularization theory is widely applied in various areas of machine learning to derive a large family of novel algorithms.
It is known as the l1-regularized regression model in regularization theory.
This property has a close relation to the use of topological degree in eigenvalue theory by the regularization method.
Generalized radial basis function (GRBF) networks which are kernel based ANNs, have the best approximation property since they represent the optimal solution of multivariate linear regularization theory.
