“…With regard to the decision about the optimal number of iterations, a stopping rule for the L-curve criterion [35] is discussed in more detail in Section 7. …”
Section: Calculating U By Regularizationmentioning
“…With regard to the decision about the optimal number of iterations, a stopping rule for the L-curve criterion [35] is discussed in more detail in Section 7. …”
Section: Calculating U By Regularizationmentioning
“…(20) for an arbitrary initial guess, f 0 [17]. In this study, for convenience and without loss of generality, we start with the initial guess of the zero function [19][20][21][22][23][24].…”
“…In fact, a direct discretization of the right-hand side in Eq. (5) creates a matrix in which the condition number is extremely large, so that the numerical inverse of the matrix does not work because the determinant of the matrix is nearly zero [16][17][18][19][20].…”
“…It is derived from a second-kind integral equation based on Banach's fixed-point theorem, which guarantees the solution's stability. For convenience, we start with the initial guess of zero function without loss of generality [16][17][18][19][20] because Landweber's regularization converges to the solution for an arbitrary initial guess u 0 [13]:…”
“…Third, we introduce a stabilization technique, known as the regularization method, to suppress the numerical instability in the solution; Landweber's regularization method is applied to stabilize the numerical solution [13][14][15][16][17][18][19][20]. The L-curve criteria [21], combined with the Landweber's regularization method, is also introduced to determine a proper choice of regularization parameters (or number of iterations) for the identification.…”
A novel procedure is proposed to identify the functional form of nonlinear restoring forces in the nonlinear oscillatory motion of a conservative system. Although the problem of identification has a unique solution, formulation results in a Volterra-type of integral equation of the "first" kind: the solution lacks stability because the integral equation is the "first" kind. Thus, the new problem at hand is ill-posed. Inevitable small errors during the identification procedure can make the prediction of nonlinear restoring forces useless. We overcome the difficulty by using a stabilization technique of Landweber's regularization in this study. The capability of the proposed procedure is investigated through numerical examples.
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