Solution of an unstable axisymmetric Cauchy–Poisson problem of dispersive water waves for a spectrum with compact support

Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone

“…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. …”

An inverse measurement of the sudden underwater movement of the sea-floor by using the time-history record of the water-wave elevation

“…(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].…”

A novel method for non-parametric identification of nonlinear restoring forces in nonlinear vibrations from noisy response data: A conservative system

“…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.…”