This chapter describes the structural damage detection of unit-linked floating structure model based on coupled free vibration and filtering theory. The structural damage identification analysis is formulated as inverse problem in computational mechanics and structural engineering. The inverse problem is analyzed under the consideration of stochastic properties of mathematical model and structural system, because the observations are usually measured in the presence of noise. The filter theory is well known as the “inverse analysis method” for considering the stochastic properties. The structural damage analysis of the unit-linked floating model using the extended Kalman and projection filtering algorithms is presented and the characteristics of both filtering algorithms are discussed in these analyses. Thus, in this chapter, the structural damage identification analyses for the unit linked floating model are carried out by using the parametric projection filter as “the third filter”, and the notable characteristics of the parametric projection filter are made clear through the inverse analyses.

This chapter discusses the way functionality of the L-curve is extended to solve an ill-posed problem using a conjugate gradient method. The curvature of the L-curve that is important to locate the corner of the curve is computed using a finite difference approach. A new coordinate rotation method to locate the corner of the L-curve is proposed and the numerical simulation shows that the method provides a more robust performance in estimating the optimal solution than the curvature method. A criterion of an optimal solution based on minimizing the norm of residual vector is reformulated. This criterion can be applied to the problem in which the accuracy is important only in a particular region, and provides an accurate solution. The method is called a “relaxed residual norm” method. All the methods proposed in this chapter are examined and evaluated comprehensively using some ill-posed inversion problems. A good improvement to obtain the accurate solution for the problems is achieved.

This chapter presents a new inverse analysis method of finding local elastic moduli from displacement data for a heterogeneous and anisotropic elastic body in state of plane stress or strain. The method chooses a small block by connecting points at which displacement is measured and analyzes its deformation to determine elastic moduli separately from the rest of the body. The procedures of the elasticity inversion are developed with some techniques to exclude measurement errors from the displacement data. It is demonstrated in numerical simulation that the elasticity inversion is able to determine local elastic moduli accurately, when displacement data of sufficient quantity and quality is given. The limitations in applying the elasticity inversion: (1) only relative values are found, (2) discretization errors cannot be fully removed from data, and (3) the prediction becomes less accurate as less smooth displacement fields are used.

This chapter provides an overview of the scheme of modeling vibrational spectra of the large size molecular systems based on stable numerical methods. New regularized algorithm for the determination of intermolecular potential U(R) on a base of second virial data B(T) is proposed. A priori information on convexity and monotonicity of potential U(R) at different distances R is also included in the regularization procedure. Mathematical formulation of the problem is reduced to minimization of a non-quadratic functional on a set of linear constraints defining a concave-convex potential function with the fixed inflection point. Position of inflection point is determined using enumerative techniques and the functional is minimized by the projection of conjugate gradients method.

Reconstruction method for X-ray tomography of 3D vector fields is developed based on series expansion method by vector spherical harmonics. An inversion method for the P-transform or the X-ray transform of 3D vector fields is developed. The chapter also highlights pecora employed scalar spherical harmonics for the inversion of scalar P-transform. For the reconstruction of a steady object at least three measurements from three different directions are needed. The use of mobile frame attached to the object and the vector spherical harmonics simplifies the matter drastically.

An inverse problem is considered to identify the geometry of discontinuities in a conductive material Ω ⊂ R2 with anisotropic conductivity (I+(K-I) χD) from Cauchy data measurements taken on the boundary ∂Ω, where D ⊂ Ω , whereby, K is known, symmetric, and positive definite tensor not equal to the identity tensor and χD is the characteristic constant of the domain D. In this chapter, a real coded genetic algorithm in conjunction with a boundary element method to detect the size and location of a cavity, or an isotropic or anisotropic inclusion, D by a single boundary measurement is proposed. The genetic algorithms based method developed in this chapter is a robust and efficient method for detecting the size and location of subsurface discontinuities.

This chapter focuses on the method to identify the shape of source wave from speech signals by inverse-filtering of the vocal tract transfer characteristics that are reconstructed from formant frequencies and bandwidths obtained by peak picking method. First an experiment is conducted using artificial voices. Then it is applied to the real speech signals. Some comparisons with the results from Linear Prediction residue method are also presented. By the proposed reconstruction method, source waves are well estimated from the speech signals except for /i/, /u/. For its speech the first formant is near to the pitch frequency.

This chapter aims at finding the optimal shapes of the sound-insulating wall for auto highways. A 2D model of the wall and the auto-highway is considered in which a sound-insulating wall stands perpendicularly to the infinite horizontal plane of the ground and a point source of noise is located at the central point on the road. A cellular automaton is applied together with the boundary element method for this purpose, to find the optimal shapes of the sound insulating wall under several important situations. It is also demonstrated that an optimal shape of the wall, which drastically reduces noise can be found if it is assumed that the cells change even in the diagonal directions.

This chapter focuses on several modifications of Tikhonov's regularization algorithm and applies it to solve the problem of image reconstruction for close quadruple gravitational lens systems. Different assumptions concerning a priori information about the unknown solution are considered for reconstructing images of gravitational lens QSO 2237+0305 (Einstein Cross). To construct the algorithms several function sets and theoretical model of light distribution in background galaxy were used. In all cases the acceptable chi square value is obtained. Numerical results presented reveal that incorporation of all available priori information leads to encouraging results in the image restoration.

This chapter focuses on a multidimensional consolidation inverse analysis. The effects of the measured pore water pressure and the assumption of anisotropy are investigated. An elastoplastic hypothetical ground is analyzed to check the proposed analytical method. The hypothetical observation data are obtained based on the Sekiguchi-Ohtaelasto-plastic model. The effectiveness of the observed pore water pressure is proved; vertical and lateral displacements and pore water pressure are predicted well with the observed pore water pressure. The laboratory anisotropic consolidation tests are also analyzed, because accurate measurements are necessary for the development of the method. The measured deformation and pore pressure obtained in the tests are used for the inverse analysis. According to the results of this analysis; it is shown that the measured pore water pressure and assumption of anisotropy work well, to predict the future consolidation behavior.