| As an important part of the inverse problem of partial differential equations,elastic wave scattering has a wide range of applications in many fields such as geological exploration,medical diagnosis,non-destructive testing,and seismology.This thesis mainly studies two types of inverse problems in elastodynamics and the.perturbation analysis of a related linear system.For the inverse problem,the study of stability is very important,which is a fine quantification of uniqueness.In chapter 2,we mainly study the inverse problem which is about the the elastic medium scatterers,i.e.,the shape stability of the elastic medium scatterers is determined from on far-field measurement.When the support is a convex polygon in Rn(n=2,3),We not only establish a double logarithmic estimate but also derive it is independent of the material parameters.We also prove that if a generic medium scatterer admits a convex polygonal point that satisfies certain properties,there must exist a positive lower bound of its far-field pattern.The latter result indicates that if an elastic material object possesses a convex polygonal point on its support,then it scatters every incident wave stably.The elastic medium scattering problem including obstacles is a very challenging subject(i.e.,the obstacle embedded in an elastic medium is determined by far-field measurements).Therefore,in chapter 3,we mainly study the time-harmonic scattering problem of an inhomogeneous elastic medium containing an embedded obstacle and show that an isotropic elastic medium with some special material parameters can well approximate the obstacle,in other words,there exists an effective ε-realization(ε<<1)for the impenetrable obstacles.According to the proposed effective medium theory,the elastic medium scattering problem including obstacles can be transformed into an elastic medium scattering problem(i.e.,the isotropic medium scatterer embedded in an elastic medium is determined by far-field measurements)with a simpler topology.At the same time,our theory enlightens the design of a new algorithm by using the effective medium theory for the elastic medium scattering problem including obstacles.The discretized linear system from the inverse scattering problem is often ill-conditioned.Such as the Shepp-Logan model in the field of ultrasonic inverse scattering is an ill-posed system,which can be reduced to a series of algebraic equations by discretization.The core of solving these algebraic equations is to solve the linear system Ax=b.The truncated total least squares(TTLS)method is an effective regularization strategy for the linear system.Therefore,in chapter 4,we consider the sensitivity analysis on the TTLS problem.We present explicit expressions for the mixed and componentwise condition numbers under component perturbations.At the same time,this thesis also propose structured and unstructured norm,mixture,and component condition number estimation algorithms based on small-sample statistical condition estimation(SCE).Based on the singular value decomposition(SVD)of the augmented matrix[A b],we can integrate these algorithms into a SVD-based direct solver for small and medium size TTLS problems and the error estimation for the numerical TTLS solution can also be derived.Finally,numerical experiments are reported to illustrate the reliability of the proposed condition estimation algorithms. |
