THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE SMOOTH MIXED BOUNDARY CONDITIONS

The spectral distribution exp( ), where {} are the eigenvalues of the negative Laplacian -△=- in the (x^1,x^2)-plane, is studied for a variety of domains, where -∞< t <∞ and i=(1/2)(-1) . The dependence of (t)on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain Ω in R^2 with a smooth boundary Ω, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts Γj(j = 1,……,n) of Ω are considered such that Some geometrical properties of Ω(e.g., the area of Ω, the total lengths of the boundary, the curvature of its boundary, etc.) are determined, from the asymptotic expansions of (t) for |t| → 0.

A NEW APPROACH TO INVERSE PROBLEMS OF WAVE EQUATIONS

We introduce a multi-cost-functional method for solving inverse problems of wave equations. This method has its simplicity, efficiency and good physical interpretation. It has the advantage of being programmed for two- or three- (space) dimensional problems as well as for one-dimensional problems.

THE PARAMETER PERTURBATION METHOD ON ELASTIC WAVE EQUATION IN INHOMOGENEOUS MEDIUM

In this paper, the medium parameters of the elastic wave equation in inhomogeneous medium are rewritten by introducing the referential variables and the perturbational variables, and the wave equation whose sources are the medium parameter perturbational term in homogeneous medium is obtained. By using the Green function theory, the integral equation of the perturbational parameters is obtained. Then the displacement field in homogeneous medium is considered the result of the first iteration, and the displacement field is solved by this integral equation. When the perturbations of medium parameters are about 50 percent, this method can solve the displacement field effectively. from the analysis of the numerical results, the characteristics of wave field in inhomogeneous medium are obtained. The results conform with the local principles of wave function in inhomogeneous medium.

The inverse problem based on a full dispersive wave equation

The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.

ON AN INVERSE PROBLEM FOR 1-DIMENSIONAL WAVE EQUATION

The inverse problem for the 1-dimensional acoustic wave equation is discussed to deter-mine propagation velocity from impulse response. A relation between the propagation velocityand the wavefield can be established from the analysis of propagation of discontinuities forhyperbolic equations. As a result, the inverse problem discussed in this paper is reduced to aparticular initial value problem of a semilinear system of P. D. E.. The Picard iteration forsolving this initial value problem is constructed and the convergence of iteration is proved.The main results are the following: (i) the propagation velocity can always be recovered fromthe impulse response, unless the inverse problem contains a singular point, where the propa-gation velocity is infinite or zero, or its total variation in the neighborhood of the singularpoint is infinite; (ii) the stability behaviour of the solutions of this inverse problem is es-sentially dependent on the total variation of logarithm of propagation velocity.

Difference inversion model of wave equation

A numerical iterative model was derived from the difference method and a perturbation assumption to calculate the coefficient function of a wave equation. The method was used to solve the disaccord problem of numerical precision between the direct problem model and inverse problem model, and its serial problems using the old method. Numerical simulation calculation shows that the method is feasible and effective.

A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.

Some Universal Properties of the Green’s Functions Associated with the Wave Equation in Bounded Partially-Homogeneous Domains and Their Use in Acoustic Tomography

Direct and inverse scattering problems connected with the wave equation in non-homogeneous bounded domains constitute challenging actual subjects for both mathematicians and engineers. Among them one can mention, for example, inverse source problems in seismology, nondestructive archeological probing, mine prospecting, inverse initial-value problems in acoustic tomography, etc. In spite of its crucial importance, almost all of the available rigorous investigations concern the case of unbounded simple domains such as layered planar or cylindrical or spherical structures. The main reason for the lack of the works related to non-homogeneous bounded structures is the extreme complexity of the explicit expressions of the Green’s functions. The aim of the present work consists in discovering some universal properties of the Green’s functions in question, which reduce enormously the difficulties arising in various applications. The universality mentioned here means that the properties are not depend on the geometrical and physical properties of the configuration. To this end one considers first the case when the domain is partially-homogeneous. Then the results are generalized to the most general case. To show the importance of the universal properties in question, they are applied to an inverse initial-value problem connected with photo-acoustic tomography.

A Method of Wide Convergence Region for Solving Inverse Problems of Wave Equations

Based on the embedding thought, a method of wide convergence region for solving the coefficient inverse problem of wave equations in the space-time variable domain is presented. The numerical simulation shows that the method is feasible and effective.

A Differential Continuation－Regularization Method for Solving Inverse Problems of Acoustic Wave Equations

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NONLINEAR INTEGRAL EQUATION OF COEFFICIENT INVERSION OF ACOUSTIC WAVE EQUATION AND ITERATION

In this paper, the 1-D, 2-D and 3-D coefficient inverse problem of the acoustic waveequation is reduced to the nonlinear integral equation. In the 1-D case, the nonlinear integralequation belongs to the second kind, while in the 2-D and 3-D cases, the nonlinear integralequation is an interesting integral geometry problem. The iteration for solving the aboveintegral equation has been considered. The nonlinear integral equation and its iterationin this paper will be useful in the theoretical and numerical analysis and in application toscience and engineering of the above inversion.

Recovering of Damping Coefficients for a System of Coupled Wave Equations with Neumann Boundary Conditions:Uniqueness and Stability

The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion F1 of the boundary F, and over a computable time interval T 〉 0. Under sharp conditions on Г0= Г／Г1, T 〉 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route ＂post-Carleman estimates＂ suggested by Isakov in 2006.

Quantifying Tectonic and Geomorphic Interpretations of Thermochronometer Data with Inverse Problem Theory

Thermochronometer data offer a powerful tool for quantifying a wide range of geologic processes,such as the deformation and erosion of mountain ranges,topographic evolution,and hydrocarbon maturation.With increasing interest to quantify a wider range of complicated geologic processes,more sophisticated techniques are needed.This paper is concerned with an inverse problem method for interpreting the thermochronometer data quantitatively.Two novel models are proposed to simulate the crustal thermal fields and paleo mountain topography as a function of tectonic and surface processes.One is a heat transport model that describes the change of temperature of rocks;while the other is surface process model which explains the change of mountain topography.New computational algorithms are presented for solving the inverse problem of the coupled system of these two models.The model successfully provides a new tool for reconstructing the kinematic and the topographic history of mountains.

Multi-scale seismic full waveform inversion in the frequency-domain with a multi-grid method

Although full waveform inversion in the frequency domain can overcome the local minima problem in the time direction, such problem still exists in the space direction because of the media subsurface complexity. Based on the optimal steep descent methods, we present an algorithm which combines the preconditioned bi-conjugated gradient stable method and the multi-grid method to compute the wave propagation and the gradient space. The multiple scale prosperity of the waveform inversion and the multi-grid method can overcome the inverse problems local minima defect and accelerate convergence. The local inhomogeneous three-hole model simulated results and the Marmousi model certify the algorithm effectiveness.

Reconstruction of density and wave velocity from reflection and transmission data

Consider an inverse problem of reconstructing the coefficient in a linearwave equation on an inhomogeneous slab with density ρ(z) and wave velocity c(z). The inversioninput information is the reflection and transmission data corresponding to a point source. Byapplying the characteristic theory for hyperbolic equations, we establish an integral system fromwhich ρ(z) and c(z) can be recovered simultaneously. In contrast to some known results, our inverseapproach is carried out for depth variable, rather than for travel-time variable. Thereforeinversion results in this paper are more appropriate for the physical interpretation of a mediumslab.

Data Recovering Problem Using a New KMF Algorithm for Annular Domain

This paper is interested at the Cauchy problem for Laplace’s equation, which is to recover both Dirichlet and Neumann conditions on the inaccessible part of the boundary (inner part) of an annular domain from the over specified conditions on the accessible one (outer part). This work is an extension of the proposed algorithm for a unit circle [1] to annular domain, where, we describe an alternating formulation of the KMF algorithm proposed by Kozlov, Mazya and Fomin, and its relationship with the standard formulation. The new KMF algorithm ameliorates the accuracy of the solution and reduces the number of iterations required to achieve convergence. In the last part, the discussion of the error estimation of solution is presented and some numerical tests, using the software Freefem are given to show the efficiency of the proposed method.

径向热传导方程源项的分数阶Tikhonov-Landweber反演方法

本文研究了时间分数阶的径向热传导方程的源项反演问题.根据径向区域的终止时刻数据,利用分数阶Tikhonov-Landweber迭代正则化方法,求得在先验和后验的条件下的正则化解并得到相应误差.最后,数值实验表明了该正则化方法是有效的.

AN ITERATIVE ALGORITHM FOR DETERMINING TIME－DEPENDENT COEFFICIENT OF TWO DIMENSIONAL LINEAR WAVE EQUATION

A new iterative algorithm is proposed to solve inverse problems of time-dependent coefficient of two-dimensional linear wave equations, which is based on regularized metrtod and the minimization of functionals of the differences between the observations and the numerical solutions of the two-dimensional wave equation to determine the time-dependent coefficients in time-domain which is demonstrated. It has the practical advantages of having the necessary data measured on a portion of the boundary and sampling time. Numerical simulations on the three examples are carried out to test the feasibility of the new algorithm.Subject classifications: 35R30, 35L05, 35A40

波动方程压制多次波的技术方法

多次波问题是海洋地震勘探中最突出的问题之一 ,如何有效地压制多次波是解决问题的关键。多次波数据处理的压制方法和技术分为运动学的滤波方法和动力学的波动方程方法两大类。波动方程方法由于几乎不需要先验信息和具有较好的压制效果 ,已经成为多次波压制方法和技术发展的主要趋势。文章首先对波动方程方法进行了较全面的综述 ;然后 ,对波动方程方法中的反馈迭代反演方法做了深入的讨论 ;最后 ,通过理论模型的正反演计算 ,验证了运用波动方程反馈迭代反演法不需要知道震源函数、地下岩层结构和岩性等先验信息。此方法适用于压制多种类型的多次波 ,具有优异的振幅保真性等独特的优点。

浅海声速剖面的匹配波束反演方法

为了快速反演获得浅海声速剖面,利用匹配波束构造了反演代价函数,在此基础上研究了浅海声速剖面的匹配波束反演方法(MBI)。并运用东中国海的实验数据进行实际反演,验证了MBI的可行性和鲁棒性。利用垂直线列阵接收的29个不同位置的爆炸声信号进行了声速剖面反演, MBI反演的结果与实验中直接测量的声速剖面一致,且反演声速剖面的均方根误差小于2 m/s。研究结果表明,MBI比常规匹配场反演(MFI)获得浅海声速剖面更快速、更准确,并且对底质参数的失配具有较高的鲁棒性。