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. ...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.展开更多
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...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.展开更多
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 f...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.展开更多
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 ...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.展开更多
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 i...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.展开更多
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 numerica...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.展开更多
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 ove...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.展开更多
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 ...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展开更多
A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing th...A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.展开更多
文摘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.
文摘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.
基金Project supported by National Natural Science Foundation of China.
文摘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.
基金supported by the China State Key Science and Technology Project on Marine Carbonate Reservoir Characterization (No. 2011ZX05004-003)the Basic Research Programs of CNPC during the 12th Five-Year Plan Period (NO.2011A-3603)+1 种基金the Natural Science Foundation of China (No.41104066)the RIPED Young Professional Innovation Fund (NO.2010-13-16-02, 2010-A-26-02)
文摘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.
基金The research of G.Bao,Y.Wang,and Z.Xu was supported in part by the NSF CMG grant EAR-0724527NSF grant EAR-0724656 to T.Ehlers and P.Li.
文摘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.
文摘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.
文摘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.
文摘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
文摘A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.
