In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were sup...In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were supposed to satisfy some additional monotonic condition. Moreover, with the assumption that the singular values of operator have power form, the improved convergence rates of the regularized solution were worked out.展开更多
The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matri...The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation(VCE) and minimum standard deviation(MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.展开更多
Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed ...Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform(FFT)algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.展开更多
In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional...In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.展开更多
The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, w...The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.展开更多
文摘In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were supposed to satisfy some additional monotonic condition. Moreover, with the assumption that the singular values of operator have power form, the improved convergence rates of the regularized solution were worked out.
基金supported by the National Natural Science Foundation of China (Nos.41374023,41131067,41474019)the National 973 Project of China (No.2013CB733302)+2 种基金the China Postdoctoral Science Foundation (No.2016M602301)the Key Laboratory of Geospace Envi-ronment and Geodesy,Ministry of Education,Wuhan University (No.15-02-08)the State Scholarship Fund from Chinese Scholarship Council (No.201306270014)
文摘The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation(VCE) and minimum standard deviation(MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.
基金supported by the National Natural Science Foundation of China(41304022,41174026,41104047)the National 973 Foundation(61322201,2013CB733303)+1 种基金the Key laboratory Foundation of Geo-space Environment and Geodesy of the Ministry of Education(13-01-08)the Youth Innovation Foundation of High Resolution Earth Observation(GFZX04060103-5-12)
文摘Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform(FFT)algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.
基金supported by the National Natural Science Foundation of China(11961044)the Doctor Fund of Lan Zhou University of Technologythe Natural Science Foundation of Gansu Provice(21JR7RA214)。
文摘In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.
文摘The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.