The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approxima...The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approximate solution obtained by the Tikhonov regularization method in general form may lack many details of the exact solution.Combining the fractional Tikhonov method with the preconditioned technique,and using the discrepancy principle for determining the regularization parameter,we present a preconditioned projected fractional Tikhonov regularization method for solving discrete ill-posed problems.Numerical experiments illustrate that the proposed algorithm has higher accuracy compared with the existing classical regularization methods.展开更多
To overcome the difficulty in directly measuring the impact force of a mechanical press, the inverse theory is employed to reconstruct the impact force from the corresponding response data in time domain. The nature o...To overcome the difficulty in directly measuring the impact force of a mechanical press, the inverse theory is employed to reconstruct the impact force from the corresponding response data in time domain. The nature of ill-posedness of impact force reconstruction is explored through singular value decomposition （SVD） and the Tikhonov regularization is utilized to deal with the ill-posedness, in which the optimal parameter is chosen in light of the L-curve criterion and the generalized cross- validation （GCV）. The experimentally measured strain responses of upper and lower dies of the press are chosen as source data for impact force reconstruction, and the corresponding numerical results are compared with the experimental measurements, which verifies the effectiveness of the reconstruction method.展开更多
EIT （electrical impedance tomography） problem should be represented by a group of partial differential equation, in numerical calculation： the nonlinear problem should be linearization approximately, and then linea...EIT （electrical impedance tomography） problem should be represented by a group of partial differential equation, in numerical calculation： the nonlinear problem should be linearization approximately, and then linear equations set is obtained, so EIT image reconstruct problem should be considered as a classical ill-posed, ill-conditioned, linear inverse problem. Its biggest problem is the number of unknown is much more than the number of the equations, this result in the low imaging quality. Especially, it can not imaging in center area. For this problem, we induce the CS technique into EIT image reconstruction algorithm. The main contributions in this paper are： firstly, built up the relationship between CS and EIT definitely; secondly, sparse reconstruction is a critical step in CS, built up a general sparse regularization model based on EIT; finally, gives out some EIT imaging models based on sparse regularization method. For different scenarios, compared with traditional Tikhonov regularization （smooth regularization） method, sparse reconstruction method is not only better at anti-noise, and imaging in center area, but also faster and better resolution.展开更多
Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective...Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective iterative projection algorithm for solving large linear equations due to its simplicity.We propose a regularized randomized extended Kaczmarz(RREK)algorithm for solving large discrete ill-posed problems via combining the Tikhonov regularization and the randomized Kaczmarz method.The convergence of the algorithm is proved.Numerical experiments illustrate that the proposed algorithm has higher accuracy and better image restoration quality compared with the existing randomized extended Kaczmarz(REK)method.展开更多
Singular value system is applied to construct a new class of improved regularizing methods for solving the first kind of Fredholm integral equations with noisy data. By a priori choosing regularizing parameters, optim...Singular value system is applied to construct a new class of improved regularizing methods for solving the first kind of Fredholm integral equations with noisy data. By a priori choosing regularizing parameters, optimal convergence order of the regularized solution is obtained. And with aids of MATLAB software, numerical results are presented which roughly coincide with the theoretical analysis.展开更多
In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n...In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n ≥ 3).展开更多
The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary...The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.展开更多
LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for...LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition(TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.展开更多
Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We first...Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We firstly establish the uniqueness for this inverse problem, which provides the theoretical basis for the reconstruction scheme. Then based on the contrast source inversion(CSI) method, we propose an algorithm determining the refraction index and the artificial wave sources alternately by a dynamic iterative scheme. The algorithm defines the iterates by solving a series of minimization problems with uniformly convex penalty terms, which are allowed to be non-smooth to include L1 and total variation like functionals, ensuring the reconstruction quality when the unknown refraction index has the special features such as sparsity and discontinuity. By choosing the regularizing parameter automatically, the algorithm is terminated in terms of discrepancy principle. The convergence property of the iterative sequence is rigorously proven. Numerical implementations demonstrate the validity of the proposed algorithm.展开更多
This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operator...This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.展开更多
Ship maneuverability, in the field of ship engineering, is often predicted by maneuvering motion group （MMG） mathematical model. Then it is necessary to determine hydrodynamic coefficients and interaction force coef...Ship maneuverability, in the field of ship engineering, is often predicted by maneuvering motion group （MMG） mathematical model. Then it is necessary to determine hydrodynamic coefficients and interaction force coefficients of the model. Based on the data of free running model test, the problem for obtaining these coefficients is called inverse one. For the inverse problem, ill-posedness is inherent, nonlinearity and great computation happen, and the computation is also insensitive, unstable and time-consuming. In the paper, a regularization method is introduced to solve ill-posed problem and genetic algorithm is used for nonlinear motion of ship maneuvering. In addition, the immunity is applied to solve the prematurity, to promote the global searching ability and to increase the converging speed. The combination of regularization method and immune genetic algorithm（RIGA） applied in MMG mathematical model, showed rapid converging speed and good stability.展开更多
Learning with coefficient-based regularization has attracted a considerable amount of attention in recent years, on both theoretical analysis and applications. In this paper, we study coefficient-based learning scheme...Learning with coefficient-based regularization has attracted a considerable amount of attention in recent years, on both theoretical analysis and applications. In this paper, we study coefficient-based learning scheme （CBLS） for regression problem with /q-regularizer （1 〈 q ≤ 2）. Our analysis is conducted under more general conditions, and particularly the kernel function is not necessarily positive definite. This paper applies concentration inequality with/2-empirical covering numbers to present an elaborate capacity dependence analysis for CBLS, which yields sharper estimates than existing bounds. Moreover, we estimate the regularization error to support our assumptions in error analysis, also provide an illustrative example to further verify the theoretical results.展开更多
基金supported in part by the National Natural Science Foundation of China(No.62073161)the Fundamental Research Funds 2019“Artificial Intelligence+Special Project”of Nanjing University of Aeronautics and Astronautics(No.2019009)
文摘The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approximate solution obtained by the Tikhonov regularization method in general form may lack many details of the exact solution.Combining the fractional Tikhonov method with the preconditioned technique,and using the discrepancy principle for determining the regularization parameter,we present a preconditioned projected fractional Tikhonov regularization method for solving discrete ill-posed problems.Numerical experiments illustrate that the proposed algorithm has higher accuracy compared with the existing classical regularization methods.
基金Transformation Program of Science and Technology Achievements of Jiangsu Province(No.BA2008030)
文摘To overcome the difficulty in directly measuring the impact force of a mechanical press, the inverse theory is employed to reconstruct the impact force from the corresponding response data in time domain. The nature of ill-posedness of impact force reconstruction is explored through singular value decomposition （SVD） and the Tikhonov regularization is utilized to deal with the ill-posedness, in which the optimal parameter is chosen in light of the L-curve criterion and the generalized cross- validation （GCV）. The experimentally measured strain responses of upper and lower dies of the press are chosen as source data for impact force reconstruction, and the corresponding numerical results are compared with the experimental measurements, which verifies the effectiveness of the reconstruction method.
基金This work was supported by Chinese Postdoctoral Science Foundation （2012M512098）, Science and Technology Research Project of Shaanxi Province （2012K13-02-10）, the National Science ＆ Technology Pillar Program （2011BAI08B13 and 2012BAI20B02）, Military Program （AWS 11 C010-8）.
文摘EIT （electrical impedance tomography） problem should be represented by a group of partial differential equation, in numerical calculation： the nonlinear problem should be linearization approximately, and then linear equations set is obtained, so EIT image reconstruct problem should be considered as a classical ill-posed, ill-conditioned, linear inverse problem. Its biggest problem is the number of unknown is much more than the number of the equations, this result in the low imaging quality. Especially, it can not imaging in center area. For this problem, we induce the CS technique into EIT image reconstruction algorithm. The main contributions in this paper are： firstly, built up the relationship between CS and EIT definitely; secondly, sparse reconstruction is a critical step in CS, built up a general sparse regularization model based on EIT; finally, gives out some EIT imaging models based on sparse regularization method. For different scenarios, compared with traditional Tikhonov regularization （smooth regularization） method, sparse reconstruction method is not only better at anti-noise, and imaging in center area, but also faster and better resolution.
基金supported by the National Natural Science Foundations of China(Nos.11571171,62073161,and 61473148)。
文摘Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective iterative projection algorithm for solving large linear equations due to its simplicity.We propose a regularized randomized extended Kaczmarz(RREK)algorithm for solving large discrete ill-posed problems via combining the Tikhonov regularization and the randomized Kaczmarz method.The convergence of the algorithm is proved.Numerical experiments illustrate that the proposed algorithm has higher accuracy and better image restoration quality compared with the existing randomized extended Kaczmarz(REK)method.
基金Natural Science Foundation of Shandong Province (Y2001E03)
文摘Singular value system is applied to construct a new class of improved regularizing methods for solving the first kind of Fredholm integral equations with noisy data. By a priori choosing regularizing parameters, optimal convergence order of the regularized solution is obtained. And with aids of MATLAB software, numerical results are presented which roughly coincide with the theoretical analysis.
文摘In this paper, the author studies the regularity of solutions to the Dirichlet problem forequation Lu = f, where L is a second order degenerate elliptic operator in divergence form inΩ, a bounded open subset of Rn (n ≥ 3).
文摘The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.
基金supported by National Basic Research Program of China (Grant No. 2011CB302400)National Natural Science Foundation of China (Grant No. 11371219)
文摘LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition(TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.
基金supported by National Natural Science Foundation of China(Grant Nos.11421110002,11531005 and 11501102)National Science Foundation of Jiangsu Province(Grant No.BK20150594)
文摘Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We firstly establish the uniqueness for this inverse problem, which provides the theoretical basis for the reconstruction scheme. Then based on the contrast source inversion(CSI) method, we propose an algorithm determining the refraction index and the artificial wave sources alternately by a dynamic iterative scheme. The algorithm defines the iterates by solving a series of minimization problems with uniformly convex penalty terms, which are allowed to be non-smooth to include L1 and total variation like functionals, ensuring the reconstruction quality when the unknown refraction index has the special features such as sparsity and discontinuity. By choosing the regularizing parameter automatically, the algorithm is terminated in terms of discrepancy principle. The convergence property of the iterative sequence is rigorously proven. Numerical implementations demonstrate the validity of the proposed algorithm.
文摘This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.
文摘Ship maneuverability, in the field of ship engineering, is often predicted by maneuvering motion group （MMG） mathematical model. Then it is necessary to determine hydrodynamic coefficients and interaction force coefficients of the model. Based on the data of free running model test, the problem for obtaining these coefficients is called inverse one. For the inverse problem, ill-posedness is inherent, nonlinearity and great computation happen, and the computation is also insensitive, unstable and time-consuming. In the paper, a regularization method is introduced to solve ill-posed problem and genetic algorithm is used for nonlinear motion of ship maneuvering. In addition, the immunity is applied to solve the prematurity, to promote the global searching ability and to increase the converging speed. The combination of regularization method and immune genetic algorithm（RIGA） applied in MMG mathematical model, showed rapid converging speed and good stability.
基金supported by National Natural Science Foundation of China (Grant Nos.11226111 and 71171166)
文摘Learning with coefficient-based regularization has attracted a considerable amount of attention in recent years, on both theoretical analysis and applications. In this paper, we study coefficient-based learning scheme （CBLS） for regression problem with /q-regularizer （1 〈 q ≤ 2）. Our analysis is conducted under more general conditions, and particularly the kernel function is not necessarily positive definite. This paper applies concentration inequality with/2-empirical covering numbers to present an elaborate capacity dependence analysis for CBLS, which yields sharper estimates than existing bounds. Moreover, we estimate the regularization error to support our assumptions in error analysis, also provide an illustrative example to further verify the theoretical results.
