In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetr...In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O（2） symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.展开更多
Three algorithms based on the bifurcation method are applied to solving the D4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bi- furcation parameter, the...Three algorithms based on the bifurcation method are applied to solving the D4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bi- furcation parameter, the D4-Σd(D4-Σ1, D4-Σ2) symmetry-breaking bifurcation points on the branch of the D4 symmetric positive solutions are found via the extended systems. Finally, Σd(Σ1, Σ2) sym- metric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.展开更多
The optimization inversion method based on derivatives is an important inversion technique in seismic data processing,where the key problem is how to compute the Jacobian matrix.The computation precision of the Jacobi...The optimization inversion method based on derivatives is an important inversion technique in seismic data processing,where the key problem is how to compute the Jacobian matrix.The computation precision of the Jacobian matrix directly influences the success of the optimization inversion method.Currently,all AVO(Amplitude Versus Offset) inversion techniques are based on approximate expressions of Zoeppritz equations to obtain derivatives.As a result,the computation precision and application range of these AVO inversions are restricted undesirably.In order to improve the computation precision and to extend the application range of AVO inversions,the partial derivative equation(Jacobian matrix equation(JME) for the P-and S-wave velocities inversion) is established with Zoeppritz equations,and the derivatives of each matrix entry with respect to Pand S-wave velocities are derived.By solving the JME,we obtain the partial derivatives of the seismic wave reflection coefficients(RCs) with respect to P-and S-wave velocities,respectively,which are then used to invert for P-and S-wave velocities.To better understand the behavior of the new method,we plot partial derivatives of the seismic wave reflection coefficients,analyze the characteristics of these curves,and present new understandings for the derivatives acquired from in-depth analysis.Because only a linear system of equations is solved in our method,the computation of Jacobian matrix is not only of high precision but also is fast and efficient.Finally,the theoretical foundation is established so that we can further study inversion problems involving layered structures(including those with large incident angle) and can further improve computational speed and precision.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10901106)the Shanghai Leading Academic Discipline Project (No. S30405)+2 种基金the Shanghai Normal University Academic Project (No. SK200936)the Natural Science Foundation of Shanghai (No. 09ZR1423200)the Innovation Program of Shanghai Municipal Education Commission (No. 09YZ150)
文摘In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O（2） symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.
基金supported by the National Natural Science Foundation of China (Grant No. 10671130)the Science Foundation of Shanghai Municipal Education Commission (Grant No. 05DZ07)+2 种基金Shanghai Leading Academic Discipline Project (Grant No. T0401)Leading Foundation of Shanghai Science and Technology Commission (Grant No. 06JC14092)the Foundation of the Scientific Computing Key Laboratory of Shang-hai Universities
文摘Three algorithms based on the bifurcation method are applied to solving the D4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bi- furcation parameter, the D4-Σd(D4-Σ1, D4-Σ2) symmetry-breaking bifurcation points on the branch of the D4 symmetric positive solutions are found via the extended systems. Finally, Σd(Σ1, Σ2) sym- metric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.
基金supported by Funding Project for Academic Human Resources Development in Institutions of Higher Learning (Grant No. PHR(20117145))National Natural Science Foundation of China (Grant No. 10705049)
文摘The optimization inversion method based on derivatives is an important inversion technique in seismic data processing,where the key problem is how to compute the Jacobian matrix.The computation precision of the Jacobian matrix directly influences the success of the optimization inversion method.Currently,all AVO(Amplitude Versus Offset) inversion techniques are based on approximate expressions of Zoeppritz equations to obtain derivatives.As a result,the computation precision and application range of these AVO inversions are restricted undesirably.In order to improve the computation precision and to extend the application range of AVO inversions,the partial derivative equation(Jacobian matrix equation(JME) for the P-and S-wave velocities inversion) is established with Zoeppritz equations,and the derivatives of each matrix entry with respect to Pand S-wave velocities are derived.By solving the JME,we obtain the partial derivatives of the seismic wave reflection coefficients(RCs) with respect to P-and S-wave velocities,respectively,which are then used to invert for P-and S-wave velocities.To better understand the behavior of the new method,we plot partial derivatives of the seismic wave reflection coefficients,analyze the characteristics of these curves,and present new understandings for the derivatives acquired from in-depth analysis.Because only a linear system of equations is solved in our method,the computation of Jacobian matrix is not only of high precision but also is fast and efficient.Finally,the theoretical foundation is established so that we can further study inversion problems involving layered structures(including those with large incident angle) and can further improve computational speed and precision.
