Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

The lower bound of nonlocal gradient for non-convex and non-smooth image patches based regularization

In the inverse problem of image processing, we have witnessed that the non-convex and non-smooth regularizers can produce clearer image edges than convex ones such as total variation (TV). This fact can be explained by the uniform lower bound theory of the local gradient in non-convex and non-smooth regularization. In recent years, although it has been numerically shown that the nonlocal regularizers of various image patches based nonlocal methods can recover image textures well, we still desire a theoretical interpretation. To this end, we propose a non-convex non-smooth and block nonlocal (NNBN) regularization model based on image patches. By integrating the advantages of the non-convex and non-smooth potential function in the regularization term, the uniform lower bound theory of the image patches based nonlocal gradient is given. This approach partially explains why the proposed method can produce clearer image textures and edges. Compared to some classical regularization methods, such as total variation (TV), non-convex and non-smooth (NN) regularization, nonlocal total variation (NLTV) and block nonlocal total variation(BNLTV), our experimental results show that the proposed method improves restoration quality.

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

Undrained seismic bearing capacity of strip footings horizontally embedded in two-layered slopes

This study investigated the undrained seismic bearing capacity of strip footings embedded in two-layered slopes by finite element limit analysis (FELA); in particular, a pseudostatic method was specified to seismic loads. Lower bound theory (LB), upper bound theory (UB), and adaptive meshing technique are employed for exploring the effect of the shear strength ratio of top layer and bottom layer, cu1/ cu2; the horizontal embedment depth of footings, b/ B; the footing locations (in bottom layer or top layer); seismic coefficient kh; and the slope gradient β on seismic bearing capacity. Results indicated that the seismic bearing capacity would increase with the growth of horizontal embedment depth of footings. The comparison between the predictions obtained from the proposed method and the existing method was presented, and the failure mechanisms are further summarized.

A BAND SELECTION METHOD FOR HIGH PRECISION REGISTRATION OF HYPERSPECTRAL IMAGE

During the registration of hyperspectral images and high spatial resolution images, too much bands in a hyperspectral image make it difficult to select bands with good registration performance. Terrible bands are possible to reduce matching speed and accuracy. To solve this problem, an algorithm based on Cram’er-Rao lower bound theory is proposed to select good matching bands in this paper. The algorithm applies the Cram’er-Rao lower bound theory to the study of registration accuracy, and selects good matching bands by CRLB parameters. Experiments show that the algorithm in this paper can choose good matching bands and provide better data for the registration of hyperspectral image and high spatial resolution image.

Study of Lower Bound Method for Plastic Limit Analysis of Wedge on Consideration of Anchor Bolt

The massive rock slope is made up of rocks and structural surface. The existence and strength of the structural surface decides the stability of the rock mass. By adopting the lower bound method for the plastic limit analysis of wedge slope, we can easily calculate the stability of the rocky slope under various circumstances. To apply this method, we first took the wedge-shaped slide block as a complex of rigid block and structural surface for the analysis of the slope stress under the function of anchor bolts, the integrated function of rocks and structural surfaces being considered. Then, on the basis of the lower bound theory for the plastic limit analysis, a mathematical programming model which takes safety factor of slope stability as the object function is established. This model has to meet the equilibrium condition of the bolt, the Mohr-Coulomb yield condition and the boundary condition of slope. In the end, a classic model of wedge is analyzed and its lower bound solution is worked out. This result is compared to the result worked out by limiting equilibrium to test the validity of the measure and procedure used in this paper.

Research of Lower Bound Method for Wedge Slope Subjected to Pore Water Pressure and Earthquake Force

Based on the lower bound theory for the plastic limit analysis, rock slope is divided into rigid block and structural surface. And the mathematical programming model which takes the safety factor as the objective function is established for the calculation of slope stability. This model has to meet the balance equations of the blocks, the Mohr-Coulomb yield conditions and the boundary conditions of slope. In the end, a classic model of rock slope on consideration of the pore water pressure and earthquake force is analyzed, and its lower bound solution is worked out. This result is compared to the result worked out by limiting equilibrium to test the validity and correctness of the method and procedure used in this paper.