Image reconstruction represents an important technique applied in various fields such as medicine, biology, materials science, nondestructive testing, and so forth. In this paper, we transform the problem of image reconstruction into the problem of solving linear systems with multiple right-hand sides. Based on the idea of K-means clustering, we propose the global randomized block Kaczmarz method, so as to solve the problem of the linear systems with multiple right-hand sides effectively and use this method to image reconstruction. Theoretical analysis proves the convergence of this method, and the simulation results demonstrate the performance of this method in image reconstruction.

Based on the K-means method, an effective block-row partitions algorithm was proposed in ^[1], which involves partitioning the rows of the coefficient matrix $A\in {\mathbb{R}}^{m\times n}$. However, with the increase of the size of the coefficient matrix, the time required for the partitioning process will increase significantly. To address this problem, we considered selecting features from the columns of the matrix $A$ to obtain a low-rank matrix $\tilde{A}\in {\mathbb{R}}^{m\times d}\left(d\ll n\right)$. Lasso is a regression analysis method for feature selection, which is simple and has excellent processing ability for high-dimensional data. In view of this, we first introduced a new criterion for selecting the projection block, and proposed the maximum residual block Kaczmarz algorithm. Then, we put forward the feature selection algorithm based on Lasso, and further presented a maximum residual block Kaczmarz algorithm based on feature selection. We analyzed the convergence of these algorithms and demonstrated their effectiveness through numerical results, while also verifying the performance of the proposed algorithms in image reconstruction.

Considering the advantages of economic benefit and cost reduction by using rank-one control, we investigated the problem of robust regional eigenvalue assignment using rank-one control for undamped gyroscopic systems. Based on the orthogonality relation, we presented a method for solving partial eigenvalue assignment problems to reassign partial undesired eigenvalues accurately. Since it is difficult to achieve robust control by assigning desired eigenvalues to precise positions with rank-one control, we assigned eigenvalues within specified regions to provide the necessary freedom. According to the sensitivity analysis theories, we derived the sensitivity of closed-loop eigenvalues to parameter perturbations to measure robustness and proposed a numerical algorithm for solving robust regional eigenvalue assignment problems so that the closed-loop eigenvalues were insensitive to parameter perturbations. Numerical experiments demonstrated the effectiveness of our method.

This work addresses active vibration control for fractional-order systems based on a multiple region eigenvalue assignment. First, we propose the selection criteria of fractional-order multiple stability regions and define multiple stability regions by generalized linear matrix inequality. Based on the inverse eigenvalue problem theory, we give the sufficient condition for solving the feedback control matrix under multiple region eigenvalue assignment and present the expression of the feedback control matrix. Then, we propose a numerical algorithm for solving this problem, which improves the control performance. Finally, we verify the feasibility and effectiveness of the proposed method through numerical examples of fractional-order simulation and actual physical systems.